From 3d0935d478955264c48bef673572d9344b8c0a1d Mon Sep 17 00:00:00 2001 From: Peter Date: Tue, 4 Jan 2022 14:52:21 +0100 Subject: [PATCH 01/49] update gitignore for vs code --- .gitignore | 9 +++++++++ 1 file changed, 9 insertions(+) diff --git a/.gitignore b/.gitignore index cfb93320..6b5a2136 100644 --- a/.gitignore +++ b/.gitignore @@ -26,3 +26,12 @@ docs/site/ # committed for packages, but should be committed for applications that require a static # environment. Manifest.toml + +# VS Code +.vscode/ + +# Local History for Visual Studio Code +.history/ + +# Built Visual Studio Code Extensions +*.vsix \ No newline at end of file From 318780ccce96059d38b9659d56239b1fbc1a57dc Mon Sep 17 00:00:00 2001 From: Peter Date: Tue, 4 Jan 2022 18:30:07 +0100 Subject: [PATCH 02/49] clean up repo, add dependencies --- Project.toml | 14 ++-- notebook/example.jl | 159 -------------------------------------------- src/STMOZOO.jl | 8 --- src/example.jl | 61 ----------------- test/example.jl | 51 -------------- test/runtests.jl | 3 +- 6 files changed, 8 insertions(+), 288 deletions(-) delete mode 100644 notebook/example.jl delete mode 100644 src/STMOZOO.jl delete mode 100644 src/example.jl delete mode 100644 test/example.jl diff --git a/Project.toml b/Project.toml index 881f78bc..ea3d1470 100644 --- a/Project.toml +++ b/Project.toml @@ -1,9 +1,9 @@ -name = "STMOZOO" -uuid = "8e7dc78f-7190-4116-bf88-3d148ae9f088" -authors = ["michielstock "] -version = "0.1.0" +name = "GradientStarvation" +uuid = "b11d05d9-c3d3-46bc-b9c9-1875ab7cf247" +authors = ["Peter Merseburger "] +version = "0.0.99" [deps] -Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" -LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" -Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" +Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c" +MLDatasets = "eb30cadb-4394-5ae3-aed4-317e484a6458" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" diff --git a/notebook/example.jl b/notebook/example.jl deleted file mode 100644 index 950417f0..00000000 --- a/notebook/example.jl +++ /dev/null @@ -1,159 +0,0 @@ -### A Pluto.jl notebook ### -# v0.14.8 - -using Markdown -using InteractiveUtils - -# ╔═╡ 5ad5c202-20f8-11eb-23f1-4f38b687c285 -using STMOZOO.Example - -# ╔═╡ d2d007b8-20f8-11eb-0ddd-1181d4565a85 -using Plots - -# ╔═╡ 45189a82-20fa-11eb-0423-05ce1b84639d -using Zygote - -# ╔═╡ 171fee18-20f6-11eb-37e5-2d04caea8c35 -md""" -# Example: solving quadratic systems - -By Michiel Stock - -## Introduction - -In this notebook, I have chosen to write code so minimize *quadratic functions* of the form: - -$$f(\mathbf{x}) = \frac{1}{2} \mathbf{x}^\intercal P\mathbf{x} + \mathbf{q} \cdot \mathbf{x} + r\,,$$ - -where $P \succ 0$. - -Hence, solving - -$$\min_\mathbf{x}\, f(\mathbf{x})\,.$$ -""" - -# ╔═╡ fb4aeb8c-20f7-11eb-0444-259de7b76883 -md""" - -## Derivation - -The gradient of the quadratic function is - -$$\nabla f(\mathbf{x})=P\mathbf{x} +\mathbf{q}\,.$$ - -Setting this to zero gives - -$$\mathbf{x}^\star=-P^{-1}\mathbf{q}\,.$$ - -We also know that $\nabla^2f(\mathbf{x}) = P \succ 0$, so $\mathbf{x}^\star$ is a minimizer. -""" - -# ╔═╡ 52e87238-20f8-11eb-2ea0-27ee1208d3c3 -md""" -## Illustration - -We will use a simple 2D example to illustrate this code. -""" - -# ╔═╡ e9a9d69e-20f8-11eb-0d9d-330ee5e9cf25 -P = [7 -1; -1 2] - -# ╔═╡ fdd4e550-20f8-11eb-227b-25f36708484d -q = [-14.0, -2.0] - -# ╔═╡ 025fd6e8-20f9-11eb-3e7d-3519f3c4b58f -r = 1 - -# ╔═╡ 096eff98-20f9-11eb-1e61-99d5714895ba -md"We have a function to generate the quadratic function." - -# ╔═╡ 165509ca-20f9-11eb-107c-550cbba0f0e9 -f_quadr = quadratic_function(P, q, r) - -# ╔═╡ 1fffc82a-20f9-11eb-198c-c160d7dac87d -f_quadr([2, 1]) - -# ╔═╡ 26ab6ce2-20f9-11eb-1836-1756b290e5e3 -md"No more need to remember the formulla for the minimizer! Just use `solve_quadratic_system`!" - -# ╔═╡ 49832a8e-20f9-11eb-0841-19a40a12db18 -x_star = solve_quadratic_system(P, q, r) - -# ╔═╡ 55e0e274-20f9-11eb-36c0-753f228f7e9b -begin - contourf(-20:0.1:20, -20:0.1:20, (x, y) -> f_quadr([x,y]), color=:speed) - scatter!([x_star[1]], [x_star[2]], label="minimizer") -end - -# ╔═╡ b1551758-20f9-11eb-3e8f-ff9a7127d7f8 -md""" -## Approximating non-quadratic functions - -We can approximate non-quadratic functions by a quadratic function: The second order Taylor approximation $\hat{f}$ of a function $f$ at $\mathbf{x}$ is -$$f(\mathbf{x}+\mathbf{v})\approx\hat{f}(\mathbf{x}+\mathbf{v}) = f(\mathbf{x}) + \nabla f(\mathbf{x})^\top \mathbf{v} + \frac{1}{2} \mathbf{v}^\top \nabla^2 f(\mathbf{x}) \mathbf{v}\,.$$ - -Let us use this idea for the Rosenbrock function. -""" - -# ╔═╡ 41d8f1dc-20fa-11eb-3586-a989427c1fd6 -f_nonquadr((x1, x2); a=1, b=5) = (a-x1)^2 + b * (x2 - x1^2)^2 - -# ╔═╡ 4ed4215e-20fa-11eb-11ee-f7741591163c -x = [0.0, 0.0] - -# ╔═╡ 56af99ee-20fa-11eb-0240-69c675efb78c -fx = f_nonquadr(x) - -# ╔═╡ 6c5473b4-20fa-11eb-327b-51ac560530eb -∇fx = f_nonquadr'(x) - -# ╔═╡ 7518c2c0-20fa-11eb-32c0-a9db2a91cbc5 -∇²fx = Zygote.hessian(f_nonquadr, x) - -# ╔═╡ 34027942-20fb-11eb-261e-3b991ce4c9f8 -v = solve_quadratic_system(∇²fx, ∇fx, fx) - -# ╔═╡ 3bbeb85c-20fc-11eb-04d0-fb12d8ace50a -f̂(x′) = quadratic_function(∇²fx, ∇fx, fx)(x′ .- x) - -# ╔═╡ 8623ac1a-20fa-11eb-2d45-49cce0fdac86 -begin - plot_nonquadr = contourf(-2:0.01:2, -2:0.01:2, (x, y) -> f_nonquadr([x,y]), color=:speed, title="non-quadratic function") - scatter!(plot_nonquadr, [x[1]], [x[2]], label="x") - scatter!(plot_nonquadr, [x[1]+v[1]], [x[2]+v[2]], label="x + v") - - plot_approx = contourf(-2:0.01:2, -2:0.01:2, (x, y) -> f̂([x,y]), color=:speed, - title="quadratic approximation") - scatter!(plot_approx, [x[1]], [x[2]], label="x") - scatter!(plot_approx, [x[1]+v[1]], [x[2]+v[2]], label="x + v") - - plot(plot_nonquadr, plot_approx, layout=(2,1), size=(600, 800)) - -end - - -# ╔═╡ Cell order: -# ╟─171fee18-20f6-11eb-37e5-2d04caea8c35 -# ╠═5ad5c202-20f8-11eb-23f1-4f38b687c285 -# ╠═d2d007b8-20f8-11eb-0ddd-1181d4565a85 -# ╟─fb4aeb8c-20f7-11eb-0444-259de7b76883 -# ╟─52e87238-20f8-11eb-2ea0-27ee1208d3c3 -# ╠═e9a9d69e-20f8-11eb-0d9d-330ee5e9cf25 -# ╠═fdd4e550-20f8-11eb-227b-25f36708484d -# ╠═025fd6e8-20f9-11eb-3e7d-3519f3c4b58f -# ╟─096eff98-20f9-11eb-1e61-99d5714895ba -# ╠═165509ca-20f9-11eb-107c-550cbba0f0e9 -# ╠═1fffc82a-20f9-11eb-198c-c160d7dac87d -# ╟─26ab6ce2-20f9-11eb-1836-1756b290e5e3 -# ╠═49832a8e-20f9-11eb-0841-19a40a12db18 -# ╠═55e0e274-20f9-11eb-36c0-753f228f7e9b -# ╟─b1551758-20f9-11eb-3e8f-ff9a7127d7f8 -# ╠═41d8f1dc-20fa-11eb-3586-a989427c1fd6 -# ╠═45189a82-20fa-11eb-0423-05ce1b84639d -# ╠═4ed4215e-20fa-11eb-11ee-f7741591163c -# ╠═56af99ee-20fa-11eb-0240-69c675efb78c -# ╠═6c5473b4-20fa-11eb-327b-51ac560530eb -# ╠═7518c2c0-20fa-11eb-32c0-a9db2a91cbc5 -# ╠═34027942-20fb-11eb-261e-3b991ce4c9f8 -# ╠═3bbeb85c-20fc-11eb-04d0-fb12d8ace50a -# ╟─8623ac1a-20fa-11eb-2d45-49cce0fdac86 diff --git a/src/STMOZOO.jl b/src/STMOZOO.jl deleted file mode 100644 index 34b94d12..00000000 --- a/src/STMOZOO.jl +++ /dev/null @@ -1,8 +0,0 @@ -module STMOZOO - - -# execute your source file and export the module you made -include("example.jl") -export Example - -end # module diff --git a/src/example.jl b/src/example.jl deleted file mode 100644 index 7fc131da..00000000 --- a/src/example.jl +++ /dev/null @@ -1,61 +0,0 @@ -# Michiel Stock -# Example of a source code file implementing a module. - - -# all your code is part of the module you are implementing -module Example - -# you have to import everything you need for your module to work -# if you use a new package, don't forget to add it in the package manager -using LinearAlgebra - -# export all functions that are relevant for the user -export solve_quadratic_system, quadratic_function - -""" - solve_quadratic_system(P, q, r=0; testPD=false) - -Find the minimizer of a canonical quadratic system: - -``f(x) = 0.5xᵀ P x + x ⋅ q + r`` - -with `P` a positive scalar or a positive-definite n x n matrix and `q` a scalar or -a n x n vector. The intercept `r` can optionally be given but does not influence -the minimizer. One can use the keyword argument `testPD` to test if `P` is positive definite, -though this comes at a computational cost. - -## Examples - -Scalar case: - -```julia-repl -julia> solve_quadratic_system(8, -4, 3) -0.5 -``` - -Vector case: - -```julia-repl -julia> P = [3 1; 1 2]; - -julia> q = [0.5, -2]; - -julia> solve_quadratic_system(P, q) -2-element Array{Float64,1}: - -0.6 - 1.2999999999999998 -``` -""" -function solve_quadratic_system(P, q, r=0; testPD=false) - testPD && @assert isposdef(P) "Provided P is not positive definite" - return - P \ q -end - -""" - quadratic_function(P, q, r) - -Give the paramters of a canonical quatratic function and returns a function. -""" -quadratic_function(P, q, r) = x -> 0.5x' * P * x + q ⋅ x + r - -end \ No newline at end of file diff --git a/test/example.jl b/test/example.jl deleted file mode 100644 index d554036c..00000000 --- a/test/example.jl +++ /dev/null @@ -1,51 +0,0 @@ -# example of a unit test in Julia -# runs a test for a certain module -# to run the tests, first open the package manager (`]` in REPL), -# activate the project if not done so and then enter `test` - -# wrap all your tests and subgroups in a `@testset` block -@testset "Example" begin - using STMOZOO.Example # load YOUR module - - @testset "solve quadratic" begin - # test some cases - @test solve_quadratic_system(8.0, -2.0, 3.0) == 0.25 - - # test for type - @test solve_quadratic_system(7.0, 27.4, 3.0) isa Number - - # test for type stability - @test solve_quadratic_system(1//2 , -2//1, 3//1) isa Rational - - P = [10 1; 1 5] - q = [100, -7] - - # test for a result - # use ≈ (`\approx`) to check approximate equality - # useful for rounding errors - # NOTE: does NOT work on ≈ 0! - @test solve_quadratic_system(P, q) ≈ - P \ q - @test solve_quadratic_system(P, q, testPD=true) isa Vector - - # test if a certain error is thrown - @test_throws AssertionError solve_quadratic_system(-P, q, testPD=true) - end - - @testset " quadratic function" begin - f_scalar = quadratic_function(3, 4, 8) - - @test f_scalar isa Function - @test f_scalar(2) isa Number - @test f_scalar(2) ≈ 0.5 * 3 * 2^2 + 4 * 2 + 8 - - P = [9.0 -2; -2 3] - q = [-3, -4] - r = -π - x = [-2.0, 2.8] - - f_vect = quadratic_function(P, q, r) - @test f_vect isa Function - @test f_vect(x) isa Number - @test f_vect(x) ≈ 0.5x' * P * x + q' * x + r - end -end \ No newline at end of file diff --git a/test/runtests.jl b/test/runtests.jl index ff9b0f7e..c9680569 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -1,4 +1,3 @@ using Test -include("example.jl") -# add here the file with your unit tests \ No newline at end of file +include("") \ No newline at end of file From be359f096e8ce0c5322fabf35d98221756e74208 Mon Sep 17 00:00:00 2001 From: Peter Date: Tue, 4 Jan 2022 18:30:37 +0100 Subject: [PATCH 03/49] provide example data from project website --- src/data.jl | 623 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 623 insertions(+) create mode 100644 src/data.jl diff --git a/src/data.jl b/src/data.jl new file mode 100644 index 00000000..ff223e7a --- /dev/null +++ b/src/data.jl @@ -0,0 +1,623 @@ +module Data + +export points_class_1, points_class_2 + +function points_class_1(offset) + x = [ + -2.717835893494339 - offset, + -2.0025756608302556 - offset, + -2.568929506049188 - offset, + -2.270602886384911 - offset, + -2.5015948773601426 - offset, + -2.330151863221954 - offset, + -0.7415685155965577 - offset, + -2.0839366412304012 - offset, + -2.8528165357526083 - offset, + -1.147008103324011 - offset, + -2.8784004738595304 - offset, + -0.8677959198389527 - offset, + -1.7589836417118123 - offset, + -2.998388218477527 - offset, + -2.122794047965966 - offset, + -0.6950154023659438 - offset, + -2.6642421732719845 - offset, + -0.4107292059512281 - offset, + -2.5650150317912725 - offset, + -0.03957624387021477 - offset, + -1.459975933353292 - offset, + -1.7826178769431 - offset, + -0.0063910086450723325 - offset, + -3.113867547530851 - offset, + 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-1.9573778533077693 - offset, + -1.0107773780231675 - offset + ] + y = [ + 0.6793610085529345, + -0.7764598781122054, + 1.3030331041149927, + 1.696103804129755, + 1.8362701766489786, + 0.5295968005263707, + 3.5042711192603146, + -0.7231575356785352, + 2.277743746352307, + 3.1121491784480697, + 1.4770243521157966, + -1.0110529899688063, + -0.21940659318914466, + 1.2783156815611991, + -0.1791718798868415, + 3.884218615064042, + 1.9439974190429932, + 3.505002569928259, + 0.5755380178279407, + -1.210873365508656, + -1.0329923799436405, + -1.0582442212859733, + 3.6142846309958205, + 1.5445673874139447, + 1.6469163448206863, + 0.2540079910444923, + -1.258967648201369, + 2.794453234058449, + -1.149826801806625, + 4.1149852157058895, + -1.4081025494488113, + 3.5570091502314423, + 3.713458077324815, + 0.9623315035843127, + 2.792146626757914, + -1.205590370248826, + 1.3344598403007284, + 3.057031649768471, + 2.578789613141873, + -0.6543242980905074, + -1.2100425673015185, + 2.9197086781500277, + 0.7696706532935776, + 1.6626933401609023, + 0.42253538652727796, + 1.4631522384301066, + 2.942466273998301, + 1.2364964850094777, + 3.152143289401243, + 3.6228480325523322, + -0.6233110354243981, + -1.3580082036894057, + -1.305693088622172, + -0.3935866480543121, + 3.111337871342845, + 2.410394433635401, + -0.7759536147862817, + 1.3340538265678492, + 2.2303142906955555, + 0.3209555538663683, + 3.89346947511081, + 0.7649583168967733, + -1.0802855176420891, + -1.2568494543447684, + 2.4944104537407057, + -0.679082202208783, + 3.693884882642762, + -0.9569209611232933, + -0.3002820219513763, + -1.2603402702853281, + -0.729087176112734, + 0.16046671198639534, + 2.8847324649501402, + 3.4142846215369556, + 2.9919434674688086, + 0.9143476976637405, + -1.0038621010673856, + 3.7992457338282373, + 0.3803212691710048, + 2.577260760939388, + 1.9925005915231553, + 3.0431663584149904, + 1.7300408024972684, + -0.46725570847050424, + -0.25749919862715076, + -1.493604284214328, + 4.396876304605113, + 1.5310130579985914, + 0.11093029172240809, + -0.6139231414423276, + 0.9208951171502733, + 0.6712563554792417, + 2.618298719018602, + 3.6804690431500036, + 0.48814363461182453, + 1.7996187499638172, + 1.1625285801183705, + 0.7389373388724324, + -1.2211299676858136, + -0.8873468825484077, + 3.4776733678417986, + 3.575493958460944, + 2.3985492607241343, + 1.71557735708596, + 3.749934520345652, + 0.3575930800531485, + 2.1434683067151066, + -0.9049053449496425, + 0.10465507554354464, + 3.1473437369575374, + 3.8373893523193847, + -0.2809886987751162, + -0.0809670563060226, + -1.0586183593166278, + 2.9422791248401476, + 3.3776803834259663, + 3.153656143587366, + 3.5152991121388566, + -1.281563505634356, + 1.7966322608299223, + 3.523561811455531, + -0.7139026560764259, + 2.830553038963384, + 1.756811334089956, + 2.7361898523968753, + -0.9036159267450711, + -0.8009280139118677, + -1.3972949916010728, + 2.4380037963785406, + -0.05613139501335074, + -0.9752089176334047, + 3.3757400103199693, + 3.2402795631690253, + 3.1005177839870184, + -1.0528879963506719, + 0.11441934583356184, + 2.5168873649410255, + 2.960577394399894, + -0.8705594922460475, + 3.573688045781485, + -0.8907098511435746, + 3.294176954534472, + -0.08783059329234524, + 2.292660484192841, + 3.4597453566827, + 0.253560604420727, + -0.39806607018651047, + 2.5778821384858093, + -0.7882243058017496, + -0.9895456553950488 + ] + + return x, y +end + +function points_class_2(offset) + x = [ + 1.2605697631874289 + offset, + 0.4428841761287988 + offset, + 2.213537711353039 + offset, + 2.2246476075616277 + offset, + 0.934302193080736 + offset, + 0.8703714425371154 + offset, + 2.6652808441973175 + offset, + 2.5868192990813768 + offset, + 2.5485659972780015 + offset, + 1.3501341878455395 + offset, + 1.3394525649634517 + offset, + 1.5703488118666078 + offset, + 2.578906600693658 + offset, + 2.6017725256967363 + offset, + 0.4855765554512216 + offset, + 1.620317196388419 + offset, + 0.6107465335988917 + offset, + 2.705677633402291 + offset, + 2.529619115397467 + offset, + 1.7205080921848972 + offset, + 2.025034051353637 + offset, + 0.9454991520471547 + offset, + 2.4474158496753415 + offset, + 0.44018725848007784 + offset, + 2.3578626209991933 + offset, + 2.305287296588136 + offset, + 2.1078203674554112 + offset, + 2.5672512309425377 + offset, + 0.76210430597206 + offset, + 0.9009064185658802 + offset, + 0.8790969921498528 + offset, + 2.834290242153208 + offset, + 1.827620864188383 + offset, + 0.9106326766024155 + offset, + 2.4986568505695774 + offset, + 1.329902276945841 + offset, + 0.30026706132425485 + offset, + 1.8709094482262436 + offset, + 2.6457300611721415 + offset, + 0.903254228522091 + offset, + 1.6008172396371303 + offset, + 3.2153091597614525 + offset, + 2.5679011437621266 + offset, + 2.359601929920219 + offset, + 2.0630126331781993 + offset, + 0.9482707644288992 + offset, + 1.318717891036563 + offset, + 2.685806352629474 + offset, + 1.8516766580167963 + offset, + 2.356710102702996 + offset, + 2.1684918727788336 + offset, + 1.5800357609867945 + offset, + 1.033126269584137 + offset, + 1.872794212999521 + offset, + 0.5235449122291833 + offset, + 0.7008785749342308 + offset, + 2.553663974985213 + offset, + 2.7411100225680163 + offset, + 2.0343480716575852 + offset, + 0.5340990535248212 + offset, + 2.4353923087262066 + offset, + 2.697340999301913 + offset, + 2.6521161619740408 + offset, + 0.8016750570795748 + offset, + 2.0888346508237094 + offset, + 0.8304001118980417 + offset, + 2.1076008673731175 + offset, + 0.3589801618753179 + offset, + 1.1229934118941256 + offset, + 1.3365490905708357 + offset, + 1.8354193619448524 + offset, + 2.718680142657365 + offset, + 1.114668177329429 + offset, + 2.4373081838238693 + offset, + 2.414593845611318 + offset, + 0.12290616957133377 + offset, + 2.432773932914067 + offset, + 2.703247300810249 + offset, + 0.9329542112336406 + offset, + 1.8387052373191828 + offset, + 2.192531994043116 + offset, + 0.9347932451321164 + offset, + 2.6506791700072787 + offset, + 1.4357888060935096 + offset, + 2.5074984951495582 + offset, + 1.452676080486036 + offset, + 2.1649169298468043 + offset, + 2.096317932113875 + offset, + 1.1626855473307327 + offset, + 0.0698015672586913 + offset, + 2.263884734272171 + offset, + 2.452097505628506 + offset, + 2.318232969547698 + offset, + 1.5708028678519623 + offset, + 1.7619872579964682 + offset, + 0.48741766950630805 + offset, + 2.2860590634348394 + offset, + 2.355561815069757 + offset, + 2.624438667886823 + offset, + 1.854267558549543 + offset, + 2.3213276658541218 + offset, + 2.745584480726928 + offset, + 0.11463689662493518 + offset, + 1.2268104662722987 + offset, + 0.4883558476641737 + offset, + 0.302830362550608 + offset, + 2.8418136908650427 + offset, + 2.4110065851996403 + offset, + 1.1883288319221876 + offset, + 0.4231560219611501 + offset, + 2.7332667742376247 + offset, + 2.4548799817800466 + offset, + 1.4492552216872536 + offset, + 0.29812737679036055 + offset, + 0.6024251082982855 + offset, + 2.4071077716684606 + offset, + 2.545623250801896 + offset, + 1.1447541103640393 + offset, + 1.6300111419299077 + offset, + 2.2144073814114598 + offset, + 1.741764282945776 + offset, + 2.584934801845699 + offset, + 2.94429370319911 + offset, + 0.04396728231852283 + offset, + 1.511277836871522 + offset, + 1.5078872546350166 + offset, + 2.2830235983573757 + offset, + 1.141833418913716 + offset, + 2.31153512617248 + offset, + 0.7930014449666454 + offset, + 1.7855431618709825 + offset, + 1.2496515034524522 + offset, + 2.494095987642121 + offset, + 0.99530344900322 + offset, + 1.6423078459022569 + offset, + 1.2698194368721236 + offset, + 2.3092312380669213 + offset, + 1.508197474828362 + offset, + 0.5282536325342755 + offset, + 2.8592120295683343 + offset, + 1.964949523096605 + offset, + 2.6235858300049006 + offset, + 1.2938945593806481 + offset, + 2.2949841542633416 + offset, + 2.347836965953938 + offset, + 2.3781640679457445 + offset, + 2.0047220333928424 + offset, + 0.5906909597410415 + offset, + 1.2986453274833423 + offset, + 2.667821411223028 + offset + ] + y = [ + 1.0458976448210704, + -3.9067277248882855, + 0.10622312924614075, + -3.131178556347437, + 1.01212401276388, + 1.14213909231317, + -1.499740679978164, + -2.3757729401741026, + -2.122671490779725, + -3.502953002856569, + -3.256081109078368, + -3.1749520646753595, + -2.155913453889749, + -0.2395214626274446, + 0.9755334524010564, + -3.471619676384854, + 0.9470748554316055, + -1.8958980011427715, + -1.750911067902624, + 0.6014389134684253, + -0.328098369799075, + -3.4065530755818263, + -0.570775344749403, + 1.2583085822944136, + 0.37952520933453554, + -1.3523455959069988, + -2.9400776134322326, + -0.931511238718386, + 0.9834249491123324, + -3.5025716943059186, + -4.248785250853195, + -0.6773777546863811, + -3.146614604919824, + 1.6069751824711314, + 0.12949305293031732, + 0.8261796714411879, + -3.569524692879105, + 0.7005037664902201, + -2.425440866187578, + -3.738092107397064, + -3.2452624729025747, + -1.368470502749851, + 0.07815030521334698, + -2.430664435412014, + -0.3937883060575643, + 0.9090318588982524, + -3.2871385464785274, + -1.0695984302841914, + 0.24434407871756103, + -2.25954235705039, + -0.7439150005837414, + -3.0397095698565275, + 0.9359563676278917, + 1.0115209086189823, + -3.3783557249585123, + 0.9594224070734308, + -0.10771629916989434, + -1.7184897675482067, + -2.708724275370595, + 1.6742944306376044, + -2.2521686832796735, + -0.3155915671715026, + -1.3799617068714918, + -3.7029324960425725, + 0.6295951027167603, + 1.3620703160533432, + -2.6320189409595427, + 1.2235177121091534, + -3.3223373231734046, + 0.7410339357717708, + 0.5026708223105636, + -1.2659346176335473, + 1.1018873624037109, + -1.7793888239691904, + -0.6335066629431028, + 0.7436758886663175, + -2.271922352203361, + -1.5681088934930953, + 1.387487453946818, + -0.18205302519419486, + 0.7509498665446229, + 1.1304461649366861, + 0.2598239003357525, + -3.638173831524141, + -0.3246237440748867, + -3.846851795839017, + 0.12598502378608517, + -2.974937206560519, + -3.611991741390597, + -3.6894705097537193, + -2.792874862433873, + -2.998549361725964, + -2.520540198719894, + 0.6171082788895208, + -3.1598996224476084, + 1.5195493259285597, + -1.9538102221406346, + -2.061630645113948, + -0.45792310107331247, + 0.07907581142523268, + -0.10696635485443987, + -1.710269012859103, + -3.60107422436533, + 1.1092227889389914, + -3.7822078275343984, + -4.21797263754794, + -1.719272711473141, + -2.58454577962155, + -3.436277286035127, + 0.9974824865464593, + -1.2960555247563033, + -2.098906157582724, + -3.2980549567958297, + -4.140712584756083, + -3.6668594006359907, + -1.9331827888832867, + -1.6997537944929542, + 1.2288964589068012, + -3.280361193419007, + -2.7915168449133825, + 0.716151381796484, + -0.4617926598386114, + -0.7751800733034873, + 1.5080467962694248, + -3.601524555444186, + 0.4038862638181287, + -2.6698628730744787, + -3.534713771642897, + -1.5036639589034162, + 0.9984399344555597, + 0.3775513743392528, + -3.61494166755954, + -0.6561789952747799, + -3.9953414791058472, + 0.8817193140291141, + 1.2639173654508489, + 0.6932869760752849, + -3.4031870404877793, + -3.6982857231300157, + -2.4260474342934755, + -3.3108893959920254, + -0.7863371260473055, + 0.7098721965355509, + 0.21815437229061818, + -2.078371734274718, + -1.1400343437416685, + -1.2769894466119114, + 1.6438700737392897, + -3.0090552847647456, + 0.2317409067300455 + ] + + return x, y +end + +end \ No newline at end of file From 55afea151bb397c0a54d695bb58647b1f28b74db Mon Sep 17 00:00:00 2001 From: Peter Date: Wed, 5 Jan 2022 14:47:50 +0100 Subject: [PATCH 04/49] first working/trainable version of neural network --- src/data.jl | 14 +++-- src/gradientstarvation.jl | 10 ++++ src/neuralnetwork.jl | 109 ++++++++++++++++++++++++++++++++++++++ 3 files changed, 128 insertions(+), 5 deletions(-) create mode 100644 src/gradientstarvation.jl create mode 100644 src/neuralnetwork.jl diff --git a/src/data.jl b/src/data.jl index ff223e7a..96173f22 100644 --- a/src/data.jl +++ b/src/data.jl @@ -2,7 +2,7 @@ module Data export points_class_1, points_class_2 -function points_class_1(offset) +function moon_class_1(offset = 0.0) x = [ -2.717835893494339 - offset, -2.0025756608302556 - offset, @@ -307,11 +307,10 @@ function points_class_1(offset) -0.7882243058017496, -0.9895456553950488 ] - - return x, y + return vcat(transpose.((x, y))...) end -function points_class_2(offset) +function moon_class_2(offset = 0.0) x = [ 1.2605697631874289 + offset, 0.4428841761287988 + offset, @@ -616,8 +615,13 @@ function points_class_2(offset) -3.0090552847647456, 0.2317409067300455 ] + return vcat(transpose.((x, y))...) +end - return x, y +function get_moons() + moons = hcat(moon_class_1(), moon_class_2()) + labels = [repeat([1], 150); repeat([-1], 150)] + return moons, labels end end \ No newline at end of file diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl new file mode 100644 index 00000000..2c0535ee --- /dev/null +++ b/src/gradientstarvation.jl @@ -0,0 +1,10 @@ +module GradientStarvation + +include("data.jl") +include("neuralnetwork.jl") + +using Plots + +NeuralNetwork.train() + +end diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl new file mode 100644 index 00000000..7ce9b703 --- /dev/null +++ b/src/neuralnetwork.jl @@ -0,0 +1,109 @@ +module NeuralNetwork + +include("data.jl") + +using Flux +using Flux: onehotbatch, onecold +using Flux.Data: DataLoader +using Flux.Losses: logitcrossentropy +using MLDatasets + +export get_example_data, get_nmist_data, get_loss_and_accuracy, train + +# WIP +# 2D classification task +# 2 layer ReLU, 500 hidden units +# cross-entropy loss training for two different arrangements of the training points + +function get_example_data(args) + x_train, y_train = Data.get_moons() + x_test, y_test = Data.get_moons() + + y_train, y_test = onehotbatch(y_train, [-1, 1]), onehotbatch(y_test, [-1, 1]) + + println("example data dimensions", size(x_train), size(y_train)) + + # create data loaders + train_loader = DataLoader((x_train, y_train), batchsize = args.batchsize, shuffle = true) + test_loader = DataLoader((x_test, y_test), batchsize = args.batchsize) + + return train_loader, test_loader +end + +function get_nmist_data(args) + # Loading Dataset + xtrain, ytrain = MLDatasets.MNIST.traindata(Float32) + xtest, ytest = MLDatasets.MNIST.testdata(Float32) + + # Reshape Data in order to flatten each image into a linear array + xtrain = Flux.flatten(xtrain) + xtest = Flux.flatten(xtest) + + # One-hot-encode the labels + ytrain, ytest = onehotbatch(ytrain, 0:9), onehotbatch(ytest, 0:9) + + println(size(xtrain), size(ytrain)) + # Create DataLoaders (mini-batch iterators) + train_loader = DataLoader((xtrain, ytrain), batchsize=args.batchsize, shuffle=true) + test_loader = DataLoader((xtest, ytest), batchsize=args.batchsize) + + return train_loader, test_loader +end + +function get_loss_and_accuracy(data_loader, model) + accuracy = 0 + loss = 0.0f0 + num = 0 + for (x, y) in data_loader + ŷ = model(y) + loss += logitcrossentropy(ŷ, y, agg = sum) + accuracy += sum(onecold(ŷ) .== onecold(y)) + num += size(x)[end] + end + return loss / num, accuracy / num +end + +function neural_network() + return Chain( + Dense(2, 300, relu), + Dense(300, 2) + # Dropout(0.7) + ) +end + +Base.@kwdef mutable struct Args + learning_rate::Float64 = 3e-2 + batchsize::Int = 50 + epochs::Int = 10 +end + +function train(args...) + args = Args(args...) + + train_loader, test_loader = get_example_data(args) # get_nmist_data(args) # + + model = neural_network() + params = Flux.params(model) + + optimizer = ADAM(args.learning_rate) + + # training + for epoch in 1:args.epochs + for (x, y) in train_loader + # println(size(x), size(y)) + # println(x, y) + # compute gradient + grad = gradient(() -> logitcrossentropy(model(x), y), params) + Flux.Optimise.update!(optimizer, params, grad) + end + + # Report on train and test + train_loss, train_accuracy = get_loss_and_accuracy(train_loader, model) + test_loss, test_accuracy = get_loss_and_accuracy(test_loader, model) + println("Epoch $epoch") + println(" train_loss = $train_loss, train_accuracy = $train_accuracy") + println(" test_loss = $test_loss, test_accuracy = $test_accuracy") + end +end + +end \ No newline at end of file From 5bac4f2a2abb7848ea1af0f87fc7b7ddbd27c754 Mon Sep 17 00:00:00 2001 From: Peter Date: Wed, 5 Jan 2022 21:21:27 +0100 Subject: [PATCH 05/49] implement moon generator using ScikitLearn; support: offset, rotation --- Project.toml | 1 + src/data.jl | 58 +++++++++++++++++++++++++++++++++++++-- src/gradientstarvation.jl | 3 -- src/neuralnetwork.jl | 25 ++++++++++------- 4 files changed, 72 insertions(+), 15 deletions(-) diff --git a/Project.toml b/Project.toml index ea3d1470..23c0a032 100644 --- a/Project.toml +++ b/Project.toml @@ -7,3 +7,4 @@ version = "0.0.99" Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c" MLDatasets = "eb30cadb-4394-5ae3-aed4-317e484a6458" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +ScikitLearn = "3646fa90-6ef7-5e7e-9f22-8aca16db6324" diff --git a/src/data.jl b/src/data.jl index 96173f22..1426de55 100644 --- a/src/data.jl +++ b/src/data.jl @@ -1,6 +1,10 @@ module Data -export points_class_1, points_class_2 +using Plots +using ScikitLearn +@sk_import datasets: make_moons + +export get_moons, get_moons_from_publication function moon_class_1(offset = 0.0) x = [ @@ -618,10 +622,60 @@ function moon_class_2(offset = 0.0) return vcat(transpose.((x, y))...) end -function get_moons() +function get_moons_from_publication() moons = hcat(moon_class_1(), moon_class_2()) labels = [repeat([1], 150); repeat([-1], 150)] return moons, labels end +function get_moons(n; noise = 0.1, offset = 0.0, rotation = 90) + X, y = make_moons(n_samples = n, noise = noise, random_state = 0) + p1 = scatter(X[:,1], X[:,2], c = y, title = "generated state") + + if rotation != 0.0 + θ = deg2rad(rotation) + R = [cos(θ) -sin(θ); sin(θ) cos(θ)] + X *= R + p2 = scatter(X[:,1], X[:,2], c = y, title = "rotated state") + end + + if offset != 0.0 + if rotation != 0.0 + # after rotation: apply offset to x-axis + X = apply_moon_offset(X, y, offset, "x") + else + # no rotation: apply offset to y-axis + X = apply_moon_offset(X, y, offset, "y") + end + end + p3 = scatter(X[:,1], X[:,2], c = y, title = "offsetted state") + + display(plot(p1, p2, p3, layout = (1, 3))) + return X, y +end + +function apply_moon_offset(X, y, offset, direction = "x") + @assert direction in ["x", "y"] "Error: direction must be either x or y" + # use offset as total offset, thus devide by number of labels; assume 2 labels + offset /= 2 + # l = length(y) # length + # b = Int(l / 2) # boundary + + direction == "x" ? direction = 1 : direction = 2 + + first_moon = true + for label in unique(y) + l_ids = findall(l -> l == label, y) + for id in l_ids + first_moon ? X[id:1:id, direction] .-= offset : X[id:1:id, direction] .+= offset + end + first_moon = false + end + + # X[:1:b, direction] .+= offset + # X[(b + 1):1:l, direction] .-= offset + + return X +end + end \ No newline at end of file diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index 2c0535ee..fc74bc06 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -1,10 +1,7 @@ module GradientStarvation -include("data.jl") include("neuralnetwork.jl") -using Plots - NeuralNetwork.train() end diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 7ce9b703..f1c62820 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -7,21 +7,21 @@ using Flux: onehotbatch, onecold using Flux.Data: DataLoader using Flux.Losses: logitcrossentropy using MLDatasets +using Plots -export get_example_data, get_nmist_data, get_loss_and_accuracy, train +export get_moon_data, get_nmist_data, get_loss_and_accuracy, train # WIP # 2D classification task # 2 layer ReLU, 500 hidden units # cross-entropy loss training for two different arrangements of the training points -function get_example_data(args) - x_train, y_train = Data.get_moons() - x_test, y_test = Data.get_moons() +function get_moon_data(args) + x_train, y_train = Data.get_moons(300, offset = 1.0) + x_test, y_test = Data.get_moons_from_publication() - y_train, y_test = onehotbatch(y_train, [-1, 1]), onehotbatch(y_test, [-1, 1]) - - println("example data dimensions", size(x_train), size(y_train)) + x_train = transpose(x_train) + y_train, y_test = onehotbatch(y_train, 0:1), onehotbatch(y_test, [-1, 1]) # create data loaders train_loader = DataLoader((x_train, y_train), batchsize = args.batchsize, shuffle = true) @@ -72,15 +72,20 @@ function neural_network() end Base.@kwdef mutable struct Args - learning_rate::Float64 = 3e-2 + learning_rate::Float64 = 1e-2 batchsize::Int = 50 - epochs::Int = 10 + epochs::Int = 50 end function train(args...) args = Args(args...) - train_loader, test_loader = get_example_data(args) # get_nmist_data(args) # + train_loader, test_loader = get_moon_data(args) # get_nmist_data(args) + + # plot train and test data sets + scatter(train_loader.data[1][1,:], train_loader.data[1][2,:], show = true) + # scatter(test_loader.data[1][1,:], test_loader.data[1][2,:], show = true) + # plot(p_train, p_test, layout = (1, 2), show = true) model = neural_network() params = Flux.params(model) From e48d009a435af4c0bf50d0ca4772467a261375ad Mon Sep 17 00:00:00 2001 From: Peter Date: Fri, 7 Jan 2022 16:41:34 +0100 Subject: [PATCH 06/49] first working version of decision boundary plot --- Project.toml | 1 + src/neuralnetwork.jl | 84 +++++++++++++++++++++++++++++++++----------- 2 files changed, 64 insertions(+), 21 deletions(-) diff --git a/Project.toml b/Project.toml index 23c0a032..24ab59c4 100644 --- a/Project.toml +++ b/Project.toml @@ -6,5 +6,6 @@ version = "0.0.99" [deps] Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c" MLDatasets = "eb30cadb-4394-5ae3-aed4-317e484a6458" +PlotlyJS = "f0f68f2c-4968-5e81-91da-67840de0976a" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" ScikitLearn = "3646fa90-6ef7-5e7e-9f22-8aca16db6324" diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index f1c62820..403dd194 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -8,14 +8,10 @@ using Flux.Data: DataLoader using Flux.Losses: logitcrossentropy using MLDatasets using Plots +using PlotlyJS export get_moon_data, get_nmist_data, get_loss_and_accuracy, train -# WIP -# 2D classification task -# 2 layer ReLU, 500 hidden units -# cross-entropy loss training for two different arrangements of the training points - function get_moon_data(args) x_train, y_train = Data.get_moons(300, offset = 1.0) x_test, y_test = Data.get_moons_from_publication() @@ -42,7 +38,6 @@ function get_nmist_data(args) # One-hot-encode the labels ytrain, ytest = onehotbatch(ytrain, 0:9), onehotbatch(ytest, 0:9) - println(size(xtrain), size(ytrain)) # Create DataLoaders (mini-batch iterators) train_loader = DataLoader((xtrain, ytrain), batchsize=args.batchsize, shuffle=true) test_loader = DataLoader((xtest, ytest), batchsize=args.batchsize) @@ -65,16 +60,23 @@ end function neural_network() return Chain( - Dense(2, 300, relu), - Dense(300, 2) - # Dropout(0.7) + Dense(2, 500, relu), + Dense(500, 2) ) end +function get_prediction(loader, model) + pred = [] + for (x, y) in loader + push!(pred, model(y)) + end + return pred +end + Base.@kwdef mutable struct Args learning_rate::Float64 = 1e-2 batchsize::Int = 50 - epochs::Int = 50 + epochs::Int = 10 end function train(args...) @@ -83,32 +85,72 @@ function train(args...) train_loader, test_loader = get_moon_data(args) # get_nmist_data(args) # plot train and test data sets - scatter(train_loader.data[1][1,:], train_loader.data[1][2,:], show = true) - # scatter(test_loader.data[1][1,:], test_loader.data[1][2,:], show = true) - # plot(p_train, p_test, layout = (1, 2), show = true) + p_train = Plots.scatter(train_loader.data[1][1,:], train_loader.data[1][2,:], c = map(x -> x == true ? 1 : 0, train_loader.data[2][1,:])) + p_test = Plots.scatter(test_loader.data[1][1,:], test_loader.data[1][2,:], c = map(x -> x == true ? 1 : 0, test_loader.data[2][1,:])) + display(Plots.plot(p_train, p_test, layout = (1, 2))) model = neural_network() params = Flux.params(model) - - optimizer = ADAM(args.learning_rate) + optimizer = ADAM(args.learning_rate) # Descent(args.learning_rate) # # training for epoch in 1:args.epochs for (x, y) in train_loader - # println(size(x), size(y)) - # println(x, y) # compute gradient - grad = gradient(() -> logitcrossentropy(model(x), y), params) - Flux.Optimise.update!(optimizer, params, grad) + grads = gradient(() -> logitcrossentropy(model(x), y), params) + Flux.Optimise.update!(optimizer, params, grads) end # Report on train and test train_loss, train_accuracy = get_loss_and_accuracy(train_loader, model) test_loss, test_accuracy = get_loss_and_accuracy(test_loader, model) println("Epoch $epoch") - println(" train_loss = $train_loss, train_accuracy = $train_accuracy") - println(" test_loss = $test_loss, test_accuracy = $test_accuracy") + println(" train loss = $train_loss, train accuracy = $train_accuracy") + println(" test loss = $test_loss, test accuracy = $test_accuracy") + + # early exit if accuracy is over 90% + acc_thres = 0.90 + if test_accuracy > acc_thres + @info "Stopped after $epoch epochs because test accuracy threshold of $(acc_thres * 100)% was exceeded." + break + end end + + return model +end + +function plot_decision_boundary(loader, model) + # grid and range step size + n = 100 + # determine limits of given data + x_max = maximum(loader.data[1][1,:]) + .25 + y_max = maximum(loader.data[1][2,:]) + .25 + x_min = minimum(loader.data[1][1,:]) - .25 + y_min = minimum(loader.data[1][2,:]) - .25 + + # upper = ceil(maximum([x_max, y_max]), digits = 1, base = 2) + # lower = floor(minimum([x_min, y_min]), digits = 1, base = 2) + + r_x = LinRange(x_min, x_max, n) + r_y = LinRange(y_min, y_max, n) + + # create grid to be used for the contours + d1 = collect(Iterators.flatten(vcat([fill(v, (n, 1)) for v in r_x]))) + d2 = collect(Iterators.flatten(vcat([reshape(r_y, 1, :) for _ in 1:n]))) + grid = hcat(d1, d2) + + # use model to predict decision boundary based on grid + gr_pred = model(grid') + gr_pred = reshape(gr_pred[1,:], (n, n)) + + # map classes (Boolean) to integers to be used as colors + cols = map(el -> el == true ? "#FF0000" : "#0000FF", loader.data[2][1,:]) + col_scale = [[-10, "blue"], [-2, "grey"], [0, "white"], [2, "white"], [10, "red"]] + + PlotlyJS.plot([ + PlotlyJS.scatter(x = loader.data[1][1,:], y = loader.data[1][2,:], mode = "markers", marker = attr(color = cols, line_width = 1)), + PlotlyJS.contour(x = r_x, y = r_y, z = gr_pred, colorscale = col_scale) + ]) end end \ No newline at end of file From 254b5107f559a95dcef49225f04c82af095eefca Mon Sep 17 00:00:00 2001 From: Peter Date: Fri, 7 Jan 2022 21:41:19 +0100 Subject: [PATCH 07/49] improve contour plot, minor refactorings --- src/data.jl | 2 +- src/neuralnetwork.jl | 61 ++++++++++++++++++++++++++++++++++---------- src/plotutils.jl | 21 +++++++++++++++ 3 files changed, 70 insertions(+), 14 deletions(-) create mode 100644 src/plotutils.jl diff --git a/src/data.jl b/src/data.jl index 1426de55..8531e68a 100644 --- a/src/data.jl +++ b/src/data.jl @@ -624,7 +624,7 @@ end function get_moons_from_publication() moons = hcat(moon_class_1(), moon_class_2()) - labels = [repeat([1], 150); repeat([-1], 150)] + labels = [repeat([0], 150); repeat([1], 150)] return moons, labels end diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 403dd194..2e560c26 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -1,14 +1,15 @@ module NeuralNetwork include("data.jl") +include("plotutils.jl") using Flux using Flux: onehotbatch, onecold using Flux.Data: DataLoader using Flux.Losses: logitcrossentropy -using MLDatasets using Plots using PlotlyJS +using MLDatasets export get_moon_data, get_nmist_data, get_loss_and_accuracy, train @@ -17,7 +18,7 @@ function get_moon_data(args) x_test, y_test = Data.get_moons_from_publication() x_train = transpose(x_train) - y_train, y_test = onehotbatch(y_train, 0:1), onehotbatch(y_test, [-1, 1]) + y_train, y_test = onehotbatch(y_train, 0:1), onehotbatch(y_test, 0:1) # create data loaders train_loader = DataLoader((x_train, y_train), batchsize = args.batchsize, shuffle = true) @@ -45,7 +46,7 @@ function get_nmist_data(args) return train_loader, test_loader end -function get_loss_and_accuracy(data_loader, model) +function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model) accuracy = 0 loss = 0.0f0 num = 0 @@ -76,7 +77,7 @@ end Base.@kwdef mutable struct Args learning_rate::Float64 = 1e-2 batchsize::Int = 50 - epochs::Int = 10 + epochs::Int = 100 end function train(args...) @@ -85,8 +86,14 @@ function train(args...) train_loader, test_loader = get_moon_data(args) # get_nmist_data(args) # plot train and test data sets - p_train = Plots.scatter(train_loader.data[1][1,:], train_loader.data[1][2,:], c = map(x -> x == true ? 1 : 0, train_loader.data[2][1,:])) - p_test = Plots.scatter(test_loader.data[1][1,:], test_loader.data[1][2,:], c = map(x -> x == true ? 1 : 0, test_loader.data[2][1,:])) + p_train = Plots.scatter( + train_loader.data[1][1,:], train_loader.data[1][2,:], + c = PlotUtils.map_bool_to_color(train_loader.data[2][1,:], "blue", "red") + ) + p_test = Plots.scatter( + test_loader.data[1][1,:], test_loader.data[1][2,:], + c = PlotUtils.map_bool_to_color(test_loader.data[2][1,:], "blue", "red") + ) display(Plots.plot(p_train, p_test, layout = (1, 2))) model = neural_network() @@ -101,14 +108,14 @@ function train(args...) Flux.Optimise.update!(optimizer, params, grads) end - # Report on train and test + # evaluate train and test loss and accuracy train_loss, train_accuracy = get_loss_and_accuracy(train_loader, model) test_loss, test_accuracy = get_loss_and_accuracy(test_loader, model) println("Epoch $epoch") println(" train loss = $train_loss, train accuracy = $train_accuracy") println(" test loss = $test_loss, test accuracy = $test_accuracy") - # early exit if accuracy is over 90% + # early exit if accuracy is over threshold acc_thres = 0.90 if test_accuracy > acc_thres @info "Stopped after $epoch epochs because test accuracy threshold of $(acc_thres * 100)% was exceeded." @@ -144,13 +151,41 @@ function plot_decision_boundary(loader, model) gr_pred = reshape(gr_pred[1,:], (n, n)) # map classes (Boolean) to integers to be used as colors - cols = map(el -> el == true ? "#FF0000" : "#0000FF", loader.data[2][1,:]) - col_scale = [[-10, "blue"], [-2, "grey"], [0, "white"], [2, "white"], [10, "red"]] + cols = PlotUtils.map_bool_to_color(loader.data[2][1,:], "#FF0000", "#0000FF") + opacity = 0.90 PlotlyJS.plot([ - PlotlyJS.scatter(x = loader.data[1][1,:], y = loader.data[1][2,:], mode = "markers", marker = attr(color = cols, line_width = 1)), - PlotlyJS.contour(x = r_x, y = r_y, z = gr_pred, colorscale = col_scale) - ]) + # data points + PlotlyJS.scatter( + x = loader.data[1][1,:], y = loader.data[1][2,:], + mode = "markers", marker = attr(color = cols, line_width = 1) + ), + # actual contours + PlotlyJS.contour( + x = r_x, y = r_y, z = gr_pred, + contours_coloring = "heatmap", colorscale = PlotUtils.get_custom_rdbu_scale(opacity), + opacity = opacity + ), + # highlight decision boundary line + PlotlyJS.contour( + x = r_x, y = r_y, z = gr_pred, + contours_start = 0, contours_end = 0, contours_size = 0, + contours_coloring = "lines", colorscale = [[0, "black"], [1, "black"]], + showscale = false, line = attr(width = 3) + )], + Layout( + width = 500, height = 500, autosize = true, + xaxis_showgrid = false, yaxis_showgrid = false, + xaxis_range = [x_min, x_max], yaxis_range = [y_min, y_max], + xaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), + yaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), + margin = attr(l = 50, r = 50, b = 50, t = 50, pad = 0), + plot_bgcolor = "rgba(0, 0, 0, 0)" + ), + config = PlotConfig( + scrollZoom = false + ) + ) end end \ No newline at end of file diff --git a/src/plotutils.jl b/src/plotutils.jl new file mode 100644 index 00000000..002e5e5f --- /dev/null +++ b/src/plotutils.jl @@ -0,0 +1,21 @@ +module PlotUtils + +export get_custom_rdbu_scale, map_bool_to_color + +function get_custom_rdbu_scale(opacity::Union{Float32, Float64}) + @assert opacity >= 0 || opacity <= 1 "Opacity must be in [0, 1]" + white = "rgba(255, 255, 255, $opacity)" + return [ + [0, "rgba(255, 0, 0, $opacity)"], + [0.475, "rgb(255, 200, 100, $opacity)"], + [0.5, white], + [0.525, "rgb(100, 200, 255, $opacity)"], + [1, "rgba(0, 0, 255, $opacity)"] + ] +end + +function map_bool_to_color(bool_vec::Union{BitVector, Vector{Bool}}, color_true::String, color_false::String) + return map(el -> el == true ? color_true : color_false, bool_vec) +end + +end \ No newline at end of file From fc4f416268090739561034cdb694f9692e97f949 Mon Sep 17 00:00:00 2001 From: Peter Date: Wed, 12 Jan 2022 17:46:39 +0100 Subject: [PATCH 08/49] fix data offset function --- src/data.jl | 50 ++++++++++++++++++++++---------------------------- 1 file changed, 22 insertions(+), 28 deletions(-) diff --git a/src/data.jl b/src/data.jl index 8531e68a..36504f9c 100644 --- a/src/data.jl +++ b/src/data.jl @@ -628,52 +628,46 @@ function get_moons_from_publication() return moons, labels end -function get_moons(n; noise = 0.1, offset = 0.0, rotation = 90) - X, y = make_moons(n_samples = n, noise = noise, random_state = 0) +function get_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand((1, 2^31))) + X, y = make_moons(n_samples = n, noise = noise, random_state = seed) p1 = scatter(X[:,1], X[:,2], c = y, title = "generated state") + if offset != 0.0 + X = apply_moon_offset(X, y, offset) + end + p2 = scatter(X[:,1], X[:,2], c = y, title = "offsetted state") + if rotation != 0.0 θ = deg2rad(rotation) R = [cos(θ) -sin(θ); sin(θ) cos(θ)] X *= R - p2 = scatter(X[:,1], X[:,2], c = y, title = "rotated state") end - - if offset != 0.0 - if rotation != 0.0 - # after rotation: apply offset to x-axis - X = apply_moon_offset(X, y, offset, "x") - else - # no rotation: apply offset to y-axis - X = apply_moon_offset(X, y, offset, "y") - end - end - p3 = scatter(X[:,1], X[:,2], c = y, title = "offsetted state") + p3 = scatter(X[:,1], X[:,2], c = y, title = "rotated state") display(plot(p1, p2, p3, layout = (1, 3))) return X, y end -function apply_moon_offset(X, y, offset, direction = "x") - @assert direction in ["x", "y"] "Error: direction must be either x or y" +function apply_moon_offset(X, y, offset) + # @assert direction in ["x", "y"] "Error: direction must be either x or y" # use offset as total offset, thus devide by number of labels; assume 2 labels offset /= 2 - # l = length(y) # length - # b = Int(l / 2) # boundary - - direction == "x" ? direction = 1 : direction = 2 + # direction == "x" ? direction = 1 : direction = 2 - first_moon = true + moons = [] for label in unique(y) - l_ids = findall(l -> l == label, y) - for id in l_ids - first_moon ? X[id:1:id, direction] .-= offset : X[id:1:id, direction] .+= offset - end - first_moon = false + push!(moons, findall(l -> l == label, y)) end - # X[:1:b, direction] .+= offset - # X[(b + 1):1:l, direction] .-= offset + max_y = [maximum(X[m, 2]) for m in moons] + + if max_y[1] > max_y[2] + X[moons[1], 2] .+= offset + X[moons[2], 2] .-= offset + else + X[moons[1], 2] .-= offset + X[moons[2], 2] .+= offset + end return X end From f67c3f012169570dfde2fdacad9e205dbb995355 Mon Sep 17 00:00:00 2001 From: Peter Date: Wed, 12 Jan 2022 17:47:16 +0100 Subject: [PATCH 09/49] adjust custom color scale --- src/plotutils.jl | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/plotutils.jl b/src/plotutils.jl index 002e5e5f..4c1f133f 100644 --- a/src/plotutils.jl +++ b/src/plotutils.jl @@ -7,9 +7,9 @@ function get_custom_rdbu_scale(opacity::Union{Float32, Float64}) white = "rgba(255, 255, 255, $opacity)" return [ [0, "rgba(255, 0, 0, $opacity)"], - [0.475, "rgb(255, 200, 100, $opacity)"], + [0.45, "rgb(255, 200, 100, $opacity)"], [0.5, white], - [0.525, "rgb(100, 200, 255, $opacity)"], + [0.55, "rgb(100, 200, 255, $opacity)"], [1, "rgba(0, 0, 255, $opacity)"] ] end From 6d4cb81621a5a67a7710fba75bd2b863795e641b Mon Sep 17 00:00:00 2001 From: Peter Date: Wed, 12 Jan 2022 17:48:26 +0100 Subject: [PATCH 10/49] simplify training; train and test on generated data; some other tweaks --- src/gradientstarvation.jl | 6 +++- src/neuralnetwork.jl | 63 +++++++++++++++++---------------------- 2 files changed, 33 insertions(+), 36 deletions(-) diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index fc74bc06..fbec8cc3 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -2,6 +2,10 @@ module GradientStarvation include("neuralnetwork.jl") -NeuralNetwork.train() +train_loader, test_loader = NeuralNetwork.get_moon_data(NeuralNetwork.Args()) +m = NeuralNetwork.train(train_loader, test_loader) + +display(NeuralNetwork.plot_decision_boundary(train_loader, m, title = "Train")) +display(NeuralNetwork.plot_decision_boundary(test_loader, m, title = "Test")) end diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 2e560c26..0aca74a6 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -14,10 +14,10 @@ using MLDatasets export get_moon_data, get_nmist_data, get_loss_and_accuracy, train function get_moon_data(args) - x_train, y_train = Data.get_moons(300, offset = 1.0) - x_test, y_test = Data.get_moons_from_publication() + x_train, y_train = Data.get_moons(300, offset = 0.5) + x_test, y_test = Data.get_moons(300, offset = 1.0, seed = 0) # Data.get_moons_from_publication() - x_train = transpose(x_train) + x_train, x_test = transpose(x_train), transpose(x_test) y_train, y_test = onehotbatch(y_train, 0:1), onehotbatch(y_test, 0:1) # create data loaders @@ -47,7 +47,7 @@ function get_nmist_data(args) end function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model) - accuracy = 0 + accuracy = 0.0f0 loss = 0.0f0 num = 0 for (x, y) in data_loader @@ -66,47 +66,38 @@ function neural_network() ) end -function get_prediction(loader, model) - pred = [] - for (x, y) in loader - push!(pred, model(y)) - end - return pred -end - Base.@kwdef mutable struct Args - learning_rate::Float64 = 1e-2 - batchsize::Int = 50 - epochs::Int = 100 + learning_rate::Float64 = 1e-1 + batchsize::Int = 300 + epochs::Int = 1000 end -function train(args...) - args = Args(args...) - - train_loader, test_loader = get_moon_data(args) # get_nmist_data(args) - +function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader) # plot train and test data sets p_train = Plots.scatter( train_loader.data[1][1,:], train_loader.data[1][2,:], - c = PlotUtils.map_bool_to_color(train_loader.data[2][1,:], "blue", "red") + c = PlotUtils.map_bool_to_color(train_loader.data[2][1,:], "blue", "red") ) p_test = Plots.scatter( test_loader.data[1][1,:], test_loader.data[1][2,:], c = PlotUtils.map_bool_to_color(test_loader.data[2][1,:], "blue", "red") ) display(Plots.plot(p_train, p_test, layout = (1, 2))) +end + +function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader; args...) + args = Args(args...) model = neural_network() params = Flux.params(model) - optimizer = ADAM(args.learning_rate) # Descent(args.learning_rate) # + # optimizer = ADAM(args.learning_rate) + optimizer = Descent(args.learning_rate) + # optimizer = Optimiser(WeightDecay(), Decent(args.learning_rate)) # training for epoch in 1:args.epochs - for (x, y) in train_loader - # compute gradient - grads = gradient(() -> logitcrossentropy(model(x), y), params) - Flux.Optimise.update!(optimizer, params, grads) - end + loss(x, y) = logitcrossentropy(model(x), y) + Flux.train!(loss, params, train_loader, optimizer) # evaluate train and test loss and accuracy train_loss, train_accuracy = get_loss_and_accuracy(train_loader, model) @@ -116,17 +107,17 @@ function train(args...) println(" test loss = $test_loss, test accuracy = $test_accuracy") # early exit if accuracy is over threshold - acc_thres = 0.90 - if test_accuracy > acc_thres - @info "Stopped after $epoch epochs because test accuracy threshold of $(acc_thres * 100)% was exceeded." - break - end + # acc_thres = 0.90 + # if test_accuracy > acc_thres + # @info "Stopped after $epoch epochs because test accuracy threshold of $(acc_thres * 100)% was exceeded." + # break + # end end return model end -function plot_decision_boundary(loader, model) +function plot_decision_boundary(loader, model; title = "") # grid and range step size n = 100 # determine limits of given data @@ -163,6 +154,7 @@ function plot_decision_boundary(loader, model) # actual contours PlotlyJS.contour( x = r_x, y = r_y, z = gr_pred, + contours_start = -50, contours_end = 50, contours_size = 5, contours_coloring = "heatmap", colorscale = PlotUtils.get_custom_rdbu_scale(opacity), opacity = opacity ), @@ -174,11 +166,12 @@ function plot_decision_boundary(loader, model) showscale = false, line = attr(width = 3) )], Layout( + title = title, width = 500, height = 500, autosize = true, xaxis_showgrid = false, yaxis_showgrid = false, xaxis_range = [x_min, x_max], yaxis_range = [y_min, y_max], - xaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), - yaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), + # xaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), + # yaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), margin = attr(l = 50, r = 50, b = 50, t = 50, pad = 0), plot_bgcolor = "rgba(0, 0, 0, 0)" ), From 02a56235c9267acba9976e28adeea48d836a0617 Mon Sep 17 00:00:00 2001 From: Peter Date: Thu, 13 Jan 2022 14:02:16 +0100 Subject: [PATCH 11/49] WIP: spectral decoupling --- src/regularization.jl | 22 ++++++++++++++++++++++ 1 file changed, 22 insertions(+) create mode 100644 src/regularization.jl diff --git a/src/regularization.jl b/src/regularization.jl new file mode 100644 index 00000000..d61e21f0 --- /dev/null +++ b/src/regularization.jl @@ -0,0 +1,22 @@ +module Regularization + +using Flux: onecold +using Flux.Losses: logitcrossentropy, crossentropy +using LinearAlgebra +using Statistics + +function spectral_decoupling(ŷ, y, λ) + # ŷ, y = onecold(ŷ), onecold(y) + # return mean(log.(1.0 .+ exp.(-ŷ .* y))) .+ λ * mean(ŷ .* 2) + # println(mean(log.(1.0 .+ exp.(-ŷ .* y))) .+ λ * mean(ŷ .* 2).^2) + + # python: torch.log(1.0 + torch.exp(-ŷ * y)).mean() + λ * (ŷ ** 2).mean() + # v = mean(log.(1.0 .+ exp.(-y .* ŷ))) .+ λ/2 * norm(ŷ, 2)^2 + # v = logitcrossentropy(ŷ, y) .+ (λ/2 * norm(ŷ, 2)^2) + v = logitcrossentropy(ŷ, y) + λ * mean(ŷ .^ 2) + # v = mean(1.0 .* log.(1.0 .+ exp.(-ŷ .* y))) + λ * mean(ŷ .^ 2) + # println(v) + return v +end + +end \ No newline at end of file From b8a1890ba54a1d96421c8d08acadc28a429a3daa Mon Sep 17 00:00:00 2001 From: Peter Date: Thu, 13 Jan 2022 14:03:11 +0100 Subject: [PATCH 12/49] adjust moon position after rotation; optionally show plots --- src/data.jl | 15 ++++++++++++--- 1 file changed, 12 insertions(+), 3 deletions(-) diff --git a/src/data.jl b/src/data.jl index 36504f9c..5db4028b 100644 --- a/src/data.jl +++ b/src/data.jl @@ -622,13 +622,16 @@ function moon_class_2(offset = 0.0) return vcat(transpose.((x, y))...) end -function get_moons_from_publication() +function get_moons_from_publication(; show_plot = false) moons = hcat(moon_class_1(), moon_class_2()) labels = [repeat([0], 150); repeat([1], 150)] + if show_plot + display(scatter(moons[1,:], moons[2,:], c = labels)) + end return moons, labels end -function get_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand((1, 2^31))) +function get_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand((1, 2^31)), show_plot = false) X, y = make_moons(n_samples = n, noise = noise, random_state = seed) p1 = scatter(X[:,1], X[:,2], c = y, title = "generated state") @@ -641,10 +644,16 @@ function get_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand((1, θ = deg2rad(rotation) R = [cos(θ) -sin(θ); sin(θ) cos(θ)] X *= R + # move center closer to (0, 0) + if offset != 0.0 + X[:,2] .+= offset/2 + end end p3 = scatter(X[:,1], X[:,2], c = y, title = "rotated state") - display(plot(p1, p2, p3, layout = (1, 3))) + if show_plot + display(plot(p1, p2, p3, layout = (1, 3))) + end return X, y end From 2f80f5886e81c8572f476aa634ee65c920f52ff5 Mon Sep 17 00:00:00 2001 From: Peter Date: Thu, 13 Jan 2022 14:04:01 +0100 Subject: [PATCH 13/49] use ProgressMeter for training; train with SD --- Project.toml | 1 + notebook/gradientstarvation.jl | 0 src/neuralnetwork.jl | 41 ++++++++++++++++++---------------- 3 files changed, 23 insertions(+), 19 deletions(-) create mode 100644 notebook/gradientstarvation.jl diff --git a/Project.toml b/Project.toml index 24ab59c4..fb9fda2a 100644 --- a/Project.toml +++ b/Project.toml @@ -8,4 +8,5 @@ Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c" MLDatasets = "eb30cadb-4394-5ae3-aed4-317e484a6458" PlotlyJS = "f0f68f2c-4968-5e81-91da-67840de0976a" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +ProgressMeter = "92933f4c-e287-5a05-a399-4b506db050ca" ScikitLearn = "3646fa90-6ef7-5e7e-9f22-8aca16db6324" diff --git a/notebook/gradientstarvation.jl b/notebook/gradientstarvation.jl new file mode 100644 index 00000000..e69de29b diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 0aca74a6..eeabd98f 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -2,20 +2,23 @@ module NeuralNetwork include("data.jl") include("plotutils.jl") +include("regularization.jl") using Flux using Flux: onehotbatch, onecold using Flux.Data: DataLoader using Flux.Losses: logitcrossentropy +using MLDatasets using Plots using PlotlyJS -using MLDatasets +using ProgressMeter +using Statistics export get_moon_data, get_nmist_data, get_loss_and_accuracy, train function get_moon_data(args) x_train, y_train = Data.get_moons(300, offset = 0.5) - x_test, y_test = Data.get_moons(300, offset = 1.0, seed = 0) # Data.get_moons_from_publication() + x_test, y_test = Data.get_moons(300, offset = 0.5, seed = 0) # Data.get_moons_from_publication() x_train, x_test = transpose(x_train), transpose(x_test) y_train, y_test = onehotbatch(y_train, 0:1), onehotbatch(y_test, 0:1) @@ -46,17 +49,18 @@ function get_nmist_data(args) return train_loader, test_loader end -function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model) +function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model; args...) + args = Args(args...) accuracy = 0.0f0 loss = 0.0f0 num = 0 for (x, y) in data_loader ŷ = model(y) - loss += logitcrossentropy(ŷ, y, agg = sum) + loss += Regularization.spectral_decoupling(ŷ, y, args.λ) # logitcrossentropy(ŷ, y, agg = sum) accuracy += sum(onecold(ŷ) .== onecold(y)) num += size(x)[end] end - return loss / num, accuracy / num + return loss, accuracy / num end function neural_network() @@ -67,9 +71,10 @@ function neural_network() end Base.@kwdef mutable struct Args - learning_rate::Float64 = 1e-1 + learning_rate::Float64 = 0.1 batchsize::Int = 300 epochs::Int = 1000 + λ::Float64 = 3e-1 end function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader) @@ -95,23 +100,21 @@ function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLo # optimizer = Optimiser(WeightDecay(), Decent(args.learning_rate)) # training + prog = Progress(args.epochs, 0.25, "Training... ", 75) for epoch in 1:args.epochs - loss(x, y) = logitcrossentropy(model(x), y) + # loss(x, y) = logitcrossentropy(model(x), y) + loss(x, y) = Regularization.spectral_decoupling(model(x), y, args.λ) Flux.train!(loss, params, train_loader, optimizer) # evaluate train and test loss and accuracy train_loss, train_accuracy = get_loss_and_accuracy(train_loader, model) test_loss, test_accuracy = get_loss_and_accuracy(test_loader, model) - println("Epoch $epoch") - println(" train loss = $train_loss, train accuracy = $train_accuracy") - println(" test loss = $test_loss, test accuracy = $test_accuracy") - - # early exit if accuracy is over threshold - # acc_thres = 0.90 - # if test_accuracy > acc_thres - # @info "Stopped after $epoch epochs because test accuracy threshold of $(acc_thres * 100)% was exceeded." - # break - # end + + ProgressMeter.next!(prog; showvalues = [ + (:epoch, epoch), + (:train_loss, train_loss), (:test_loss, test_loss), + (:train_accuracy, train_accuracy), (:test_accuracy, test_accuracy) + ]) end return model @@ -170,8 +173,8 @@ function plot_decision_boundary(loader, model; title = "") width = 500, height = 500, autosize = true, xaxis_showgrid = false, yaxis_showgrid = false, xaxis_range = [x_min, x_max], yaxis_range = [y_min, y_max], - # xaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), - # yaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), + xaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), + yaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), margin = attr(l = 50, r = 50, b = 50, t = 50, pad = 0), plot_bgcolor = "rgba(0, 0, 0, 0)" ), From 2e857c690497fb8fe7ba895fc70916b8009bd6c7 Mon Sep 17 00:00:00 2001 From: Peter Date: Thu, 13 Jan 2022 16:38:09 +0100 Subject: [PATCH 14/49] indicate WIP status in README --- README.md | 15 ++------------- 1 file changed, 2 insertions(+), 13 deletions(-) diff --git a/README.md b/README.md index e2ced9a6..bbf84e3e 100644 --- a/README.md +++ b/README.md @@ -1,14 +1,3 @@ -# STMO-ZOO +# Gradient Starvation -Welcome to the STMO zoo! This is your final assignment for the course Selected Topics in Mathematical Optimization. Your goal is to implement an optimization method in Julia and contribute this to this repository. To pass, you have to: - -- fork this repo and create a pull request; -- add a module to `src` with **at least one function** -- add at least one unit test to the folder `test`; -- document all your functions and add a page to the documentation page; -- make a notebook in [Pluto](https://github.com/fonsp/Pluto.jl) and add it to `notebooks`; -- perform a small code review of two other students. - -Depending on the project you choose some of these individual assignments might be really minimalistic, with other parts larger. For example, if you want to develop an application, say solving the graph coloring problem with Tabu Search, you might have only a single function in the source code (e.g., generating an instance) but have a fairly large notebook with a tutorial. On the other hand, if you work on a method, e.g., implementing Bee Colony Optimization, you might have many functions in the source code, while your notebook is only a demonstration on the test functions. - -[![Build Status](https://travis-ci.org/MichielStock/STMOZOO.svg?branch=master)](https://travis-ci.org/MichielStock/STMOZOO)[![Coverage Status](https://coveralls.io/repos/github/MichielStock/STMOZOO/badge.svg?branch=master)](https://coveralls.io/github/MichielStock/STMOZOO?branch=master) \ No newline at end of file +This project is still under construction. \ No newline at end of file From 075a750480fff457ba699e6e2dad57931128015f Mon Sep 17 00:00:00 2001 From: Peter Date: Thu, 13 Jan 2022 16:38:30 +0100 Subject: [PATCH 15/49] add stochastic gradient descent optimizer --- src/neuralnetwork.jl | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index eeabd98f..2fa2c828 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -71,7 +71,7 @@ function neural_network() end Base.@kwdef mutable struct Args - learning_rate::Float64 = 0.1 + learning_rate::Float64 = 1e-2 batchsize::Int = 300 epochs::Int = 1000 λ::Float64 = 3e-1 @@ -96,7 +96,9 @@ function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLo model = neural_network() params = Flux.params(model) # optimizer = ADAM(args.learning_rate) - optimizer = Descent(args.learning_rate) + # optimizer = Descent(args.learning_rate) + + optimizer = Momentum(args.learning_rate, 0.9) # stochastic gradient descent # optimizer = Optimiser(WeightDecay(), Decent(args.learning_rate)) # training From 35af33b484838b6f1cb52cedf7e3471c1d754571 Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 16 Jan 2022 20:26:45 +0100 Subject: [PATCH 16/49] add transformation features to moons from publication; refactor method names --- src/data.jl | 18 +++++++++--------- src/gradientstarvation.jl | 2 +- src/neuralnetwork.jl | 14 +++++++------- 3 files changed, 17 insertions(+), 17 deletions(-) diff --git a/src/data.jl b/src/data.jl index 5db4028b..5c7fe17a 100644 --- a/src/data.jl +++ b/src/data.jl @@ -4,9 +4,9 @@ using Plots using ScikitLearn @sk_import datasets: make_moons -export get_moons, get_moons_from_publication +export generate_moons, get_moons_from_publication -function moon_class_1(offset = 0.0) +function moon_class_1(offset = 0.0, coord_down_scale = 1) x = [ -2.717835893494339 - offset, -2.0025756608302556 - offset, @@ -311,10 +311,10 @@ function moon_class_1(offset = 0.0) -0.7882243058017496, -0.9895456553950488 ] - return vcat(transpose.((x, y))...) + return vcat(transpose.((x, y))...) / coord_down_scale end -function moon_class_2(offset = 0.0) +function moon_class_2(offset = 0.0, coord_down_scale = 1) x = [ 1.2605697631874289 + offset, 0.4428841761287988 + offset, @@ -618,12 +618,12 @@ function moon_class_2(offset = 0.0) 1.6438700737392897, -3.0090552847647456, 0.2317409067300455 - ] - return vcat(transpose.((x, y))...) + ] + return vcat(transpose.((x, y))...) / coord_down_scale end -function get_moons_from_publication(; show_plot = false) - moons = hcat(moon_class_1(), moon_class_2()) +function get_moons_from_publication(;offset = 0.0, coord_down_scale = 1, show_plot = false) + moons = hcat(moon_class_1(offset, coord_down_scale), moon_class_2(offset, coord_down_scale)) labels = [repeat([0], 150); repeat([1], 150)] if show_plot display(scatter(moons[1,:], moons[2,:], c = labels)) @@ -631,7 +631,7 @@ function get_moons_from_publication(; show_plot = false) return moons, labels end -function get_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand((1, 2^31)), show_plot = false) +function generate_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand((1, 2^31)), show_plot = false) X, y = make_moons(n_samples = n, noise = noise, random_state = seed) p1 = scatter(X[:,1], X[:,2], c = y, title = "generated state") diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index fbec8cc3..8a4fd311 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -2,7 +2,7 @@ module GradientStarvation include("neuralnetwork.jl") -train_loader, test_loader = NeuralNetwork.get_moon_data(NeuralNetwork.Args()) +train_loader, test_loader = NeuralNetwork.get_moon_data_loader(NeuralNetwork.Args()) m = NeuralNetwork.train(train_loader, test_loader) display(NeuralNetwork.plot_decision_boundary(train_loader, m, title = "Train")) diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 2fa2c828..3c60ce28 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -14,11 +14,11 @@ using PlotlyJS using ProgressMeter using Statistics -export get_moon_data, get_nmist_data, get_loss_and_accuracy, train +export get_moon_data_loader, get_nmist_data_loader, get_loss_and_accuracy, train -function get_moon_data(args) - x_train, y_train = Data.get_moons(300, offset = 0.5) - x_test, y_test = Data.get_moons(300, offset = 0.5, seed = 0) # Data.get_moons_from_publication() +function get_moon_data_loader(args) + x_train, y_train = Data.generate_moons(300, offset = 0.5) + x_test, y_test = Data.generate_moons(300, offset = 0.5, seed = 0) x_train, x_test = transpose(x_train), transpose(x_test) y_train, y_test = onehotbatch(y_train, 0:1), onehotbatch(y_test, 0:1) @@ -30,7 +30,7 @@ function get_moon_data(args) return train_loader, test_loader end -function get_nmist_data(args) +function get_nmist_data_loader(args) # Loading Dataset xtrain, ytrain = MLDatasets.MNIST.traindata(Float32) xtest, ytest = MLDatasets.MNIST.testdata(Float32) @@ -72,7 +72,7 @@ end Base.@kwdef mutable struct Args learning_rate::Float64 = 1e-2 - batchsize::Int = 300 + batchsize::Int = 50 epochs::Int = 1000 λ::Float64 = 3e-1 end @@ -159,7 +159,7 @@ function plot_decision_boundary(loader, model; title = "") # actual contours PlotlyJS.contour( x = r_x, y = r_y, z = gr_pred, - contours_start = -50, contours_end = 50, contours_size = 5, + contours_start = -10, contours_end = 10, contours_size = 1, contours_coloring = "heatmap", colorscale = PlotUtils.get_custom_rdbu_scale(opacity), opacity = opacity ), From bd8bd4f39a7f9bdb253747de6a58fedaf132aa47 Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 16 Jan 2022 21:36:57 +0100 Subject: [PATCH 17/49] refactor: better separation of NN and data, improve configurability of training --- src/args.jl | 6 ++ src/data.jl | 171 +++++++++++++++++++++++++------------- src/gradientstarvation.jl | 6 +- src/neuralnetwork.jl | 92 ++++++-------------- 4 files changed, 150 insertions(+), 125 deletions(-) create mode 100644 src/args.jl diff --git a/src/args.jl b/src/args.jl new file mode 100644 index 00000000..3ffa32db --- /dev/null +++ b/src/args.jl @@ -0,0 +1,6 @@ +Base.@kwdef mutable struct Args + learning_rate::Float64 = 1e-2 + batchsize::Int = 50 + epochs::Int = 1000 + λ::Float64 = 3e-1 +end diff --git a/src/data.jl b/src/data.jl index 5c7fe17a..56ced7c3 100644 --- a/src/data.jl +++ b/src/data.jl @@ -1,10 +1,111 @@ module Data +include("args.jl") + +using Flux +using Flux: onehotbatch +using Flux.Data: DataLoader using Plots using ScikitLearn @sk_import datasets: make_moons -export generate_moons, get_moons_from_publication +export get_moon_data_loader, get_nmist_data_loader, plot_train_and_test_data + +function get_moon_data_loader(; args...) + args = Args(args...) + + x_train, y_train = generate_moons(300, offset = 0.5) + x_test, y_test = generate_moons(300, offset = 0.5, seed = 0) + + x_train, x_test = transpose(x_train), transpose(x_test) + y_train, y_test = onehotbatch(y_train, 0:1), onehotbatch(y_test, 0:1) + + # create data loaders + train_loader = DataLoader((x_train, y_train), batchsize = args.batchsize, shuffle = true) + test_loader = DataLoader((x_test, y_test), batchsize = args.batchsize) + + return train_loader, test_loader +end + +function get_nmist_data_loader(; args...) + args = Args(args...) + + # Loading Dataset + xtrain, ytrain = MLDatasets.MNIST.traindata(Float32) + xtest, ytest = MLDatasets.MNIST.testdata(Float32) + + # Reshape Data in order to flatten each image into a linear array + xtrain = Flux.flatten(xtrain) + xtest = Flux.flatten(xtest) + + # One-hot-encode the labels + ytrain, ytest = onehotbatch(ytrain, 0:9), onehotbatch(ytest, 0:9) + + # Create DataLoaders (mini-batch iterators) + train_loader = DataLoader((xtrain, ytrain), batchsize=args.batchsize, shuffle=true) + test_loader = DataLoader((xtest, ytest), batchsize=args.batchsize) + + return train_loader, test_loader +end + +function generate_moons_from_publication(; offset = 0.0, coord_down_scale = 1, show_plot = false) + moons = hcat(moon_class_1(offset, coord_down_scale), moon_class_2(offset, coord_down_scale)) + labels = [repeat([0], 150); repeat([1], 150)] + if show_plot + display(scatter(moons[1,:], moons[2,:], c = labels)) + end + return moons, labels +end + +function generate_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand((1, 2^31)), show_plot = false) + X, y = make_moons(n_samples = n, noise = noise, random_state = seed) + p1 = scatter(X[:,1], X[:,2], c = y, title = "generated state") + + if offset != 0.0 + X = apply_moon_offset(X, y, offset) + end + p2 = scatter(X[:,1], X[:,2], c = y, title = "offsetted state") + + if rotation != 0.0 + θ = deg2rad(rotation) + R = [cos(θ) -sin(θ); sin(θ) cos(θ)] + X *= R + # move center closer to (0, 0) + if offset != 0.0 + X[:,2] .+= offset/2 + end + end + p3 = scatter(X[:,1], X[:,2], c = y, title = "rotated state") + + if show_plot + display(plot(p1, p2, p3, layout = (1, 3))) + end + return X, y +end + +function apply_moon_offset(X, y, offset) + # @assert direction in ["x", "y"] "Error: direction must be either x or y" + # use offset as total offset, thus devide by number of labels; assume 2 labels + offset /= 2 + # direction == "x" ? direction = 1 : direction = 2 + + moons = [] + for label in unique(y) + push!(moons, findall(l -> l == label, y)) + end + + max_y = [maximum(X[m, 2]) for m in moons] + + if max_y[1] > max_y[2] + X[moons[1], 2] .+= offset + X[moons[2], 2] .-= offset + else + X[moons[1], 2] .-= offset + X[moons[2], 2] .+= offset + end + + return X +end function moon_class_1(offset = 0.0, coord_down_scale = 1) x = [ @@ -622,63 +723,17 @@ function moon_class_2(offset = 0.0, coord_down_scale = 1) return vcat(transpose.((x, y))...) / coord_down_scale end -function get_moons_from_publication(;offset = 0.0, coord_down_scale = 1, show_plot = false) - moons = hcat(moon_class_1(offset, coord_down_scale), moon_class_2(offset, coord_down_scale)) - labels = [repeat([0], 150); repeat([1], 150)] - if show_plot - display(scatter(moons[1,:], moons[2,:], c = labels)) - end - return moons, labels -end - -function generate_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand((1, 2^31)), show_plot = false) - X, y = make_moons(n_samples = n, noise = noise, random_state = seed) - p1 = scatter(X[:,1], X[:,2], c = y, title = "generated state") - - if offset != 0.0 - X = apply_moon_offset(X, y, offset) - end - p2 = scatter(X[:,1], X[:,2], c = y, title = "offsetted state") - - if rotation != 0.0 - θ = deg2rad(rotation) - R = [cos(θ) -sin(θ); sin(θ) cos(θ)] - X *= R - # move center closer to (0, 0) - if offset != 0.0 - X[:,2] .+= offset/2 - end - end - p3 = scatter(X[:,1], X[:,2], c = y, title = "rotated state") - - if show_plot - display(plot(p1, p2, p3, layout = (1, 3))) - end - return X, y -end - -function apply_moon_offset(X, y, offset) - # @assert direction in ["x", "y"] "Error: direction must be either x or y" - # use offset as total offset, thus devide by number of labels; assume 2 labels - offset /= 2 - # direction == "x" ? direction = 1 : direction = 2 - - moons = [] - for label in unique(y) - push!(moons, findall(l -> l == label, y)) - end - - max_y = [maximum(X[m, 2]) for m in moons] - - if max_y[1] > max_y[2] - X[moons[1], 2] .+= offset - X[moons[2], 2] .-= offset - else - X[moons[1], 2] .-= offset - X[moons[2], 2] .+= offset - end - - return X +function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader) + # plot train and test data sets + p_train = Plots.scatter( + train_loader.data[1][1,:], train_loader.data[1][2,:], + c = PlotUtils.map_bool_to_color(train_loader.data[2][1,:], "blue", "red") + ) + p_test = Plots.scatter( + test_loader.data[1][1,:], test_loader.data[1][2,:], + c = PlotUtils.map_bool_to_color(test_loader.data[2][1,:], "blue", "red") + ) + display(Plots.plot(p_train, p_test, layout = (1, 2))) end end \ No newline at end of file diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index 8a4fd311..c29a9ead 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -1,9 +1,11 @@ module GradientStarvation +include("args.jl") +include("data.jl") include("neuralnetwork.jl") -train_loader, test_loader = NeuralNetwork.get_moon_data_loader(NeuralNetwork.Args()) -m = NeuralNetwork.train(train_loader, test_loader) +train_loader, test_loader = Data.get_moon_data_loader() +m = NeuralNetwork.train(train_loader, test_loader, "GD", false) display(NeuralNetwork.plot_decision_boundary(train_loader, m, title = "Train")) display(NeuralNetwork.plot_decision_boundary(test_loader, m, title = "Test")) diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 3c60ce28..3aae6c5e 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -1,12 +1,11 @@ module NeuralNetwork -include("data.jl") +include("args.jl") include("plotutils.jl") include("regularization.jl") using Flux -using Flux: onehotbatch, onecold -using Flux.Data: DataLoader +using Flux: onecold using Flux.Losses: logitcrossentropy using MLDatasets using Plots @@ -14,40 +13,7 @@ using PlotlyJS using ProgressMeter using Statistics -export get_moon_data_loader, get_nmist_data_loader, get_loss_and_accuracy, train - -function get_moon_data_loader(args) - x_train, y_train = Data.generate_moons(300, offset = 0.5) - x_test, y_test = Data.generate_moons(300, offset = 0.5, seed = 0) - - x_train, x_test = transpose(x_train), transpose(x_test) - y_train, y_test = onehotbatch(y_train, 0:1), onehotbatch(y_test, 0:1) - - # create data loaders - train_loader = DataLoader((x_train, y_train), batchsize = args.batchsize, shuffle = true) - test_loader = DataLoader((x_test, y_test), batchsize = args.batchsize) - - return train_loader, test_loader -end - -function get_nmist_data_loader(args) - # Loading Dataset - xtrain, ytrain = MLDatasets.MNIST.traindata(Float32) - xtest, ytest = MLDatasets.MNIST.testdata(Float32) - - # Reshape Data in order to flatten each image into a linear array - xtrain = Flux.flatten(xtrain) - xtest = Flux.flatten(xtest) - - # One-hot-encode the labels - ytrain, ytest = onehotbatch(ytrain, 0:9), onehotbatch(ytest, 0:9) - - # Create DataLoaders (mini-batch iterators) - train_loader = DataLoader((xtrain, ytrain), batchsize=args.batchsize, shuffle=true) - test_loader = DataLoader((xtest, ytest), batchsize=args.batchsize) - - return train_loader, test_loader -end +export get_loss_and_accuracy, train function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model; args...) args = Args(args...) @@ -70,42 +36,22 @@ function neural_network() ) end -Base.@kwdef mutable struct Args - learning_rate::Float64 = 1e-2 - batchsize::Int = 50 - epochs::Int = 1000 - λ::Float64 = 3e-1 -end - -function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader) - # plot train and test data sets - p_train = Plots.scatter( - train_loader.data[1][1,:], train_loader.data[1][2,:], - c = PlotUtils.map_bool_to_color(train_loader.data[2][1,:], "blue", "red") - ) - p_test = Plots.scatter( - test_loader.data[1][1,:], test_loader.data[1][2,:], - c = PlotUtils.map_bool_to_color(test_loader.data[2][1,:], "blue", "red") - ) - display(Plots.plot(p_train, p_test, layout = (1, 2))) -end - -function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader; args...) +function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spectral_decoupling::Bool = false; args...) args = Args(args...) + @info "opt = $optimizer, sd = $spectral_decoupling, lr = $(args.learning_rate), bs = $(args.batchsize)" * (spectral_decoupling ? "λ = $(args.λ)" : "") + model = neural_network() params = Flux.params(model) - # optimizer = ADAM(args.learning_rate) - # optimizer = Descent(args.learning_rate) - - optimizer = Momentum(args.learning_rate, 0.9) # stochastic gradient descent - # optimizer = Optimiser(WeightDecay(), Decent(args.learning_rate)) + optimizer = get_optimizer(args.learning_rate, optimizer) # training prog = Progress(args.epochs, 0.25, "Training... ", 75) for epoch in 1:args.epochs - # loss(x, y) = logitcrossentropy(model(x), y) - loss(x, y) = Regularization.spectral_decoupling(model(x), y, args.λ) + loss(x, y) = spectral_decoupling ? + Regularization.spectral_decoupling(model(x), y, args.λ) : + logitcrossentropy(model(x), y) + Flux.train!(loss, params, train_loader, optimizer) # evaluate train and test loss and accuracy @@ -122,6 +68,22 @@ function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLo return model end +function get_optimizer(learning_rate, optimizer = "SGD") + optimizer = uppercase(optimizer) + @assert optimizer in ["ADAM", "GD", "SGD", "WD"] "Requested optimizer '$optimizer' is not supported." + + if optimizer == "ADAM" + return ADAM(learning_rate) + elseif optimizer == "GD" + return Descent(learning_rate) + elseif optimizer == "SGD" + # stochastic gradient descent + return Momentum(learning_rate, 0.9) + elseif optimizer == "WD" + return Optimiser(WeightDecay(), Decent(learning_rate)) + end +end + function plot_decision_boundary(loader, model; title = "") # grid and range step size n = 100 From ccf0bcc4d86805528e34ddfc78b6327c054427f1 Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 16 Jan 2022 21:39:04 +0100 Subject: [PATCH 18/49] WIP: notebook --- notebook/gradientstarvation.jl | 149 +++++++++++++++++++++++++++++++++ 1 file changed, 149 insertions(+) diff --git a/notebook/gradientstarvation.jl b/notebook/gradientstarvation.jl index e69de29b..e83f8f5d 100644 --- a/notebook/gradientstarvation.jl +++ b/notebook/gradientstarvation.jl @@ -0,0 +1,149 @@ +### A Pluto.jl notebook ### +# v0.17.5 + +using Markdown +using InteractiveUtils + +# ╔═╡ df218f5d-042d-4ccb-8005-56b2ec073869 +html""" +This is still work in progress! +""" + +# ╔═╡ 8bd774c7-1acf-4beb-bc99-4c303882d262 +md""" +# Hungry Neurons - Neural Networks and Gradient Starvation + +## Introduction + +Artificial intelligence (AI) and machine learning are disciplines whose methods have become ubiquitously applied in a broad range of scientific domains, enabling the implicit learning of features inherent to a dataset. A common AI tool for self-learning of data characteristics by examples are neural networks (NNs) which attempt to mimic biological neurons. NNs are broadly applicable, e.g. for tasks such as image and speech recognition, weather forecasting, stock market prediction, robotics and data mining among many others. But how do these artificial neural networks learn? +There is a myriad of neural network subtypes and topologies, but in this notebook, I want to focus on the multi-layer perceptron (MLP). Basically speaking, a NN consists of multiple layers: an input layer, one or more hidden layers and an output layer (see figure 1). + +""" + +# ╔═╡ 8f0cdbf1-0174-4d18-9072-2a8c5f015177 +html""" +An image visualizing a basic NN topology with an input layer, a single hidden layer and an output layer + +""" + +# ╔═╡ 6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b +md"""Figure 1: Basic NN/MLP topology with input, a single hidden and output layer. + +CC-BY-SA 3.0. Source: [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Artificial_neural_network.svg)""" + +# ╔═╡ 512e6240-76e7-11ec-2511-07ffc4b74d27 +md""" +The input layer encodes the data presented to the network and feeds it weighted into the first hidden layer; all other layers consist of artificial neurons with non-linear activation functions. MLPs are fully connected, meaning that every node of one layer connects to every node of the next layer and each connection has a weight assigned to it. The weights can be understood as the actual learned parameters being adjusted as they get passed from layer to layer. The MLP learns supervised, i.e. by presentation of labelled training data. Without going into too much detail, I want to briefly formalize the learning process by the example of a classification task, that is deciding to which discrete class a given input belongs. +""" + +# ╔═╡ 93bc80a8-673d-457b-96d8-e1bb0b09682e +md""" +### Supervised Learning – Learning from Examples + +Supervised learning tries to learn a function which maps input to output values based on known input-output pairs (i.e. the pair (x, y); a correct classification y for a given input x). The observed outputs $y_i$ originate from an unknown function $f(x) = y$. Thus, the goal of the learning process is to approximate $f(x)$ by a hypothesis $h(x) \approx f(x)$. The loss is defined as the utility lost by substituting the correct classification $f(x) = y$ by the hypothesis $h(x) = \hat{y}$: + +$L(x, y, ŷ) = +Utility\\(result\ of\ using\ y\ given\ x\\) +- Utility\\(result\ of\ using\ ŷ\ given\ x\\)$ + +Commonly applied loss functions are mean absolute error and mean squared error loss. Especially popular for classification tasks is the cross-entropy loss, also called logarithmic loss: + +$L(y, ŷ) = -\sum_{i}y_i\cdot log(ŷ_i)$ + +The learning process maximizes the expected utility through choosing the optimal hypothesis by minimizing the loss function. In practice, this is estimated by the empirical loss observed on the set $X$ of $N$ training examples. + +$EmpiricalLoss(h) = \frac{1}{N}\cdot\sum_{(x, y) \in X} L(y, h(x))$ + +But what about the weights I’ve introduced earlier as the “actual learned parameters”? Keep in mind that a hypothesis $h$ is a function and the weights simply represent the function coefficients. For the linear case that would mean $h(x) = w_1 x + w_0$. +In the multivariate case there is no closed-form solution for the minimum loss. Therefore, we are confronted with an optimization problem which can be solved by gradient descent. The weight update with learning rate $\alpha$ and the vector of weights $\bf{w}$: + +$w_i = w_i - \alpha\cdot \frac{\partial}{\partial w_i}\cdot Loss(\bf{w})$ + +""" + +# ╔═╡ 66a3b5cb-a465-42fb-a8c8-404d574312d2 +md""" +### Regularization + + +> "Entities should not be multiplied beyond necessity." – William of Ockham +This quote, known as Ockham’s razor and usually paraphrased as “simple solutions are better solutions” captures the core idea of regularization. The aim is to find the hypothesis which minimizes the total cost, defined as the sum of empirical loss and its complexity: + +$Cost(h) = EmpiricalLoss(h) + \lambda\cdot Complexity(h)$ + +Regularization penalizes complex hypotheses and favors more regular functions. The estimated best hypothesis $ĥ^*$ from the hypothesis space $\mathcal{H}$ as determined by regularization is: + +$ĥ^* = \arg\min_{h \in \mathcal{H}} Cost(h)$ +""" + +# ╔═╡ 4a14831d-7736-4af5-875f-05ae6e3cf7c4 +md""" +### Side Effects of Gradient-based Learning + +Given the major workings of supervised learning, it should have become clear that learning is a mathematical optimization problem. A classic way to achieve the minimization of the loss is the application of gradient descent. + +Despite its popularity, gradient descent learning in NNs can come with some unwanted side effects such as the vanishing gradient problem, which condenses down to the gradient being too small to effectively update a weight and thus preventing learning. + +Another not yet published but currently in preprint described observation is the effect of **Gradient Starvation (GS)**. Pezeshki et al. coined and formalized this phenomenon of over-parameterized NNs in their work. According to them, _Gradient Starvation_ is a predisposition of cross-entropy loss learning NNs, which occurs when the loss minimization is driven by only a feature subset regardless of the presence of other meaningful features. Thus, the NN becomes biased towards superficial features. Beyond demonstrating the effects of GS, Pezeshki et al. introduce _Spectral Decoupling_ (SD) regularization as a means to counteract gradient starvation. +""" + +# ╔═╡ 00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 +md""" +### Objective + +My project and thereby this notebook aims to reproduce and demonstrate the effects of Gradient Starvation on a simple 2D classification task. Furthermore, the proposed regularization method of Spectral Decoupling is implemented and compared to other regularization methods. Finally, the framework is extended to a more complex task on the well-known MNIST dataset. + +""" + +# ╔═╡ ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 +md""" +## Experiments + +### 2D Classification +""" + +# ╔═╡ 85591d79-dec2-45fa-a0a9-d95d84edda51 +md""" +## Implementation +The implementation of the NN is closely related to the one of Pezeshki et al. who realized their NN with Python and PyTorch. My implementation is based on a ML framework for Julia called [Flux](https://fluxml.ai/). The NN consists of two hidden layers with 500 nodes each. The applied activation function is Rectified Linear Unit (ReLu) and learning is facilitated with cross-entropy loss. +""" + +# ╔═╡ 92cca2d8-c2b6-4046-9b86-26c5c9d64ee3 +md"## Conclusion" + +# ╔═╡ 88b30b55-4f98-474a-aaca-a1197d2886fd +md"## Code" + +# ╔═╡ c95db58c-9d47-4ef0-9ede-234d7a693f41 +md""" +### References +Pezeshki, M., Kaba, S. O., Bengio, Y., Courville, A., Precup, D., & Lajoie, G. (2020). Gradient starvation: A learning proclivity in neural networks. arXiv preprint arXiv:2011.09468. + +Pizurica, A. (2020). E016330: Artificial Intelligence – Learning from examples [PDF course notes]. Ufora. + +Pizurica, A. (2020). E016330: Artificial Intelligence - Neural networks [PDF course notes]. Ufora. + +Russell, S., & Norvig, P. (2002). Artificial intelligence: a modern approach. +""" + +# ╔═╡ Cell order: +# ╟─df218f5d-042d-4ccb-8005-56b2ec073869 +# ╟─8bd774c7-1acf-4beb-bc99-4c303882d262 +# ╟─8f0cdbf1-0174-4d18-9072-2a8c5f015177 +# ╟─6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b +# ╟─512e6240-76e7-11ec-2511-07ffc4b74d27 +# ╟─93bc80a8-673d-457b-96d8-e1bb0b09682e +# ╟─66a3b5cb-a465-42fb-a8c8-404d574312d2 +# ╟─4a14831d-7736-4af5-875f-05ae6e3cf7c4 +# ╟─00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 +# ╟─ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 +# ╠═85591d79-dec2-45fa-a0a9-d95d84edda51 +# ╠═92cca2d8-c2b6-4046-9b86-26c5c9d64ee3 +# ╠═88b30b55-4f98-474a-aaca-a1197d2886fd +# ╟─c95db58c-9d47-4ef0-9ede-234d7a693f41 From 6dc7d9481f036384f3f1114fed669efadc38c9dd Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 16 Jan 2022 23:09:04 +0100 Subject: [PATCH 19/49] fix args; create example pipeline running multiple NNs; start documenting --- src/data.jl | 4 +-- src/gradientstarvation.jl | 34 ++++++++++++++++++-- src/neuralnetwork.jl | 66 +++++++++++++++++++++++++++++++++------ 3 files changed, 89 insertions(+), 15 deletions(-) diff --git a/src/data.jl b/src/data.jl index 56ced7c3..bd871082 100644 --- a/src/data.jl +++ b/src/data.jl @@ -12,7 +12,7 @@ using ScikitLearn export get_moon_data_loader, get_nmist_data_loader, plot_train_and_test_data function get_moon_data_loader(; args...) - args = Args(args...) + args = Args(; args...) x_train, y_train = generate_moons(300, offset = 0.5) x_test, y_test = generate_moons(300, offset = 0.5, seed = 0) @@ -28,7 +28,7 @@ function get_moon_data_loader(; args...) end function get_nmist_data_loader(; args...) - args = Args(args...) + args = Args(; args...) # Loading Dataset xtrain, ytrain = MLDatasets.MNIST.traindata(Float32) diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index c29a9ead..d9883b43 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -1,13 +1,41 @@ module GradientStarvation +# activate environment +using Pkg +if isfile("Project.toml") && isfile("Manifest.toml") + Pkg.activate(".") +end + include("args.jl") include("data.jl") include("neuralnetwork.jl") train_loader, test_loader = Data.get_moon_data_loader() -m = NeuralNetwork.train(train_loader, test_loader, "GD", false) -display(NeuralNetwork.plot_decision_boundary(train_loader, m, title = "Train")) -display(NeuralNetwork.plot_decision_boundary(test_loader, m, title = "Test")) +m_gd = NeuralNetwork.train(train_loader, test_loader, "GD", false) + +p1 = NeuralNetwork.plot_decision_boundary(train_loader, m_gd, title = "GD Train") +p2 = NeuralNetwork.plot_decision_boundary(test_loader, m_gd, title = "GD Test") + +m_sgd = NeuralNetwork.train(train_loader, test_loader, "SGD", false) + +p3 = NeuralNetwork.plot_decision_boundary(train_loader, m_sgd, title = "SGD Train") +p4 = NeuralNetwork.plot_decision_boundary(test_loader, m_sgd, title = "SGD Test") + +# same with SD +m_gd_sd = NeuralNetwork.train(train_loader, test_loader, "GD", true) + +p5 = NeuralNetwork.plot_decision_boundary(train_loader, m_gd_sd, title = "GD + SD Train") +p6 = NeuralNetwork.plot_decision_boundary(test_loader, m_gd_sd, title = "GD + SD Test") + +m_sgd_sd = NeuralNetwork.train(train_loader, test_loader, "SGD", true) + +p7 = NeuralNetwork.plot_decision_boundary(train_loader, m_sgd_sd, title = "SGD + SD Train") +p8 = NeuralNetwork.plot_decision_boundary(test_loader, m_sgd_sd, title = "SGD + SD Test") + +display([p1 p2 +p3 p4 +p5 p6 +p7 p8]) end diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 3aae6c5e..07d78dae 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -15,20 +15,41 @@ using Statistics export get_loss_and_accuracy, train -function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model; args...) - args = Args(args...) +""" + get_loss_and_accuracy(data_loader, model, spectral_decoupling; args...) + +Calculates loss and accuracy of the given data for the input model. + +# Examples +```julia-repl +julia> get_loss_and_accuracy(train_loader, model) +(0.1337, 1.0f0) +``` +""" +function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model, spectral_decoupling::Bool = false; args...) + args = Args(; args...) + @assert spectral_decoupling && args.λ "λ must be specified if spectral decoupling is used" + accuracy = 0.0f0 loss = 0.0f0 num = 0 for (x, y) in data_loader ŷ = model(y) - loss += Regularization.spectral_decoupling(ŷ, y, args.λ) # logitcrossentropy(ŷ, y, agg = sum) + loss += spectral_decoupling ? + Regularization.spectral_decoupling(ŷ, y, args.λ) : + logitcrossentropy(ŷ, y, agg = sum) accuracy += sum(onecold(ŷ) .== onecold(y)) num += size(x)[end] end - return loss, accuracy / num + return spectral_decoupling ? loss : loss / num, accuracy / num end +""" + neural_network() + +Creates a Flux neural network according to the topology used by Pezeshki et al. +That is a NN with 2 hidden layers, 500 units each and ReLU activation. +""" function neural_network() return Chain( Dense(2, 500, relu), @@ -36,10 +57,28 @@ function neural_network() ) end +""" + train(train_loader, test_loader, optimizer, spectral_decoupling; args...) + +Creates and trains a neural network as defined in [`neural_network`](@ref) using cross-entropy loss with +the specified optimizer and additionally spectral decoupling as a loss regularization. + +# Examples +```julia-repl +julia> train(train_loader, test_loader, "SGD", false) +[ Info: opt = SGD, sd = false, lr = 0.01, bs = 50 +Training... 100%|███████████████████████████████████████████████████████████████████████████| Time: 0:00:42 + epoch: 1000 + train_loss: 0.1337 + test_loss: 0.1337 + train_accuracy: 1.0 + test_accuracy: 1.0 +``` +""" function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spectral_decoupling::Bool = false; args...) - args = Args(args...) + args = Args(; args...) - @info "opt = $optimizer, sd = $spectral_decoupling, lr = $(args.learning_rate), bs = $(args.batchsize)" * (spectral_decoupling ? "λ = $(args.λ)" : "") + @info "opt = $optimizer, sd = $spectral_decoupling, lr = $(args.learning_rate), bs = $(args.batchsize)" * (spectral_decoupling ? ", λ = $(args.λ)" : "") model = neural_network() params = Flux.params(model) @@ -68,6 +107,16 @@ function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLo return model end +""" + get_optimizer(learning_rate, optimizer) + +Returns a Flux.Optimiser according to the one requested by acronym with fixed parameters except +for the given learning rate. Supported optimizers are: + - Adaptive Moment Estimation (ADAM) + - Gradient Descent (GD) + - Stochastic Gradient Descent (SGD) + - Weight Decay (WD) +""" function get_optimizer(learning_rate, optimizer = "SGD") optimizer = uppercase(optimizer) @assert optimizer in ["ADAM", "GD", "SGD", "WD"] "Requested optimizer '$optimizer' is not supported." @@ -93,9 +142,6 @@ function plot_decision_boundary(loader, model; title = "") x_min = minimum(loader.data[1][1,:]) - .25 y_min = minimum(loader.data[1][2,:]) - .25 - # upper = ceil(maximum([x_max, y_max]), digits = 1, base = 2) - # lower = floor(minimum([x_min, y_min]), digits = 1, base = 2) - r_x = LinRange(x_min, x_max, n) r_y = LinRange(y_min, y_max, n) @@ -139,7 +185,7 @@ function plot_decision_boundary(loader, model; title = "") xaxis_range = [x_min, x_max], yaxis_range = [y_min, y_max], xaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), yaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), - margin = attr(l = 50, r = 50, b = 50, t = 50, pad = 0), + margin = attr(l = 0, r = 0, b = 0, t = 0, pad = 0), plot_bgcolor = "rgba(0, 0, 0, 0)" ), config = PlotConfig( From 6732558780d368c5834202284ebb5c37f9335fd7 Mon Sep 17 00:00:00 2001 From: Peter Date: Sat, 29 Jan 2022 19:55:34 +0100 Subject: [PATCH 20/49] add Experiment struct, move structs to one file --- src/data.jl | 2 +- src/gradientstarvation.jl | 1 + src/neuralnetwork.jl | 3 +-- src/{args.jl => structs.jl} | 8 +++++++- 4 files changed, 10 insertions(+), 4 deletions(-) rename src/{args.jl => structs.jl} (52%) diff --git a/src/data.jl b/src/data.jl index bd871082..2f018cf7 100644 --- a/src/data.jl +++ b/src/data.jl @@ -1,6 +1,6 @@ module Data -include("args.jl") +include("structs.jl") using Flux using Flux: onehotbatch diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index d9883b43..6de853c3 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -9,6 +9,7 @@ end include("args.jl") include("data.jl") include("neuralnetwork.jl") +include("structs.jl") train_loader, test_loader = Data.get_moon_data_loader() diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 07d78dae..6470109e 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -1,7 +1,6 @@ module NeuralNetwork -include("args.jl") -include("plotutils.jl") +include("structs.jl") include("regularization.jl") using Flux diff --git a/src/args.jl b/src/structs.jl similarity index 52% rename from src/args.jl rename to src/structs.jl index 3ffa32db..8818eea3 100644 --- a/src/args.jl +++ b/src/structs.jl @@ -1,6 +1,12 @@ Base.@kwdef mutable struct Args learning_rate::Float64 = 1e-2 batchsize::Int = 50 - epochs::Int = 1000 + epochs::Int = 500 λ::Float64 = 3e-1 end + +struct Experiment + optimizer::String + offset::Float64 + sd::Bool +end From 7a65f4aaf4d76c3f1c4bc4ea939395c4d53251cb Mon Sep 17 00:00:00 2001 From: Peter Date: Sat, 29 Jan 2022 19:55:57 +0100 Subject: [PATCH 21/49] move plots to dedicated file, simplify data loaders, plot loss and accuracy, clean up regularization, lots of minor improvements --- Project.toml | 1 + docs/Project.toml | 2 +- docs/make.jl | 20 ++++--- notebook/gradientstarvation.jl | 26 ++++++-- src/data.jl | 54 +++++------------ src/gradientstarvation.jl | 44 ++++++-------- src/neuralnetwork.jl | 101 ++++++++----------------------- src/plotlib.jl | 105 +++++++++++++++++++++++++++++++++ src/regularization.jl | 17 +----- 9 files changed, 201 insertions(+), 169 deletions(-) create mode 100644 src/plotlib.jl diff --git a/Project.toml b/Project.toml index fb9fda2a..50c6b4c6 100644 --- a/Project.toml +++ b/Project.toml @@ -4,6 +4,7 @@ authors = ["Peter Merseburger "] version = "0.0.99" [deps] +DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8" Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c" MLDatasets = "eb30cadb-4394-5ae3-aed4-317e484a6458" PlotlyJS = "f0f68f2c-4968-5e81-91da-67840de0976a" diff --git a/docs/Project.toml b/docs/Project.toml index 16656868..f1f63797 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -1,3 +1,3 @@ [deps] Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" -STMOZOO = "8e7dc78f-7190-4116-bf88-3d148ae9f088" +GradientStarvation = "8e7dc78f-7190-4116-bf88-3d148ae9f088" diff --git a/docs/make.jl b/docs/make.jl index 21d4327e..3b1a8a61 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -1,17 +1,21 @@ using Documenter -using STMOZOO -using STMOZOO.Example +using GradientStarvation +using GradientStarvation.Data +using GradientStarvation.NeuralNetwork -makedocs(sitename="STMO ZOO", +makedocs( + sitename = "Gradient Starvation - An STMO ZOO Project", format = Documenter.HTML(), - modules=[Example], # add your module - pages=Any[ - "Example"=> "man/example.md", # add the page to your documentation - ]) + modules = [Data, NeuralNetwork], + pages = Any[ + "Data" => "man/data.md", + "Neural Network" => "man/neuralnetwork.md" + ] +) #= deploydocs( - repo = "github.com/michielstock/STMOZOO.jl.git", + repo = "github.com/justinsane1337/GradientStarvation.git", ) =# \ No newline at end of file diff --git a/notebook/gradientstarvation.jl b/notebook/gradientstarvation.jl index e83f8f5d..0e4cd3e4 100644 --- a/notebook/gradientstarvation.jl +++ b/notebook/gradientstarvation.jl @@ -90,7 +90,7 @@ Given the major workings of supervised learning, it should have become clear tha Despite its popularity, gradient descent learning in NNs can come with some unwanted side effects such as the vanishing gradient problem, which condenses down to the gradient being too small to effectively update a weight and thus preventing learning. -Another not yet published but currently in preprint described observation is the effect of **Gradient Starvation (GS)**. Pezeshki et al. coined and formalized this phenomenon of over-parameterized NNs in their work. According to them, _Gradient Starvation_ is a predisposition of cross-entropy loss learning NNs, which occurs when the loss minimization is driven by only a feature subset regardless of the presence of other meaningful features. Thus, the NN becomes biased towards superficial features. Beyond demonstrating the effects of GS, Pezeshki et al. introduce _Spectral Decoupling_ (SD) regularization as a means to counteract gradient starvation. +Another not yet published but currently in preprint described observation is the effect of **Gradient Starvation (GS)**. Pezeshki et al. coined and formalized this phenomenon of over-parameterized NNs in their work. According to them, _Gradient Starvation_ is a predisposition of NNs learned by cross-entropy, which occurs when the loss minimization is driven by only a feature subset regardless of the presence of other meaningful features. Thus, the NN becomes biased towards superficial features. Beyond demonstrating the effects of GS, Pezeshki et al. introduce _Spectral Decoupling_ (SD) regularization as a means to counteract gradient starvation. """ # ╔═╡ 00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 @@ -106,12 +106,13 @@ md""" ## Experiments ### 2D Classification +A modified version of the [Moons dataset](https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_moons.html) was used with two different topologies. The first represents the fully linearly separable case, thus allows to differentiate the two moons by a straight line along one axis. The second topolgy uses an offset to interleave both moons which only allows to separate them by learning a curved classification boundary. Pezeshki et al. """ # ╔═╡ 85591d79-dec2-45fa-a0a9-d95d84edda51 md""" ## Implementation -The implementation of the NN is closely related to the one of Pezeshki et al. who realized their NN with Python and PyTorch. My implementation is based on a ML framework for Julia called [Flux](https://fluxml.ai/). The NN consists of two hidden layers with 500 nodes each. The applied activation function is Rectified Linear Unit (ReLu) and learning is facilitated with cross-entropy loss. +The implementation of the NN is closely related to the one of Pezeshki et al. who realized theirs with Python and PyTorch. The implementation provided here is based on [Flux](https://fluxml.ai/), a ML framework for Julia. The NN consists of two hidden layers with 500 nodes each. Rectified Linear Unit (ReLu) is used as an activation function and learning is facilitated with cross-entropy loss. """ # ╔═╡ 92cca2d8-c2b6-4046-9b86-26c5c9d64ee3 @@ -132,6 +133,21 @@ Pizurica, A. (2020). E016330: Artificial Intelligence - Neural networks [PDF cou Russell, S., & Norvig, P. (2002). Artificial intelligence: a modern approach. """ +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.7.0" +manifest_format = "2.0" + +[deps] +""" + # ╔═╡ Cell order: # ╟─df218f5d-042d-4ccb-8005-56b2ec073869 # ╟─8bd774c7-1acf-4beb-bc99-4c303882d262 @@ -140,10 +156,12 @@ Russell, S., & Norvig, P. (2002). Artificial intelligence: a modern approach. # ╟─512e6240-76e7-11ec-2511-07ffc4b74d27 # ╟─93bc80a8-673d-457b-96d8-e1bb0b09682e # ╟─66a3b5cb-a465-42fb-a8c8-404d574312d2 -# ╟─4a14831d-7736-4af5-875f-05ae6e3cf7c4 +# ╠═4a14831d-7736-4af5-875f-05ae6e3cf7c4 # ╟─00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 -# ╟─ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 +# ╠═ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 # ╠═85591d79-dec2-45fa-a0a9-d95d84edda51 # ╠═92cca2d8-c2b6-4046-9b86-26c5c9d64ee3 # ╠═88b30b55-4f98-474a-aaca-a1197d2886fd # ╟─c95db58c-9d47-4ef0-9ede-234d7a693f41 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/src/data.jl b/src/data.jl index 2f018cf7..7ffe9320 100644 --- a/src/data.jl +++ b/src/data.jl @@ -9,43 +9,30 @@ using Plots using ScikitLearn @sk_import datasets: make_moons -export get_moon_data_loader, get_nmist_data_loader, plot_train_and_test_data +export get_moon_data_loader, get_nmist_data_loader -function get_moon_data_loader(; args...) +function get_moon_data_loader(; n = 300, offset = 0.5, shuffle = true, args...) args = Args(; args...) - x_train, y_train = generate_moons(300, offset = 0.5) - x_test, y_test = generate_moons(300, offset = 0.5, seed = 0) + # offset: linearly separable => 1.0; not lin. sep. => 0.5 + x, y = generate_moons(n, offset = offset) + x, y = transpose(x), onehotbatch(y, 0:1) - x_train, x_test = transpose(x_train), transpose(x_test) - y_train, y_test = onehotbatch(y_train, 0:1), onehotbatch(y_test, 0:1) - - # create data loaders - train_loader = DataLoader((x_train, y_train), batchsize = args.batchsize, shuffle = true) - test_loader = DataLoader((x_test, y_test), batchsize = args.batchsize) - - return train_loader, test_loader + # create data loader + return DataLoader((x, y), batchsize = args.batchsize, shuffle = shuffle) end -function get_nmist_data_loader(; args...) +function get_nmist_data_loader(; shuffle = true, args...) args = Args(; args...) - # Loading Dataset - xtrain, ytrain = MLDatasets.MNIST.traindata(Float32) - xtest, ytest = MLDatasets.MNIST.testdata(Float32) + x, y = MLDatasets.MNIST.traindata(Float32) - # Reshape Data in order to flatten each image into a linear array - xtrain = Flux.flatten(xtrain) - xtest = Flux.flatten(xtest) - - # One-hot-encode the labels - ytrain, ytest = onehotbatch(ytrain, 0:9), onehotbatch(ytest, 0:9) + # reshape Data in order to flatten each image into a linear array + # encode labels with one-hot + x, y = Flux.flatten(x), onehotbatch(y, 0:9) - # Create DataLoaders (mini-batch iterators) - train_loader = DataLoader((xtrain, ytrain), batchsize=args.batchsize, shuffle=true) - test_loader = DataLoader((xtest, ytest), batchsize=args.batchsize) - - return train_loader, test_loader + # create data loader + return DataLoader((x, y), batchsize = args.batchsize, shuffle = shuffle) end function generate_moons_from_publication(; offset = 0.0, coord_down_scale = 1, show_plot = false) @@ -723,17 +710,4 @@ function moon_class_2(offset = 0.0, coord_down_scale = 1) return vcat(transpose.((x, y))...) / coord_down_scale end -function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader) - # plot train and test data sets - p_train = Plots.scatter( - train_loader.data[1][1,:], train_loader.data[1][2,:], - c = PlotUtils.map_bool_to_color(train_loader.data[2][1,:], "blue", "red") - ) - p_test = Plots.scatter( - test_loader.data[1][1,:], test_loader.data[1][2,:], - c = PlotUtils.map_bool_to_color(test_loader.data[2][1,:], "blue", "red") - ) - display(Plots.plot(p_train, p_test, layout = (1, 2))) -end - end \ No newline at end of file diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index 6de853c3..1af17654 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -6,37 +6,29 @@ if isfile("Project.toml") && isfile("Manifest.toml") Pkg.activate(".") end -include("args.jl") +using DataStructures +using PlotlyJS + include("data.jl") include("neuralnetwork.jl") +include("plotlib.jl") include("structs.jl") -train_loader, test_loader = Data.get_moon_data_loader() - -m_gd = NeuralNetwork.train(train_loader, test_loader, "GD", false) - -p1 = NeuralNetwork.plot_decision_boundary(train_loader, m_gd, title = "GD Train") -p2 = NeuralNetwork.plot_decision_boundary(test_loader, m_gd, title = "GD Test") - -m_sgd = NeuralNetwork.train(train_loader, test_loader, "SGD", false) - -p3 = NeuralNetwork.plot_decision_boundary(train_loader, m_sgd, title = "SGD Train") -p4 = NeuralNetwork.plot_decision_boundary(test_loader, m_sgd, title = "SGD Test") +experiments = OrderedDict( + "Δ1.0" => Experiment("SGD", 1.0, false), + "Δ0.5" => Experiment("SGD", 0.5, false), + "Δ0.5 + SD" => Experiment("SGD", 0.5, true), +) -# same with SD -m_gd_sd = NeuralNetwork.train(train_loader, test_loader, "GD", true) +plots = [] +for (name, def) in experiments + train_loader = Data.get_moon_data_loader(offset = def.offset) + model, stats = NeuralNetwork.train(train_loader, def.optimizer, def.sd) -p5 = NeuralNetwork.plot_decision_boundary(train_loader, m_gd_sd, title = "GD + SD Train") -p6 = NeuralNetwork.plot_decision_boundary(test_loader, m_gd_sd, title = "GD + SD Test") - -m_sgd_sd = NeuralNetwork.train(train_loader, test_loader, "SGD", true) - -p7 = NeuralNetwork.plot_decision_boundary(train_loader, m_sgd_sd, title = "SGD + SD Train") -p8 = NeuralNetwork.plot_decision_boundary(test_loader, m_sgd_sd, title = "SGD + SD Test") - -display([p1 p2 -p3 p4 -p5 p6 -p7 p8]) + p = PlotLib.plot_decision_boundary(train_loader, model, title = name) + p_stats = PlotLib.plot_loss_and_accuracy(stats["loss"], stats["accuracy"]) + global plots = push!(plots, p, p_stats) +end +display([plots[1] plots[2]; plots[3] plots[4]; plots[5] plots[6]]) end diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 6470109e..f3df43dc 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -5,10 +5,9 @@ include("regularization.jl") using Flux using Flux: onecold -using Flux.Losses: logitcrossentropy +using Flux: Optimiser +using Flux.Losses: binarycrossentropy, crossentropy, logitcrossentropy, logitbinarycrossentropy using MLDatasets -using Plots -using PlotlyJS using ProgressMeter using Statistics @@ -27,19 +26,23 @@ julia> get_loss_and_accuracy(train_loader, model) """ function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model, spectral_decoupling::Bool = false; args...) args = Args(; args...) - @assert spectral_decoupling && args.λ "λ must be specified if spectral decoupling is used" + @assert !(spectral_decoupling && isa(args.λ, Float64)) "λ must be specified if spectral decoupling is used" accuracy = 0.0f0 loss = 0.0f0 num = 0 + + # iterate over batches for (x, y) in data_loader - ŷ = model(y) + ŷ = model(y) loss += spectral_decoupling ? Regularization.spectral_decoupling(ŷ, y, args.λ) : logitcrossentropy(ŷ, y, agg = sum) accuracy += sum(onecold(ŷ) .== onecold(y)) num += size(x)[end] end + + # divide metrics cumulated over batches by number of total data points return spectral_decoupling ? loss : loss / num, accuracy / num end @@ -74,7 +77,7 @@ Training... 100%|█████████████████████ test_accuracy: 1.0 ``` """ -function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spectral_decoupling::Bool = false; args...) +function train(data_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spectral_decoupling::Bool = false; args...) args = Args(; args...) @info "opt = $optimizer, sd = $spectral_decoupling, lr = $(args.learning_rate), bs = $(args.batchsize)" * (spectral_decoupling ? ", λ = $(args.λ)" : "") @@ -83,6 +86,9 @@ function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLo params = Flux.params(model) optimizer = get_optimizer(args.learning_rate, optimizer) + # record loss and accuracy progression over epochs + loss_prog, accuracy_prog = [], [] + # training prog = Progress(args.epochs, 0.25, "Training... ", 75) for epoch in 1:args.epochs @@ -90,20 +96,24 @@ function train(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLo Regularization.spectral_decoupling(model(x), y, args.λ) : logitcrossentropy(model(x), y) - Flux.train!(loss, params, train_loader, optimizer) + Flux.train!(loss, params, data_loader, optimizer) - # evaluate train and test loss and accuracy - train_loss, train_accuracy = get_loss_and_accuracy(train_loader, model) - test_loss, test_accuracy = get_loss_and_accuracy(test_loader, model) + # evaluate loss and accuracy + loss, accuracy = get_loss_and_accuracy(data_loader, model) + push!(loss_prog, loss) + push!(accuracy_prog, accuracy) + + # show neat progress bar ProgressMeter.next!(prog; showvalues = [ (:epoch, epoch), - (:train_loss, train_loss), (:test_loss, test_loss), - (:train_accuracy, train_accuracy), (:test_accuracy, test_accuracy) - ]) + (:loss, loss), + (:accuracy, accuracy) + ] + ) end - return model + return model, Dict("loss" => loss_prog, "accuracy" => accuracy_prog) end """ @@ -128,69 +138,8 @@ function get_optimizer(learning_rate, optimizer = "SGD") # stochastic gradient descent return Momentum(learning_rate, 0.9) elseif optimizer == "WD" - return Optimiser(WeightDecay(), Decent(learning_rate)) + return Optimiser(WeightDecay(), Descent(learning_rate)) end end -function plot_decision_boundary(loader, model; title = "") - # grid and range step size - n = 100 - # determine limits of given data - x_max = maximum(loader.data[1][1,:]) + .25 - y_max = maximum(loader.data[1][2,:]) + .25 - x_min = minimum(loader.data[1][1,:]) - .25 - y_min = minimum(loader.data[1][2,:]) - .25 - - r_x = LinRange(x_min, x_max, n) - r_y = LinRange(y_min, y_max, n) - - # create grid to be used for the contours - d1 = collect(Iterators.flatten(vcat([fill(v, (n, 1)) for v in r_x]))) - d2 = collect(Iterators.flatten(vcat([reshape(r_y, 1, :) for _ in 1:n]))) - grid = hcat(d1, d2) - - # use model to predict decision boundary based on grid - gr_pred = model(grid') - gr_pred = reshape(gr_pred[1,:], (n, n)) - - # map classes (Boolean) to integers to be used as colors - cols = PlotUtils.map_bool_to_color(loader.data[2][1,:], "#FF0000", "#0000FF") - opacity = 0.90 - - PlotlyJS.plot([ - # data points - PlotlyJS.scatter( - x = loader.data[1][1,:], y = loader.data[1][2,:], - mode = "markers", marker = attr(color = cols, line_width = 1) - ), - # actual contours - PlotlyJS.contour( - x = r_x, y = r_y, z = gr_pred, - contours_start = -10, contours_end = 10, contours_size = 1, - contours_coloring = "heatmap", colorscale = PlotUtils.get_custom_rdbu_scale(opacity), - opacity = opacity - ), - # highlight decision boundary line - PlotlyJS.contour( - x = r_x, y = r_y, z = gr_pred, - contours_start = 0, contours_end = 0, contours_size = 0, - contours_coloring = "lines", colorscale = [[0, "black"], [1, "black"]], - showscale = false, line = attr(width = 3) - )], - Layout( - title = title, - width = 500, height = 500, autosize = true, - xaxis_showgrid = false, yaxis_showgrid = false, - xaxis_range = [x_min, x_max], yaxis_range = [y_min, y_max], - xaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), - yaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), - margin = attr(l = 0, r = 0, b = 0, t = 0, pad = 0), - plot_bgcolor = "rgba(0, 0, 0, 0)" - ), - config = PlotConfig( - scrollZoom = false - ) - ) -end - end \ No newline at end of file diff --git a/src/plotlib.jl b/src/plotlib.jl new file mode 100644 index 00000000..efbb0efa --- /dev/null +++ b/src/plotlib.jl @@ -0,0 +1,105 @@ +module PlotLib + +include("plotutils.jl") +include("structs.jl") + +using Flux +using PlotlyJS +using Plots + +export plot_train_and_test_data, plot_loss_and_accuracy, plot_decision_boundary + +function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader) + # plot train and test data sets next to each other + p_train = Plots.scatter( + train_loader.data[1][1,:], train_loader.data[1][2,:], + c = PlotUtils.map_bool_to_color(train_loader.data[2][1,:], "blue", "red") + ) + p_test = Plots.scatter( + test_loader.data[1][1,:], test_loader.data[1][2,:], + c = PlotUtils.map_bool_to_color(test_loader.data[2][1,:], "blue", "red") + ) + display(Plots.plot(p_train, p_test, layout = (1, 2))) +end + +function plot_loss_and_accuracy(loss, accuracy; args...) + args = Args(; args...) + PlotlyJS.plot([ + PlotlyJS.scatter(y = loss, x = 1:args.epochs, mode = "lines", name = "loss"), + PlotlyJS.scatter(y = accuracy, x = 1:args.epochs, mode = "lines", name = "accuracy") + ], + Layout( + title = "Loss and Accuracy", + width = 500, height = 500, autosize = true, + xaxis_showgrid = false, + yaxis = attr(range = [0, 1], dtick = 0.1, showgrid = false), + legend = attr(x = -1, y = -1, bgcolor = "rgba(0, 0, 0, 0)"), + margin = attr(l = 0, r = 0, b = 0, t = 0, pad = 0), + plot_bgcolor = "rgba(0, 0, 0, 0)" + ) + ) +end + +function plot_decision_boundary(loader, model; title = "") + # grid and range step size + n = 100 + # determine limits of given data + x_max = maximum(loader.data[1][1,:]) + .25 + y_max = maximum(loader.data[1][2,:]) + .25 + x_min = minimum(loader.data[1][1,:]) - .25 + y_min = minimum(loader.data[1][2,:]) - .25 + + r_x = LinRange(x_min, x_max, n) + r_y = LinRange(y_min, y_max, n) + + # create grid to be used for the contours + d1 = collect(Iterators.flatten(vcat([fill(v, (n, 1)) for v in r_x]))) + d2 = collect(Iterators.flatten(vcat([reshape(r_y, 1, :) for _ in 1:n]))) + grid = hcat(d1, d2) + + # use model to predict decision boundary based on grid + gr_pred = model(grid') + gr_pred = reshape(gr_pred[1,:], (n, n)) + + # map classes (Boolean) to integers to be used as colors + cols = PlotUtils.map_bool_to_color(loader.data[2][1,:], "#FF0000", "#0000FF") + opacity = 0.90 + + PlotlyJS.plot([ + # data points + PlotlyJS.scatter( + x = loader.data[1][1,:], y = loader.data[1][2,:], + mode = "markers", marker = attr(color = cols, line_width = 1), + showlegend = false + ), + # actual contours + PlotlyJS.contour( + x = r_x, y = r_y, z = gr_pred, + contours_start = -10, contours_end = 10, contours_size = 1, + contours_coloring = "heatmap", colorscale = PlotUtils.get_custom_rdbu_scale(opacity), + colorbar = attr(orientation = "h"), opacity = opacity, showlegend = false + ), + # highlight decision boundary line + PlotlyJS.contour( + x = r_x, y = r_y, z = gr_pred, + contours_start = 0, contours_end = 0, contours_size = 0, + contours_coloring = "lines", colorscale = [[0, "black"], [1, "black"]], + showscale = false, showlegend = false, line = attr(width = 3) + )], + Layout( + title = title, + width = 500, height = 500, autosize = true, + xaxis_showgrid = false, yaxis_showgrid = false, + xaxis_range = [x_min, x_max], yaxis_range = [y_min, y_max], + xaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), + yaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), + margin = attr(l = 0, r = 0, b = 0, t = 0, pad = 0), + plot_bgcolor = "rgba(0, 0, 0, 0)" + ), + config = PlotConfig( + scrollZoom = false + ) + ) +end + +end \ No newline at end of file diff --git a/src/regularization.jl b/src/regularization.jl index d61e21f0..172db5a4 100644 --- a/src/regularization.jl +++ b/src/regularization.jl @@ -1,22 +1,11 @@ module Regularization -using Flux: onecold -using Flux.Losses: logitcrossentropy, crossentropy -using LinearAlgebra +using Flux.Losses: crossentropy, logitcrossentropy, logitbinarycrossentropy using Statistics function spectral_decoupling(ŷ, y, λ) - # ŷ, y = onecold(ŷ), onecold(y) - # return mean(log.(1.0 .+ exp.(-ŷ .* y))) .+ λ * mean(ŷ .* 2) - # println(mean(log.(1.0 .+ exp.(-ŷ .* y))) .+ λ * mean(ŷ .* 2).^2) - - # python: torch.log(1.0 + torch.exp(-ŷ * y)).mean() + λ * (ŷ ** 2).mean() - # v = mean(log.(1.0 .+ exp.(-y .* ŷ))) .+ λ/2 * norm(ŷ, 2)^2 - # v = logitcrossentropy(ŷ, y) .+ (λ/2 * norm(ŷ, 2)^2) - v = logitcrossentropy(ŷ, y) + λ * mean(ŷ .^ 2) - # v = mean(1.0 .* log.(1.0 .+ exp.(-ŷ .* y))) + λ * mean(ŷ .^ 2) - # println(v) - return v + # return mean(1.0 .* log.(1.0 .+ exp.(-ŷ .* y))) + λ * mean(ŷ .^ 2) + return logitcrossentropy(ŷ, y) + λ * mean(ŷ .^ 2) end end \ No newline at end of file From eca1fffbe582ad5ed139abda57e2e514b9298b7c Mon Sep 17 00:00:00 2001 From: Peter Date: Sat, 29 Jan 2022 23:15:19 +0100 Subject: [PATCH 22/49] adjust plots for better readability --- src/plotlib.jl | 48 ++++++++++++++++++++++++++++++++++-------------- src/plotutils.jl | 3 +-- 2 files changed, 35 insertions(+), 16 deletions(-) diff --git a/src/plotlib.jl b/src/plotlib.jl index efbb0efa..02335652 100644 --- a/src/plotlib.jl +++ b/src/plotlib.jl @@ -9,6 +9,9 @@ using Plots export plot_train_and_test_data, plot_loss_and_accuracy, plot_decision_boundary +title_font_size = 18 +tick_font_size = 16 + function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader) # plot train and test data sets next to each other p_train = Plots.scatter( @@ -29,13 +32,21 @@ function plot_loss_and_accuracy(loss, accuracy; args...) PlotlyJS.scatter(y = accuracy, x = 1:args.epochs, mode = "lines", name = "accuracy") ], Layout( - title = "Loss and Accuracy", - width = 500, height = 500, autosize = true, - xaxis_showgrid = false, - yaxis = attr(range = [0, 1], dtick = 0.1, showgrid = false), - legend = attr(x = -1, y = -1, bgcolor = "rgba(0, 0, 0, 0)"), + title = attr(text = "Loss and Accuracy", font = attr(size = title_font_size)), + width = 1000, height = 1000, autosize = false, + xaxis = attr( + showgrid = false, + ticks = "outside", + tickfont = attr(size = tick_font_size)), + yaxis = attr( + range = [0, 1], dtick = 0.1, showgrid = false, + ticks = "outside", + tickfont = attr(size = tick_font_size)), + legend = attr(font = attr(size = tick_font_size)), + automargin = false, margin = attr(l = 0, r = 0, b = 0, t = 0, pad = 0), - plot_bgcolor = "rgba(0, 0, 0, 0)" + plot_bgcolor = "rgba(0, 0, 0, 0)", + paper_bgcolor = "rgba(0, 0, 0, 0)" ) ) end @@ -77,7 +88,8 @@ function plot_decision_boundary(loader, model; title = "") x = r_x, y = r_y, z = gr_pred, contours_start = -10, contours_end = 10, contours_size = 1, contours_coloring = "heatmap", colorscale = PlotUtils.get_custom_rdbu_scale(opacity), - colorbar = attr(orientation = "h"), opacity = opacity, showlegend = false + colorbar = attr(thickness = 25, len = 0.9, y = 0.425), + opacity = opacity, showlegend = false ), # highlight decision boundary line PlotlyJS.contour( @@ -87,14 +99,22 @@ function plot_decision_boundary(loader, model; title = "") showscale = false, showlegend = false, line = attr(width = 3) )], Layout( - title = title, - width = 500, height = 500, autosize = true, - xaxis_showgrid = false, yaxis_showgrid = false, - xaxis_range = [x_min, x_max], yaxis_range = [y_min, y_max], - xaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), - yaxis = attr(zeroline = true, zerolinewidth = 1, zerolinecolor = "black", automargin = true), + title = attr(text = title, font = attr(size = title_font_size)), + width = 1000, height = 1000, autosize = false, + xaxis = attr( + range = [x_min, x_max], + zeroline = true, zerolinewidth = 1, zerolinecolor = "black", + automargin = false, showgrid = false, + tickfont = attr(size = tick_font_size)), + yaxis = attr( + range = [y_min, y_max], + zeroline = true, zerolinewidth = 1, zerolinecolor = "black", + automargin = false, showgrid = false, + tickfont = attr(size = tick_font_size)), + legend = attr(font = attr(size = tick_font_size)), margin = attr(l = 0, r = 0, b = 0, t = 0, pad = 0), - plot_bgcolor = "rgba(0, 0, 0, 0)" + plot_bgcolor = "rgba(0, 0, 0, 0)", + paper_bgcolor = "rgba(0, 0, 0, 0)" ), config = PlotConfig( scrollZoom = false diff --git a/src/plotutils.jl b/src/plotutils.jl index 4c1f133f..04deb329 100644 --- a/src/plotutils.jl +++ b/src/plotutils.jl @@ -4,11 +4,10 @@ export get_custom_rdbu_scale, map_bool_to_color function get_custom_rdbu_scale(opacity::Union{Float32, Float64}) @assert opacity >= 0 || opacity <= 1 "Opacity must be in [0, 1]" - white = "rgba(255, 255, 255, $opacity)" return [ [0, "rgba(255, 0, 0, $opacity)"], [0.45, "rgb(255, 200, 100, $opacity)"], - [0.5, white], + [0.5, "rgba(255, 255, 255, $opacity)"], [0.55, "rgb(100, 200, 255, $opacity)"], [1, "rgba(0, 0, 255, $opacity)"] ] From f51272b1fda6cc7c62c1eb307b68da7eae0e9b44 Mon Sep 17 00:00:00 2001 From: Peter Date: Sat, 29 Jan 2022 23:16:22 +0100 Subject: [PATCH 23/49] implement function to run experiments on nn --- src/experiments.jl | 56 +++++++++++++++++++++++++++++++++++++++ src/gradientstarvation.jl | 33 ++++++++++++----------- src/structs.jl | 2 +- 3 files changed, 74 insertions(+), 17 deletions(-) create mode 100644 src/experiments.jl diff --git a/src/experiments.jl b/src/experiments.jl new file mode 100644 index 00000000..31b134e4 --- /dev/null +++ b/src/experiments.jl @@ -0,0 +1,56 @@ +module Experiments + +using PlotlyJS + +include("data.jl") +include("neuralnetwork.jl") +include("plotlib.jl") +include("structs.jl") + +function run_experiments(experiments; prefix = "exp", show_plots = true, save_plots = true, plot_size = "medium") + plot_width, plot_height = get_plot_resolution(plot_size) + + for (name, def) in experiments + # load data + train_loader = Data.get_moon_data_loader(offset = def.offset) + + # create and train the model + model, stats = NeuralNetwork.train(train_loader, def.optimizer, def.sd) + + # create plots + p1 = PlotLib.plot_decision_boundary(train_loader, model, title = name) + p2 = PlotLib.plot_loss_and_accuracy(stats["loss"], stats["accuracy"]) + p = [p1 p2] + PlotlyJS.relayout!(p, paper_bgcolor = "rgba(0, 0, 0, 0)") + + # plot/save decision boundary and performance stats + if save_plots + p_name = "plots/$(prefix)_" * replace(name, " " => "_") * ".png" + PlotlyJS.savefig(p, p_name, width = plot_width, height = plot_height) + end + + if show_plots + display(p) + end + end + @info "Finished $(length(experiments)) experiments." +end + +function get_plot_resolution(size) + size = lowercase(size) + @assert size in ["small", "medium", "large", "xlarge"] "Requested plot size '$size' is not supported." + + if size == "small" + # 16:9 360p + return 640, 360 + elseif size == "medium" + # 16:9 480p + return 854, 480 + elseif size == "large" + # 16:9 720p + return 1280, 720 + elseif size == "xlarge" + # 16:9 1080p + return 1920, 1080 + end +end \ No newline at end of file diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index 1af17654..9aac21a0 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -9,26 +9,27 @@ end using DataStructures using PlotlyJS -include("data.jl") -include("neuralnetwork.jl") -include("plotlib.jl") +include("experiments.jl") include("structs.jl") experiments = OrderedDict( - "Δ1.0" => Experiment("SGD", 1.0, false), - "Δ0.5" => Experiment("SGD", 0.5, false), - "Δ0.5 + SD" => Experiment("SGD", 0.5, true), -) + "GD Δ1.0" => Experiment("GD", 1.0, false), + "GD Δ0.5" => Experiment("GD", 0.5, false), + "GD Δ0.5 + SD" => Experiment("GD", 0.5, true), -plots = [] -for (name, def) in experiments - train_loader = Data.get_moon_data_loader(offset = def.offset) - model, stats = NeuralNetwork.train(train_loader, def.optimizer, def.sd) + "SGD Δ1.0" => Experiment("SGD", 1.0, false), + "SGD Δ0.5" => Experiment("SGD", 0.5, false), + "SGD Δ0.5 + SD" => Experiment("SGD", 0.5, true), - p = PlotLib.plot_decision_boundary(train_loader, model, title = name) - p_stats = PlotLib.plot_loss_and_accuracy(stats["loss"], stats["accuracy"]) - global plots = push!(plots, p, p_stats) -end + "WD Δ1.0" => Experiment("WD", 1.0, false), + "WD Δ0.5" => Experiment("WD", 0.5, false), + "WD Δ0.5 + SD" => Experiment("WD", 0.5, true), + + "ADAM Δ1.0" => Experiment("ADAM", 1.0, false), + "ADAM Δ0.5" => Experiment("ADAM", 0.5, false), + "ADAM Δ0.5 + SD" => Experiment("ADAM", 0.5, true), +) + +Experiments.run_experiments(experiments) -display([plots[1] plots[2]; plots[3] plots[4]; plots[5] plots[6]]) end diff --git a/src/structs.jl b/src/structs.jl index 8818eea3..5fef18e1 100644 --- a/src/structs.jl +++ b/src/structs.jl @@ -1,7 +1,7 @@ Base.@kwdef mutable struct Args learning_rate::Float64 = 1e-2 batchsize::Int = 50 - epochs::Int = 500 + epochs::Int = 1000 λ::Float64 = 3e-1 end From 05262bda28b7912e8428b8dc31c88661d4f1ac12 Mon Sep 17 00:00:00 2001 From: Peter Date: Sat, 29 Jan 2022 23:21:08 +0100 Subject: [PATCH 24/49] add plots from first experiment --- "plots/ADAM_\316\2240.5.png" | Bin 0 -> 338011 bytes "plots/ADAM_\316\2240.5_+_SD.png" | Bin 0 -> 250833 bytes "plots/ADAM_\316\2241.0.png" | Bin 0 -> 254428 bytes "plots/GD_\316\2240.5.png" | Bin 0 -> 320598 bytes "plots/GD_\316\2240.5_+_SD.png" | Bin 0 -> 241641 bytes "plots/GD_\316\2241.0.png" | Bin 0 -> 323438 bytes "plots/SGD_\316\2240.5.png" | Bin 0 -> 378034 bytes "plots/SGD_\316\2240.5_+_SD.png" | Bin 0 -> 234635 bytes "plots/SGD_\316\2241.0.png" | Bin 0 -> 342546 bytes "plots/WD_\316\2240.5.png" | Bin 0 -> 315291 bytes "plots/WD_\316\2240.5_+_SD.png" | Bin 0 -> 241611 bytes "plots/WD_\316\2241.0.png" | Bin 0 -> 323870 bytes src/experiments.jl | 4 ++-- 13 files changed, 2 insertions(+), 2 deletions(-) create mode 100644 "plots/ADAM_\316\2240.5.png" create mode 100644 "plots/ADAM_\316\2240.5_+_SD.png" create mode 100644 "plots/ADAM_\316\2241.0.png" create mode 100644 "plots/GD_\316\2240.5.png" create mode 100644 "plots/GD_\316\2240.5_+_SD.png" create mode 100644 "plots/GD_\316\2241.0.png" create mode 100644 "plots/SGD_\316\2240.5.png" create mode 100644 "plots/SGD_\316\2240.5_+_SD.png" create mode 100644 "plots/SGD_\316\2241.0.png" create mode 100644 "plots/WD_\316\2240.5.png" create mode 100644 "plots/WD_\316\2240.5_+_SD.png" create mode 100644 "plots/WD_\316\2241.0.png" diff --git "a/plots/ADAM_\316\2240.5.png" "b/plots/ADAM_\316\2240.5.png" new file mode 100644 index 0000000000000000000000000000000000000000..e42e4673a9a49a0f125cf4ea14bd3d1699073a00 GIT binary patch literal 338011 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zhxr(jUZXPw4C*SgbH#|UnnhcZ0N7A&|>sn$E|;NF7pj z=f0J5)(^_R5KSJPE~<3GI!?arQZbqsLub=nNx-da2j06?_2N=-OWgZgp;~?da%t@3 zkj3iY{;TSs*DE{!`_0w;ki66*_Z9ZkAANP|TBw=YQ8eCd)JY^?oIMlKQ+#&1w1(E~ z*QyAlPEV70myi82vNYF@J2`xTOJ{+rb!7H-gs#~rs}q4u6}9GERR?(_crlGa z2*?STJUG7o$mgPgk6EfB;9%Ct9~t(WRZ*Sir~##2h|x;Tf#2v#Q#B%0m`5FgiWMSZ z0qz&ACgi4^dGo#Je4)YOwL1GxdY~^TpxNFI zET7I2=}>V#pJzqJkX9(ME$@&a4qq<;=9l(C#~=ne2J%zKAh#s*+dTaz82uHKs8vr# zNBLi)^IJSr_k-kvp#>saEiV4sO8)&_;41)C@IS8oW_L4yAJqQ>v;`?6f4lx)(f7|^ z0;oU!AJ={h(?}e+T$pij*Ob_QyZ$SW|F4gNu&(p}zV`dWk`#mC9=`UyxBK_&|9%CU tiR9ncely4and#Kf|7WKEpV{R1lWoq(b< Date: Sun, 30 Jan 2022 13:50:53 +0100 Subject: [PATCH 25/49] fix plot naming; make background transparent --- src/experiments.jl | 10 +++++++--- 1 file changed, 7 insertions(+), 3 deletions(-) diff --git a/src/experiments.jl b/src/experiments.jl index f2cb0f5e..ff318e66 100644 --- a/src/experiments.jl +++ b/src/experiments.jl @@ -20,12 +20,14 @@ function run_experiments(experiments; prefix = "", show_plots = true, save_plots # create plots p1 = PlotLib.plot_decision_boundary(train_loader, model, title = name) p2 = PlotLib.plot_loss_and_accuracy(stats["loss"], stats["accuracy"]) + # combine subplots p = [p1 p2] - PlotlyJS.relayout!(p, paper_bgcolor = "rgba(0, 0, 0, 0)") + # make background and plots transparent (looks better in dark-styled browsers) + PlotlyJS.relayout!(p, paper_bgcolor = "rgba(0, 0, 0, 0)", plot_bgcolor = "rgba(0, 0, 0, 0)") - # plot/save decision boundary and performance stats + # save and/or display decision boundary and performance stats if save_plots - p_name = "plots/$(prefix)_" * replace(name, " " => "_") * ".png" + p_name = "plots/$(prefix)" * replace(name, " " => "_") * ".png" PlotlyJS.savefig(p, p_name, width = plot_width, height = plot_height) end @@ -53,4 +55,6 @@ function get_plot_resolution(size) # 16:9 1080p return 1920, 1080 end +end + end \ No newline at end of file From c95ed730f136483145490d64c2248e9682759def Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 13:51:21 +0100 Subject: [PATCH 26/49] attempt to visualize features --- src/neuralnetwork.jl | 77 ++++++++++++++++++++++++++++++-------------- src/plotlib.jl | 25 +++++++++++++- 2 files changed, 76 insertions(+), 26 deletions(-) diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index f3df43dc..237f6a6f 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -7,6 +7,7 @@ using Flux using Flux: onecold using Flux: Optimiser using Flux.Losses: binarycrossentropy, crossentropy, logitcrossentropy, logitbinarycrossentropy +using LinearAlgebra using MLDatasets using ProgressMeter using Statistics @@ -47,15 +48,16 @@ function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model, spectra end """ - neural_network() + neural_network(500) Creates a Flux neural network according to the topology used by Pezeshki et al. -That is a NN with 2 hidden layers, 500 units each and ReLU activation. +That is a NN with 2 hidden layers, 500 units each and ReLU activation. The hidden +layer dimensions can be adjusted. """ -function neural_network() +function neural_network(hidden_dim = 500) return Chain( - Dense(2, 500, relu), - Dense(500, 2) + Dense(2, hidden_dim, relu), + Dense(hidden_dim, 2) ) end @@ -88,32 +90,57 @@ function train(data_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spe # record loss and accuracy progression over epochs loss_prog, accuracy_prog = [], [] + # record weight evolution of learned features + z1_prog, z2_prog = [], [] # training prog = Progress(args.epochs, 0.25, "Training... ", 75) for epoch in 1:args.epochs - loss(x, y) = spectral_decoupling ? - Regularization.spectral_decoupling(model(x), y, args.λ) : - logitcrossentropy(model(x), y) - - Flux.train!(loss, params, data_loader, optimizer) - - # evaluate loss and accuracy - loss, accuracy = get_loss_and_accuracy(data_loader, model) - - push!(loss_prog, loss) - push!(accuracy_prog, accuracy) - - # show neat progress bar - ProgressMeter.next!(prog; showvalues = [ - (:epoch, epoch), - (:loss, loss), - (:accuracy, accuracy) - ] - ) + for (x, y) in data_loader + grads = Flux.gradient(params) do + spectral_decoupling ? + Regularization.spectral_decoupling(model(x), y, args.λ) : + logitcrossentropy(model(x), y) + end + + Flux.Optimise.update!(optimizer, params, grads) + + # evaluate loss and accuracy + loss, accuracy = get_loss_and_accuracy(data_loader, model) + push!(loss_prog, loss) + push!(accuracy_prog, accuracy) + + # evaluate learned features + # Y = Diagonal(y) + # Φ = ? --> NTK regime, unable to obtain NTRF matrix to perform SVD + # U = U of singular value decomposition + + # workaround + # can only be done if hidden_dim = batchsize + if (size(model.layers[1].W, 1) == size(x, 2)) + U = transpose(model.layers[1].W) .* model.layers[2].W + zs = U .* model(x) + # U1 = transpose(model.layers[1].W) + # U2 = model.layers[2].W + # z1 = U1 .* model(x) + # z2 = U2 .* model(x) + # push!(z1_prog, sum(abs.(z1))) + # push!(z2_prog, sum(abs.(z2))) + push!(z1_prog, sum(abs.(zs[1,:]))) + push!(z2_prog, sum(abs.(zs[2,:]))) + end + + # show neat progress bar + ProgressMeter.next!(prog; showvalues = [ + (:epoch, epoch), + (:loss, loss), + (:accuracy, accuracy) + ] + ) + end end - return model, Dict("loss" => loss_prog, "accuracy" => accuracy_prog) + return model, Dict("loss" => loss_prog, "accuracy" => accuracy_prog, "z1" => z1_prog, "z2" => z2_prog) end """ diff --git a/src/plotlib.jl b/src/plotlib.jl index 02335652..ff0d0c66 100644 --- a/src/plotlib.jl +++ b/src/plotlib.jl @@ -7,7 +7,7 @@ using Flux using PlotlyJS using Plots -export plot_train_and_test_data, plot_loss_and_accuracy, plot_decision_boundary +export plot_train_and_test_data, plot_loss_and_accuracy, plot_features, plot_decision_boundary title_font_size = 18 tick_font_size = 16 @@ -51,6 +51,29 @@ function plot_loss_and_accuracy(loss, accuracy; args...) ) end +function plot_features(z1, z2; args...) + args = Args(; args...) + PlotlyJS.plot([ + PlotlyJS.scatter(y = z1, x = 1:args.epochs, mode = "lines", name = "z1"), + PlotlyJS.scatter(y = z2, x = 1:args.epochs, mode = "lines", name = "z2") + ], + Layout( + title = attr(text = "Features", font = attr(size = title_font_size)), + width = 1000, height = 1000, autosize = false, + xaxis = attr( + ticks = "outside", + tickfont = attr(size = tick_font_size)), + yaxis = attr( + ticks = "outside", + tickfont = attr(size = tick_font_size)), + legend = attr(font = attr(size = tick_font_size)), + margin = attr(l = 0, r = 0, b = 0, t = 0, pad = 0), + plot_bgcolor = "rgba(0, 0, 0, 0)", + paper_bgcolor = "rgba(0, 0, 0, 0)" + ) + ) +end + function plot_decision_boundary(loader, model; title = "") # grid and range step size n = 100 From ed78589caaf4674ca28473b155f4ab8b7b49f3a9 Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 13:52:51 +0100 Subject: [PATCH 27/49] clean up other alternative of feature vis --- src/neuralnetwork.jl | 8 +------- 1 file changed, 1 insertion(+), 7 deletions(-) diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 237f6a6f..9b1bcc86 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -116,16 +116,10 @@ function train(data_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spe # U = U of singular value decomposition # workaround - # can only be done if hidden_dim = batchsize + # can only be done if hidden_dim == batchsize if (size(model.layers[1].W, 1) == size(x, 2)) U = transpose(model.layers[1].W) .* model.layers[2].W zs = U .* model(x) - # U1 = transpose(model.layers[1].W) - # U2 = model.layers[2].W - # z1 = U1 .* model(x) - # z2 = U2 .* model(x) - # push!(z1_prog, sum(abs.(z1))) - # push!(z2_prog, sum(abs.(z2))) push!(z1_prog, sum(abs.(zs[1,:]))) push!(z2_prog, sum(abs.(zs[2,:]))) end From 3ef07edda21bbeff31723baa0c94b8c8181fd296 Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 21:31:47 +0100 Subject: [PATCH 28/49] moon creation: propagte seed param to exported methods --- src/data.jl | 11 ++++++++--- 1 file changed, 8 insertions(+), 3 deletions(-) diff --git a/src/data.jl b/src/data.jl index 7ffe9320..b1c5c412 100644 --- a/src/data.jl +++ b/src/data.jl @@ -11,11 +11,11 @@ using ScikitLearn export get_moon_data_loader, get_nmist_data_loader -function get_moon_data_loader(; n = 300, offset = 0.5, shuffle = true, args...) +function get_moon_data_loader(; n = 300, offset = 0.5, shuffle = true, seed = rand((1, 2^31)), args...) args = Args(; args...) # offset: linearly separable => 1.0; not lin. sep. => 0.5 - x, y = generate_moons(n, offset = offset) + x, y = generate_moons(n, offset = offset, seed = seed) x, y = transpose(x), onehotbatch(y, 0:1) # create data loader @@ -38,13 +38,15 @@ end function generate_moons_from_publication(; offset = 0.0, coord_down_scale = 1, show_plot = false) moons = hcat(moon_class_1(offset, coord_down_scale), moon_class_2(offset, coord_down_scale)) labels = [repeat([0], 150); repeat([1], 150)] + if show_plot display(scatter(moons[1,:], moons[2,:], c = labels)) end + return moons, labels end -function generate_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand((1, 2^31)), show_plot = false) +function generate_moons(n; noise = 0.09, offset = 0.0, rotation = 90, seed = rand((1, 2^31)), show_plot = false) X, y = make_moons(n_samples = n, noise = noise, random_state = seed) p1 = scatter(X[:,1], X[:,2], c = y, title = "generated state") @@ -67,6 +69,7 @@ function generate_moons(n; noise = 0.1, offset = 0.0, rotation = 90, seed = rand if show_plot display(plot(p1, p2, p3, layout = (1, 3))) end + return X, y end @@ -399,6 +402,7 @@ function moon_class_1(offset = 0.0, coord_down_scale = 1) -0.7882243058017496, -0.9895456553950488 ] + return vcat(transpose.((x, y))...) / coord_down_scale end @@ -707,6 +711,7 @@ function moon_class_2(offset = 0.0, coord_down_scale = 1) -3.0090552847647456, 0.2317409067300455 ] + return vcat(transpose.((x, y))...) / coord_down_scale end From 0667dfe06887037a097632697c81c4618884a1d4 Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 21:32:13 +0100 Subject: [PATCH 29/49] create module for special figures --- src/figures.jl | 21 +++++++++++++++++++++ 1 file changed, 21 insertions(+) create mode 100644 src/figures.jl diff --git a/src/figures.jl b/src/figures.jl new file mode 100644 index 00000000..9de0c916 --- /dev/null +++ b/src/figures.jl @@ -0,0 +1,21 @@ +module Figures + +include("data.jl") +include("plotlib.jl") +include("plotutils.jl") + +using PlotlyJS + +export create_figure_1 + +plot_dir = "plots" + +function create_figure_1(plot_size) + width, height = PlotUtils.get_plot_resolution(plot_size) + loader_1 = Data.get_moon_data_loader(offset = 1.0, seed = 1337) + loader_2 = Data.get_moon_data_loader(offset = 0.5, seed = 1337) + p = PlotLib.plot_train_and_test_data(loader_1, loader_2; train_title = "Δ0.5", test_title = "Δ1.0") + PlotlyJS.savefig(p, "$(plot_dir)/figure_1.png", width = width, height = height) +end + +end \ No newline at end of file From ce676c22c4f25374bd0a197b111fc275cbacaf8d Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 21:33:33 +0100 Subject: [PATCH 30/49] move plot resolution method to plotutils --- src/experiments.jl | 19 ------------------- src/plotutils.jl | 21 ++++++++++++++++++++- 2 files changed, 20 insertions(+), 20 deletions(-) diff --git a/src/experiments.jl b/src/experiments.jl index ff318e66..3b3a3083 100644 --- a/src/experiments.jl +++ b/src/experiments.jl @@ -38,23 +38,4 @@ function run_experiments(experiments; prefix = "", show_plots = true, save_plots @info "Finished $(length(experiments)) experiments." end -function get_plot_resolution(size) - size = lowercase(size) - @assert size in ["small", "medium", "large", "xlarge"] "Requested plot size '$size' is not supported." - - if size == "small" - # 16:9 360p - return 640, 360 - elseif size == "medium" - # 16:9 480p - return 854, 480 - elseif size == "large" - # 16:9 720p - return 1280, 720 - elseif size == "xlarge" - # 16:9 1080p - return 1920, 1080 - end -end - end \ No newline at end of file diff --git a/src/plotutils.jl b/src/plotutils.jl index 04deb329..ee0bed7b 100644 --- a/src/plotutils.jl +++ b/src/plotutils.jl @@ -1,6 +1,6 @@ module PlotUtils -export get_custom_rdbu_scale, map_bool_to_color +export get_custom_rdbu_scale, get_plot_resolution, map_bool_to_color function get_custom_rdbu_scale(opacity::Union{Float32, Float64}) @assert opacity >= 0 || opacity <= 1 "Opacity must be in [0, 1]" @@ -13,6 +13,25 @@ function get_custom_rdbu_scale(opacity::Union{Float32, Float64}) ] end +function get_plot_resolution(size) + size = lowercase(size) + @assert size in ["small", "medium", "large", "xlarge"] "Requested plot size '$size' is not supported." + + if size == "small" + # 16:9 360p + return 640, 360 + elseif size == "medium" + # 16:9 480p + return 854, 480 + elseif size == "large" + # 16:9 720p + return 1280, 720 + elseif size == "xlarge" + # 16:9 1080p + return 1920, 1080 + end +end + function map_bool_to_color(bool_vec::Union{BitVector, Vector{Bool}}, color_true::String, color_false::String) return map(el -> el == true ? color_true : color_false, bool_vec) end From 7a9bcf9200c6d5954c5f44f2a6808972a0561abc Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 21:34:14 +0100 Subject: [PATCH 31/49] allow run_experiments save subplots --- src/experiments.jl | 45 ++++++++++++++++++++++++++++++++------------- 1 file changed, 32 insertions(+), 13 deletions(-) diff --git a/src/experiments.jl b/src/experiments.jl index 3b3a3083..1c084ce0 100644 --- a/src/experiments.jl +++ b/src/experiments.jl @@ -1,38 +1,57 @@ module Experiments -using PlotlyJS - include("data.jl") include("neuralnetwork.jl") include("plotlib.jl") +include("plotutils.jl") include("structs.jl") -function run_experiments(experiments; prefix = "", show_plots = true, save_plots = true, plot_size = "medium") - plot_width, plot_height = get_plot_resolution(plot_size) +using PlotlyJS + +export run_experiments + +plot_dir = "plots" + +function run_experiments(experiments; prefix = "", random_moons = true, show_plots = true, save_plots = true, plot_size = "medium", plot_type = "both", args...) + plot_type = lowercase(plot_type) + @assert plot_type in ["boundary", "stats", "both"] "Requested plot type '$plot_type' is not supported." + + plot_width, plot_height = PlotUtils.get_plot_resolution(plot_size) for (name, def) in experiments # load data - train_loader = Data.get_moon_data_loader(offset = def.offset) + if random_moons + train_loader = Data.get_moon_data_loader(offset = def.offset) + else + train_loader = Data.get_moon_data_loader(offset = def.offset, seed = 1337) + end # create and train the model - model, stats = NeuralNetwork.train(train_loader, def.optimizer, def.sd) + model, stats = NeuralNetwork.train(train_loader, def.optimizer, def.sd; args...) # create plots p1 = PlotLib.plot_decision_boundary(train_loader, model, title = name) p2 = PlotLib.plot_loss_and_accuracy(stats["loss"], stats["accuracy"]) - # combine subplots - p = [p1 p2] - # make background and plots transparent (looks better in dark-styled browsers) - PlotlyJS.relayout!(p, paper_bgcolor = "rgba(0, 0, 0, 0)", plot_bgcolor = "rgba(0, 0, 0, 0)") + + if plot_type == "both" + # combine subplots + p = [p1 p2] + # make background and plots transparent (looks better in dark-styled browsers) + PlotlyJS.relayout!(p, paper_bgcolor = "rgba(0, 0, 0, 0)", plot_bgcolor = "rgba(0, 0, 0, 0)") + elseif plot_type == "boundary" + p = p1 + elseif plot_type == "stats" + p = p2 + end - # save and/or display decision boundary and performance stats + # save requested plot if save_plots - p_name = "plots/$(prefix)" * replace(name, " " => "_") * ".png" + p_name = "$plot_dir/$prefix" * (prefix != "" ? "_" : "") * replace(name, " " => "_") * (plot_type == "both" ? "" : plot_type) * ".png" PlotlyJS.savefig(p, p_name, width = plot_width, height = plot_height) end if show_plots - display(p) + display(p) end end @info "Finished $(length(experiments)) experiments." From 1cf78d53bd84c11be584061770668cede6be704e Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 21:34:55 +0100 Subject: [PATCH 32/49] refactor plot_train_and_test_data to plotlyjs --- src/plotlib.jl | 48 ++++++++++++++++++++++++++++++++++++------------ 1 file changed, 36 insertions(+), 12 deletions(-) diff --git a/src/plotlib.jl b/src/plotlib.jl index ff0d0c66..5ec1e27a 100644 --- a/src/plotlib.jl +++ b/src/plotlib.jl @@ -12,17 +12,41 @@ export plot_train_and_test_data, plot_loss_and_accuracy, plot_features, plot_dec title_font_size = 18 tick_font_size = 16 -function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader) - # plot train and test data sets next to each other - p_train = Plots.scatter( - train_loader.data[1][1,:], train_loader.data[1][2,:], - c = PlotUtils.map_bool_to_color(train_loader.data[2][1,:], "blue", "red") +function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader; train_title = "", test_title = "") + # define common layout + layout = Layout( + width = 1000, height = 1000, autosize = false, + plot_bgcolor = "rgba(0, 0, 0, 0)", paper_bgcolor = "rgba(0, 0, 0, 0)", + xaxis = attr( + showgrid = true, gridcolor = "#e2e2e2", + ticks = "outside", zeroline = false, + tickfont = attr(size = tick_font_size)), + yaxis = attr( + showgrid = true, gridcolor = "#e2e2e2", + ticks = "outside", zeroline = false, + tickfont = attr(size = tick_font_size)), ) - p_test = Plots.scatter( - test_loader.data[1][1,:], test_loader.data[1][2,:], - c = PlotUtils.map_bool_to_color(test_loader.data[2][1,:], "blue", "red") - ) - display(Plots.plot(p_train, p_test, layout = (1, 2))) + + # create two independent plots + p_train = PlotlyJS.plot(PlotlyJS.scatter( + x = train_loader.data[1][1,:], y = train_loader.data[1][2,:], mode = "markers", + marker_color = PlotUtils.map_bool_to_color(train_loader.data[2][1,:], "red", "blue"), + showlegend = false + ), layout) + p_test = PlotlyJS.plot(PlotlyJS.scatter( + x = test_loader.data[1][1,:], y = test_loader.data[1][2,:], mode = "markers", + marker_color = PlotUtils.map_bool_to_color(test_loader.data[2][1,:], "red", "blue"), + showlegend = false, showgrid = true + ), layout) + + # adjust common layout with individual titles + PlotlyJS.relayout!(p_train, title = attr(text = train_title, font = attr(size = title_font_size))) + PlotlyJS.relayout!(p_test, title = attr(text = test_title, font = attr(size = title_font_size))) + + # combine plots into one + p = [p_train p_test] + PlotlyJS.relayout!(p, plot_bgcolor = "rgba(0, 0, 0, 0)", paper_bgcolor = "rgba(0, 0, 0, 0)") + p end function plot_loss_and_accuracy(loss, accuracy; args...) @@ -126,12 +150,12 @@ function plot_decision_boundary(loader, model; title = "") width = 1000, height = 1000, autosize = false, xaxis = attr( range = [x_min, x_max], - zeroline = true, zerolinewidth = 1, zerolinecolor = "black", + zeroline = false, zerolinewidth = 1, zerolinecolor = "black", automargin = false, showgrid = false, tickfont = attr(size = tick_font_size)), yaxis = attr( range = [y_min, y_max], - zeroline = true, zerolinewidth = 1, zerolinecolor = "black", + zeroline = false, zerolinewidth = 1, zerolinecolor = "black", automargin = false, showgrid = false, tickfont = attr(size = tick_font_size)), legend = attr(font = attr(size = tick_font_size)), From e1794c826dcf3f956204ca51b3c7a3e9bdd14b08 Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 21:36:30 +0100 Subject: [PATCH 33/49] update args with wd param, fix progress bar --- src/neuralnetwork.jl | 48 ++++++++++++++++++++++++-------------------- src/structs.jl | 3 ++- 2 files changed, 28 insertions(+), 23 deletions(-) diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 9b1bcc86..3d261e67 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -27,7 +27,7 @@ julia> get_loss_and_accuracy(train_loader, model) """ function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model, spectral_decoupling::Bool = false; args...) args = Args(; args...) - @assert !(spectral_decoupling && isa(args.λ, Float64)) "λ must be specified if spectral decoupling is used" + @assert !(spectral_decoupling && isa(args.sd_λ, Float64)) "sd_λ must be specified if spectral decoupling is used" accuracy = 0.0f0 loss = 0.0f0 @@ -37,7 +37,7 @@ function get_loss_and_accuracy(data_loader::Flux.Data.DataLoader, model, spectra for (x, y) in data_loader ŷ = model(y) loss += spectral_decoupling ? - Regularization.spectral_decoupling(ŷ, y, args.λ) : + Regularization.spectral_decoupling(ŷ, y, args.sd_λ) : logitcrossentropy(ŷ, y, agg = sum) accuracy += sum(onecold(ŷ) .== onecold(y)) num += size(x)[end] @@ -82,11 +82,12 @@ Training... 100%|█████████████████████ function train(data_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spectral_decoupling::Bool = false; args...) args = Args(; args...) - @info "opt = $optimizer, sd = $spectral_decoupling, lr = $(args.learning_rate), bs = $(args.batchsize)" * (spectral_decoupling ? ", λ = $(args.λ)" : "") + @info "opt = $optimizer, sd = $spectral_decoupling, lr = $(args.learning_rate), bs = $(args.batchsize)" * + (spectral_decoupling ? ", sd_λ = $(args.sd_λ)" : "") * (optimizer == "WD" ? ", wd_λ = $(args.wd_λ)" : "") model = neural_network() params = Flux.params(model) - optimizer = get_optimizer(args.learning_rate, optimizer) + optimizer = get_optimizer(args.learning_rate, optimizer; args.wd_λ) # record loss and accuracy progression over epochs loss_prog, accuracy_prog = [], [] @@ -99,17 +100,11 @@ function train(data_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spe for (x, y) in data_loader grads = Flux.gradient(params) do spectral_decoupling ? - Regularization.spectral_decoupling(model(x), y, args.λ) : + Regularization.spectral_decoupling(model(x), y, args.sd_λ) : logitcrossentropy(model(x), y) end Flux.Optimise.update!(optimizer, params, grads) - - # evaluate loss and accuracy - loss, accuracy = get_loss_and_accuracy(data_loader, model) - push!(loss_prog, loss) - push!(accuracy_prog, accuracy) - # evaluate learned features # Y = Diagonal(y) # Φ = ? --> NTK regime, unable to obtain NTRF matrix to perform SVD @@ -123,22 +118,27 @@ function train(data_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spe push!(z1_prog, sum(abs.(zs[1,:]))) push!(z2_prog, sum(abs.(zs[2,:]))) end - - # show neat progress bar - ProgressMeter.next!(prog; showvalues = [ - (:epoch, epoch), - (:loss, loss), - (:accuracy, accuracy) - ] - ) end + + # evaluate loss and accuracy + loss, accuracy = get_loss_and_accuracy(data_loader, model) + push!(loss_prog, loss) + push!(accuracy_prog, accuracy) + + # show neat progress bar + ProgressMeter.next!(prog; showvalues = [ + (:epoch, epoch), + (:loss, loss), + (:accuracy, accuracy) + ] + ) end return model, Dict("loss" => loss_prog, "accuracy" => accuracy_prog, "z1" => z1_prog, "z2" => z2_prog) end """ - get_optimizer(learning_rate, optimizer) + get_optimizer(learning_rate, optimizer, wd_λ) Returns a Flux.Optimiser according to the one requested by acronym with fixed parameters except for the given learning rate. Supported optimizers are: @@ -147,9 +147,13 @@ for the given learning rate. Supported optimizers are: - Stochastic Gradient Descent (SGD) - Weight Decay (WD) """ -function get_optimizer(learning_rate, optimizer = "SGD") +function get_optimizer(learning_rate, optimizer = "SGD"; args...) + args = Args(; args...) optimizer = uppercase(optimizer) @assert optimizer in ["ADAM", "GD", "SGD", "WD"] "Requested optimizer '$optimizer' is not supported." + if optimizer == "WD" + @assert isa(args.wd_λ, Float64) "wd_λ must be specified if WD is used" + end if optimizer == "ADAM" return ADAM(learning_rate) @@ -159,7 +163,7 @@ function get_optimizer(learning_rate, optimizer = "SGD") # stochastic gradient descent return Momentum(learning_rate, 0.9) elseif optimizer == "WD" - return Optimiser(WeightDecay(), Descent(learning_rate)) + return Optimiser(WeightDecay(args.wd_λ), Descent(learning_rate)) end end diff --git a/src/structs.jl b/src/structs.jl index 5fef18e1..568518ac 100644 --- a/src/structs.jl +++ b/src/structs.jl @@ -2,7 +2,8 @@ Base.@kwdef mutable struct Args learning_rate::Float64 = 1e-2 batchsize::Int = 50 epochs::Int = 1000 - λ::Float64 = 3e-1 + sd_λ::Float64 = 3e-3 + wd_λ::Float64 = 1e-2 end struct Experiment From 85b79083bd9800561a8930de4cb1bce06c09504f Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 21:36:46 +0100 Subject: [PATCH 34/49] WIP: notebook --- notebook/gradientstarvation.jl | 428 ++++++++++++++++++++++++++++++--- 1 file changed, 389 insertions(+), 39 deletions(-) diff --git a/notebook/gradientstarvation.jl b/notebook/gradientstarvation.jl index 0e4cd3e4..c7111da9 100644 --- a/notebook/gradientstarvation.jl +++ b/notebook/gradientstarvation.jl @@ -4,19 +4,19 @@ using Markdown using InteractiveUtils -# ╔═╡ df218f5d-042d-4ccb-8005-56b2ec073869 -html""" -This is still work in progress! -""" +# ╔═╡ 5c563230-df6e-4f65-a106-a707d2454fe7 +using PlutoUI # ╔═╡ 8bd774c7-1acf-4beb-bc99-4c303882d262 md""" -# Hungry Neurons - Neural Networks and Gradient Starvation +# Starving Neurons - Neural Networks and Gradient Starvation ## Introduction Artificial intelligence (AI) and machine learning are disciplines whose methods have become ubiquitously applied in a broad range of scientific domains, enabling the implicit learning of features inherent to a dataset. A common AI tool for self-learning of data characteristics by examples are neural networks (NNs) which attempt to mimic biological neurons. NNs are broadly applicable, e.g. for tasks such as image and speech recognition, weather forecasting, stock market prediction, robotics and data mining among many others. But how do these artificial neural networks learn? -There is a myriad of neural network subtypes and topologies, but in this notebook, I want to focus on the multi-layer perceptron (MLP). Basically speaking, a NN consists of multiple layers: an input layer, one or more hidden layers and an output layer (see figure 1). +There is a myriad of neural network subtypes and topologies, but in this notebook, I want to focus on the multi-layer perceptron (MLP). + +Basically speaking, a NN consists of multiple layers: an input layer, one or more hidden layers and an output layer (see fig. 1). """ @@ -32,27 +32,29 @@ html""" """ -# ╔═╡ 6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b -md"""Figure 1: Basic NN/MLP topology with input, a single hidden and output layer. -CC-BY-SA 3.0. Source: [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Artificial_neural_network.svg)""" +# ╔═╡ 6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b +md""" +Figure 1: Basic NN/MLP topology with input, a single hidden and output layer. +CC-BY-SA 3.0. Source: [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Artificial_neural_network.svg) +""" # ╔═╡ 512e6240-76e7-11ec-2511-07ffc4b74d27 md""" -The input layer encodes the data presented to the network and feeds it weighted into the first hidden layer; all other layers consist of artificial neurons with non-linear activation functions. MLPs are fully connected, meaning that every node of one layer connects to every node of the next layer and each connection has a weight assigned to it. The weights can be understood as the actual learned parameters being adjusted as they get passed from layer to layer. The MLP learns supervised, i.e. by presentation of labelled training data. Without going into too much detail, I want to briefly formalize the learning process by the example of a classification task, that is deciding to which discrete class a given input belongs. +The input layer encodes the data presented to the network and feeds it weighted into the first hidden layer; all other layers consist of artificial neurons with non-linear activation functions. MLPs are fully connected, meaning that every node of one layer connects to every node of the next layer and each connection has a weight assigned to it. The weights can be understood as the actual learned parameters being adjusted as they get passed from layer to layer. The MLP learns supervised, i.e. by presentation of labelled training data. Without going into too much detail, I want to briefly formalize the learning process by the example of a classification task, that is deciding to which discrete class a given input belongs. This is exactly the use case we will evaluate later on. """ # ╔═╡ 93bc80a8-673d-457b-96d8-e1bb0b09682e md""" ### Supervised Learning – Learning from Examples -Supervised learning tries to learn a function which maps input to output values based on known input-output pairs (i.e. the pair (x, y); a correct classification y for a given input x). The observed outputs $y_i$ originate from an unknown function $f(x) = y$. Thus, the goal of the learning process is to approximate $f(x)$ by a hypothesis $h(x) \approx f(x)$. The loss is defined as the utility lost by substituting the correct classification $f(x) = y$ by the hypothesis $h(x) = \hat{y}$: +Supervised learning tries to learn a function which maps input to output values based on known input-output pairs (i.e. the pair ($x$, $y$); a correct classification $y$ for a given input $x$). The observed outputs $y$ originate from an unknown function $f(x) = y$. Thus, the goal of the learning process is to approximate $f(x)$ by a hypothesis $h(x) \approx f(x)$ while minimizing the error of the approximation, refered to as loss. More formally, the loss is defined as the utility lost by substituting the correct classification $f(x) = y$ by the hypothesis $h(x) = \hat{y}$: $L(x, y, ŷ) = Utility\\(result\ of\ using\ y\ given\ x\\) - Utility\\(result\ of\ using\ ŷ\ given\ x\\)$ -Commonly applied loss functions are mean absolute error and mean squared error loss. Especially popular for classification tasks is the cross-entropy loss, also called logarithmic loss: +Commonly applied loss functions are the mean absolute error and mean squared error loss. Especially popular for classification tasks is the cross-entropy loss, also called logarithmic loss: $L(y, ŷ) = -\sum_{i}y_i\cdot log(ŷ_i)$ @@ -61,7 +63,7 @@ The learning process maximizes the expected utility through choosing the optimal $EmpiricalLoss(h) = \frac{1}{N}\cdot\sum_{(x, y) \in X} L(y, h(x))$ But what about the weights I’ve introduced earlier as the “actual learned parameters”? Keep in mind that a hypothesis $h$ is a function and the weights simply represent the function coefficients. For the linear case that would mean $h(x) = w_1 x + w_0$. -In the multivariate case there is no closed-form solution for the minimum loss. Therefore, we are confronted with an optimization problem which can be solved by gradient descent. The weight update with learning rate $\alpha$ and the vector of weights $\bf{w}$: +In the multivariate case there is no closed-form solution for the minimum loss. Therefore, we are confronted with an optimization problem which can be solved by gradient descent. The weight update with learning rate $\alpha$ and the vector of weights $\bf{w}$ looks like this: $w_i = w_i - \alpha\cdot \frac{\partial}{\partial w_i}\cdot Loss(\bf{w})$ @@ -71,55 +73,181 @@ $w_i = w_i - \alpha\cdot \frac{\partial}{\partial w_i}\cdot Loss(\bf{w})$ md""" ### Regularization - > "Entities should not be multiplied beyond necessity." – William of Ockham -This quote, known as Ockham’s razor and usually paraphrased as “simple solutions are better solutions” captures the core idea of regularization. The aim is to find the hypothesis which minimizes the total cost, defined as the sum of empirical loss and its complexity: +This quote, known as Ockham’s razor and usually paraphrased as “simple solutions are better solutions” captures the core idea of regularization. The aim of regularization is to find the hypothesis which minimizes the total cost, defined as the sum of empirical loss and its complexity: $Cost(h) = EmpiricalLoss(h) + \lambda\cdot Complexity(h)$ -Regularization penalizes complex hypotheses and favors more regular functions. The estimated best hypothesis $ĥ^*$ from the hypothesis space $\mathcal{H}$ as determined by regularization is: +Regularization penalizes complex hypotheses (mind the added regularization term in the formula $\lambda\cdot Complexity(h)$) and favors more regular functions. It limits the values that the weights can take during the learning process. Therefore, regularization avoids overfitting of the data. +The estimated best hypothesis $ĥ^*$ from the hypothesis space $\mathcal{H}$ now becomes: $ĥ^* = \arg\min_{h \in \mathcal{H}} Cost(h)$ + +### Optimization +Given the major workings of supervised learning, it should have become clear that learning is a mathematical optimization problem. A classic way to achieve the minimization of the cost is the application of gradient descent (GD). However, the path the gradient take on the surface of the solution space can limit its ability to find the global optimum. To conquer that, the steps can be modified by additional optimization methods. Common choices for that purpose are the addition of momentum, resulting in stochastic gradient descent (SGD); an adaptive learning rate, different forms of parameter initialization as well as normalization techniques. """ # ╔═╡ 4a14831d-7736-4af5-875f-05ae6e3cf7c4 md""" ### Side Effects of Gradient-based Learning +Despite its popularity, gradient descent learning in NNs can come with some unwanted side effects such as the vanishing gradient problem, which condenses down to the gradient being too small to effectively update a weight and thus preventing learning. This phenomenon is complemented in the other direction by exploding gradients. -Given the major workings of supervised learning, it should have become clear that learning is a mathematical optimization problem. A classic way to achieve the minimization of the loss is the application of gradient descent. - -Despite its popularity, gradient descent learning in NNs can come with some unwanted side effects such as the vanishing gradient problem, which condenses down to the gradient being too small to effectively update a weight and thus preventing learning. - -Another not yet published but currently in preprint described observation is the effect of **Gradient Starvation (GS)**. Pezeshki et al. coined and formalized this phenomenon of over-parameterized NNs in their work. According to them, _Gradient Starvation_ is a predisposition of NNs learned by cross-entropy, which occurs when the loss minimization is driven by only a feature subset regardless of the presence of other meaningful features. Thus, the NN becomes biased towards superficial features. Beyond demonstrating the effects of GS, Pezeshki et al. introduce _Spectral Decoupling_ (SD) regularization as a means to counteract gradient starvation. +Another not yet published but currently in preprint described observation is the effect of [**_Gradient Starvation_ (GS)**](https://arxiv.org/abs/2011.09468). Pezeshki et al. coined and formalized this phenomenon of over-parameterized NNs in their work. According to them, _Gradient Starvation_ is a predisposition of NNs learned by cross-entropy, which occurs when the loss minimization is driven by only a feature subset regardless of the presence of other meaningful features. Thus, the NN becomes biased towards superficial features. Beyond demonstrating the effects of GS, Pezeshki et al. introduce **_Spectral Decoupling_ (SD)** regularization as a means to counteract gradient starvation. """ # ╔═╡ 00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 md""" ### Objective -My project and thereby this notebook aims to reproduce and demonstrate the effects of Gradient Starvation on a simple 2D classification task. Furthermore, the proposed regularization method of Spectral Decoupling is implemented and compared to other regularization methods. Finally, the framework is extended to a more complex task on the well-known MNIST dataset. - +My project and thereby this notebook aims to reproduce and demonstrate the effects of GS on a simple 2D classification task. Furthermore, the proposed SD regularization method is implemented and compared to other methods. On top of that, spectral decoupling is not only applied in the context of SGD but all other investigated optimization approaches. """ # ╔═╡ ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 md""" -## Experiments +### Waning or Waxing Moon? A 2D Classification Experiment +A modified version of the [Moons dataset](https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_moons.html) was used with two different topologies as exemplified in figure 1. +The first topology allows to draw a line by a very small margin between the two moons. Therefore, it represents the fully linearly separable case (fig. 1, left). The second one applies a smaller offset to interleave both moons which only allows to separate them by learning a curved classification boundary (fig. 1, right). The task of the NN is to discriminate if a given data point belongs to the waning (blue) or waxing moon (red). +""" + +# ╔═╡ 2f766a9c-de6e-4f95-aa73-aacb486aaf61 +md""" +$(LocalResource("../plots/figure_1.png")) +Figure 1: Two moons topologies; one linear separable (offset Δ1.0), the other separable by a curved boundary (offset Δ0.5). +""" + +# ╔═╡ 7f7ed965-fa08-4d6f-8908-dfc8876bdd02 +md""" +The idea of Pezeshki et al. was that in the linear separable case the NN is not encouraged to learn a curved decision boundary because the loss becomes neglibile by only discriminating the two moons along one axis, therefore learning only one feature and neglecting the other. In contrast to that, the interleaved moons can only be separated by learning both features which better accommodates the data structure in general. +They observed that a different choice of regularization and optimization methods like Weight Decay (WD) and Adaptive Moment Estimation (ADAM) among others are not able to +stimulate the NN to learn a curved decision boundary. +Moreover, Pezeshki et al. explain that circumstance by cross-entropy loss learning which starves the gradients from the other feature, only reinforcing the superficial one. + +#### Repercussions of Gradient Starvation +Why does the shape of the decision boundary even matter as long as it discriminates the data points confidently? As stated by the authors, GS results in a very small distance between the data points and the boundary which translates into a lack of robustness when generalizing to new data. Contrarily, GS could also have the positive effect of the prevention of overfitting by not learning non-dominant features. Pezeshki et al. make the comparison to the aformentioned Ockham's razor: GS leads to simpler decision boundaries and hence NNs that generalize better, thus being a form of implicit regularization. + +#### Implementation +The implementation is based on [Flux](https://fluxml.ai/), a ML framework for Julia. The NN consists of two hidden layers with 500 nodes each. Rectified Linear Unit (ReLu) is used as an activation function and learning is facilitated with cross-entropy loss for 1000 epochs on a data set with 300 observations (150 blue and 150 red points). In contrast to Pezeshki et al., I opted to train in batches of 50 (6 batches in total) since this is a common practice for training NNs. +All networks are trained with gradient descent but with different regularization and optimization approaches. This yields the following different setups: +- Gradient descent (GD; no regularization) +- Gradient descent with momentum (stochastic gradient descent (SGD); no regularization) +- Gradient descent with weight decay (WD; λ⋅L2 regularization) +- Gradient descent with _Adaptive Moment Estimation_ (ADAM; no regularization) + +Different sets of hyper parameters were evaluated. However, since the main objective is to reproduce the research conducted by Pezeshki et al., their set of hyperparameters were used: +- learning rate = 0.01 +- weight decay coefficient = 0.01 +- spectral decoupling coefficient = 0.003 +""" + +# ╔═╡ 88ddf822-622b-4d46-a0a4-a7d749c8bc48 +md""" +#### Results +""" + +# ╔═╡ 884bc572-9daa-40ad-8d76-56ec58728f63 +md""" +##### Linear separable case +$(LocalResource("../plots/GD_Δ1.0.png")) +Figure 2: Gradient descent (GD) on the moons with offset = Δ1.0. The decision boundary is almost linear and the loss continuously falls while training. Accuracy reaches 100%. + +$(LocalResource("../plots/SGD_Δ1.0.png")) +Figure 3: Stochastic gradient descent (SGD) on the moons with offset = Δ1.0. The decision boundary is similar to the one learned with GD but equally distant from both classes. Loss declines quicker and reaches an overall lower level compared to GD. Accuracy reaches 100% after less training. + +$(LocalResource("../plots/WD_Δ1.0.png")) +Figure 4: Gradient descent with weight decay (WD) on the moons with offset = Δ1.0. Decay coefficient = 0.01. The boundary is similar to GD without WD. + +$(LocalResource("../plots/ADAM_Δ1.0.png")) +Figure 5: Gradient descent with Adaptive Moment Estimation (ADAM) on the moons with offset = Δ1.0. The decision boundary is slightly curved around the data points closest to the boundary. Loss converges to zero almost immediately while full accuracy is attained. +""" + +# ╔═╡ ac453412-0f71-45ee-aefb-0cc519467c6c +md""" +As we can see from figures 2-5, the decision boundaries of all networks exhibit some specificites. The GD network learned an almost linear decision boundary which appears to be closer to the blue-labeled moon. The boundary looks similar with SGD, but the distance to both moons is equal. Only the network with ADAM optimization learns a slight curvature around the points closest to the boundary. Furthermore, the ADAM classification learned the fastest as can be seen from the rapid decrease in loss and increase in accuracy. At the same time, the contour lines all closely condense around the decision boundary which indicates that the ADAM network is most confident in discriminating both classes. +""" + +# ╔═╡ 5e282e1a-62fc-4911-9b7e-f2631eeb2669 +md""" +##### Linear inseparable case -### 2D Classification -A modified version of the [Moons dataset](https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_moons.html) was used with two different topologies. The first represents the fully linearly separable case, thus allows to differentiate the two moons by a straight line along one axis. The second topolgy uses an offset to interleave both moons which only allows to separate them by learning a curved classification boundary. Pezeshki et al. +$(LocalResource("../plots/GD_Δ0.5.png")) +Figure 6: GD on the moons with offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes. + +$(LocalResource("../plots/SGD_Δ0.5.png")) +Figure 7: SGD on the moons with offset = Δ0.5. The network learns a curved decision boundary which is very close to the interleaving data points. Loss becomes negligible during learning. + +$(LocalResource("../plots/WD_Δ0.5.png")) +Figure 8: WD on the moons with offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes. Again, similar result compared to GD. + +$(LocalResource("../plots/ADAM_Δ0.5.png")) +Figure 9: ADAM on the moons with offset = Δ0.5. The network learns a curved decision boundary. Loss fluctuates especially during the early iterations of training. """ -# ╔═╡ 85591d79-dec2-45fa-a0a9-d95d84edda51 +# ╔═╡ f7bae71f-885a-404d-985b-92d612e1f3d6 md""" -## Implementation -The implementation of the NN is closely related to the one of Pezeshki et al. who realized theirs with Python and PyTorch. The implementation provided here is based on [Flux](https://fluxml.ai/), a ML framework for Julia. The NN consists of two hidden layers with 500 nodes each. Rectified Linear Unit (ReLu) is used as an activation function and learning is facilitated with cross-entropy loss. +SGD and ADAM are able to learn a curved decision boundary (fig. 7 and 9) while GD and WD fail to do so (fig. 6 and 8). Moreover, the decision boundary learned by ADAM better resembles the data structure. +""" + +# ╔═╡ 2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f +md""" +#### Spectral Decoupling + +$(LocalResource("../plots/GD_Δ0.5_+_SD.png")) +Figure 10: GD with SD on the moons with offset = Δ0.5. No curved decision boundary is learned but the loss also did not converge. + +$(LocalResource("../plots/SGD_Δ0.5_+_SD.png")) +Figure 11: SGD with SD on the moons with offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the moon data structure. Loss converges to ~15%. + +$(LocalResource("../plots/WD_Δ0.5_+_SD.png")) +Figure 12: WD with SD on the moons with offset = Δ0.5 fails to separate the two data classes. + +$(LocalResource("../plots/ADAM_Δ0.5_+_SD.png")) +Figure 13: ADAM with SD on the moons with offset = Δ0.5 learns a curved decision boundary which greatly resembles the data structure. Loss heavily oscillates and steadily increases with longer training. +""" + +# ╔═╡ 7460f609-c32b-4704-a361-71c98bb626cd +md""" +Spectral decoupling, originally applied as a regularization by Pezeshki et al. only with SGD, is able to yield curved decision boundaries which greatly resemble the data structure of the two moons (SGD and ADAM). While SGD learns a more angular boundary with low loss, ADAM learns a smoother one but exhibits a high loss with strong oscillations. Interestingly, ADAM learns to group all data points of a class within the same contour (fig. 13). GD and WD do not profit from SD regularization and don't learn a curved boundary. +""" + +# ╔═╡ 87cc8172-d5cc-49ff-9a0e-fa734e1010c1 +md""" +#### Additional experiments +Given that the loss did not converge for GD + SD after 1000 epochs, the network was trained for 10000 iterations. + +$(LocalResource("../plots/GD_Δ0.5_+_SD_10000.png")) +Figure 14: GD with SD on the moons with offset = Δ0.5 trained 10 times longer (10000 iterations). The learned curved decision boundary resembles the result obtained with SGD + SD. + +To conquer the overfitting observed with ADAM + SD lower learning rates were evaluated. + +$(LocalResource("../plots/1e-3_ADAM_Δ0.5_+_SD.png")) +Figure 15: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.001 still suffers from severe overfitting. + +$(LocalResource("../plots/1e-4_ADAM_Δ0.5_+_SD.png")) +Figure 16: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.0001 shows lower loss but a decision boundary with a smaller margin. +""" + +# ╔═╡ bd4f515c-1e41-48bc-8c36-233535a3022c +md""" +### Summary +This section summarizes the decision boundaries of all figures from above for easier visual comparison in one plot. + + +Figure 14: +""" + +# ╔═╡ 2c740919-1fd4-45c4-91d1-ef3d3e081139 +md""" +## Discussion +Pezeshki et al. described gradient starvation as a learning proclivity that can exhibit both negative and positive effects on neural network learning. Considering the 2D classification exmample discussed in this notebook, the effect of learning a decision boundary with a small margin, thus hardly robust, could be exemplified for stochastic gradient descent. Combining SGD with spectral decoupling regularization was greatly able to enlarge the margin, therefore yielding a more robust network that not only focussed on a single dominant feature. On the other hand, the potential negative effect of overfitting by precisely learning less dominant features, as well became evident through higher losses (SGD: 8.61% without vs. 13.37% with SD; ADAM: 2.00% without vs. 68.16%). +Contrarily to the results from Pezeshki et al., the here used neural network was perfectly able to learn curved decision boundaries featuring a large margin between the data points using ADAM without SD. This result could potentially be explained by the differing learning rates. While Pezeshki et al. trained their ADAM network with 0.0001 and 0.001, I trained all networks under the same conditions, hence both SGD and ADAM with a learning rate of 0.01. Despite the higher learning rate, ADAM showed no sign of overfitting without SD. However, with SD the overfitting was extreme but ADAM with SD was also a combination of methods not assessed by the researchers. +Regarding the same matter, my approach to apply SD not only to SGD but all other optimizers simultaneously offered only limited insight. First of all, the combination of WD (L2 regularization) and SD was expected to not work well since both methods are somewhat conflicting. ADAM performed well with SD but suffered from overfitting which can be conquered by a lower learning rate as demonstrated in fig. 16. Nontheless, the network with purportedly lower loss learns a boundary with a smaller margin. """ # ╔═╡ 92cca2d8-c2b6-4046-9b86-26c5c9d64ee3 -md"## Conclusion" +md""" +## Conclusion -# ╔═╡ 88b30b55-4f98-474a-aaca-a1197d2886fd -md"## Code" +""" # ╔═╡ c95db58c-9d47-4ef0-9ede-234d7a693f41 md""" @@ -131,11 +259,25 @@ Pizurica, A. (2020). E016330: Artificial Intelligence – Learning from examples Pizurica, A. (2020). E016330: Artificial Intelligence - Neural networks [PDF course notes]. Ufora. Russell, S., & Norvig, P. (2002). Artificial intelligence: a modern approach. + +Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. """ +# ╔═╡ 9441b599-cb0d-4892-8151-4c40125214be +md""" +### Appendix +""" + +# ╔═╡ c7331867-28b5-4a3f-b95c-926a3639dad9 + + # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" + +[compat] +PlutoUI = "~0.7.32" """ # ╔═╡ 00000000-0000-0000-0000-000000000002 @@ -145,23 +287,231 @@ PLUTO_MANIFEST_TOML_CONTENTS = """ julia_version = "1.7.0" manifest_format = "2.0" -[deps] +[[deps.AbstractPlutoDingetjes]] +deps = ["Pkg"] +git-tree-sha1 = "8eaf9f1b4921132a4cff3f36a1d9ba923b14a481" +uuid = "6e696c72-6542-2067-7265-42206c756150" +version = "1.1.4" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" + +[[deps.Artifacts]] +uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" + +[[deps.ColorTypes]] +deps = ["FixedPointNumbers", "Random"] +git-tree-sha1 = "024fe24d83e4a5bf5fc80501a314ce0d1aa35597" +uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f" 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+ +[[deps.Sockets]] +uuid = "6462fe0b-24de-5631-8697-dd941f90decc" + +[[deps.SparseArrays]] +deps = ["LinearAlgebra", "Random"] +uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" + +[[deps.Statistics]] +deps = ["LinearAlgebra", "SparseArrays"] +uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" + +[[deps.TOML]] +deps = ["Dates"] +uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76" + +[[deps.Tar]] +deps = ["ArgTools", "SHA"] +uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e" + +[[deps.Test]] +deps = ["InteractiveUtils", "Logging", "Random", "Serialization"] +uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40" + +[[deps.UUIDs]] +deps = ["Random", "SHA"] +uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4" + +[[deps.Unicode]] +uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5" + +[[deps.Zlib_jll]] +deps = ["Libdl"] +uuid = "83775a58-1f1d-513f-b197-d71354ab007a" + +[[deps.libblastrampoline_jll]] +deps = ["Artifacts", "Libdl", "OpenBLAS_jll"] +uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" + +[[deps.p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" """ # ╔═╡ Cell order: -# ╟─df218f5d-042d-4ccb-8005-56b2ec073869 +# ╟─5c563230-df6e-4f65-a106-a707d2454fe7 # ╟─8bd774c7-1acf-4beb-bc99-4c303882d262 # ╟─8f0cdbf1-0174-4d18-9072-2a8c5f015177 # ╟─6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b # ╟─512e6240-76e7-11ec-2511-07ffc4b74d27 -# ╟─93bc80a8-673d-457b-96d8-e1bb0b09682e -# ╟─66a3b5cb-a465-42fb-a8c8-404d574312d2 +# ╠═93bc80a8-673d-457b-96d8-e1bb0b09682e +# ╠═66a3b5cb-a465-42fb-a8c8-404d574312d2 # ╠═4a14831d-7736-4af5-875f-05ae6e3cf7c4 -# ╟─00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 +# ╠═00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 # ╠═ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 -# ╠═85591d79-dec2-45fa-a0a9-d95d84edda51 +# ╠═2f766a9c-de6e-4f95-aa73-aacb486aaf61 +# ╠═7f7ed965-fa08-4d6f-8908-dfc8876bdd02 +# ╟─88ddf822-622b-4d46-a0a4-a7d749c8bc48 +# ╟─884bc572-9daa-40ad-8d76-56ec58728f63 +# ╟─ac453412-0f71-45ee-aefb-0cc519467c6c +# ╟─5e282e1a-62fc-4911-9b7e-f2631eeb2669 +# ╠═f7bae71f-885a-404d-985b-92d612e1f3d6 +# ╠═2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f +# ╠═7460f609-c32b-4704-a361-71c98bb626cd +# ╠═87cc8172-d5cc-49ff-9a0e-fa734e1010c1 +# ╠═bd4f515c-1e41-48bc-8c36-233535a3022c +# ╠═2c740919-1fd4-45c4-91d1-ef3d3e081139 # ╠═92cca2d8-c2b6-4046-9b86-26c5c9d64ee3 -# ╠═88b30b55-4f98-474a-aaca-a1197d2886fd # ╟─c95db58c-9d47-4ef0-9ede-234d7a693f41 +# ╠═9441b599-cb0d-4892-8151-4c40125214be +# ╠═c7331867-28b5-4a3f-b95c-926a3639dad9 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002 From b4212d237f1f44d28180c51b6491ef9375975abc Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 21:38:13 +0100 Subject: [PATCH 35/49] update plots --- "plots/1e-3_ADAM_\316\2240.5_+_SD.png" | Bin 0 -> 165987 bytes "plots/1e-4_ADAM_\316\2240.5_+_SD.png" | Bin 0 -> 176038 bytes "plots/ADAM_\316\2240.5.png" | Bin 338011 -> 205313 bytes "plots/ADAM_\316\2240.5_+_SD.png" | Bin 250833 -> 169679 bytes "plots/ADAM_\316\2241.0.png" | Bin 254428 -> 171278 bytes "plots/ADAM_\316\2241.0_+_SD.png" | Bin 0 -> 124480 bytes "plots/GD_\316\2240.5.png" | Bin 320598 -> 174849 bytes "plots/GD_\316\2240.5_+_SD.png" | Bin 241641 -> 156889 bytes "plots/GD_\316\2240.5_+_SD_10000.png" | Bin 0 -> 178474 bytes "plots/GD_\316\2241.0.png" | Bin 323438 -> 181235 bytes "plots/GD_\316\2241.0_+_SD.png" | Bin 0 -> 164387 bytes "plots/SGD_\316\2240.5.png" | Bin 378034 -> 217596 bytes "plots/SGD_\316\2240.5_+_SD.png" | Bin 234635 -> 180443 bytes "plots/SGD_\316\2241.0.png" | Bin 342546 -> 190246 bytes "plots/SGD_\316\2241.0_+_SD.png" | Bin 0 -> 159539 bytes "plots/WD_\316\2240.5.png" | Bin 315291 -> 157168 bytes "plots/WD_\316\2240.5_+_SD.png" | Bin 241611 -> 154612 bytes "plots/WD_\316\2241.0.png" | Bin 323870 -> 161952 bytes "plots/WD_\316\2241.0_+_SD.png" | Bin 0 -> 158569 bytes plots/figure_1.png | Bin 0 -> 54746 bytes src/gradientstarvation.jl 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b/src/gradientstarvation.jl index 9aac21a0..f3db9b31 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -10,26 +10,44 @@ using DataStructures using PlotlyJS include("experiments.jl") +include("figures.jl") include("structs.jl") -experiments = OrderedDict( - "GD Δ1.0" => Experiment("GD", 1.0, false), - "GD Δ0.5" => Experiment("GD", 0.5, false), - "GD Δ0.5 + SD" => Experiment("GD", 0.5, true), - - "SGD Δ1.0" => Experiment("SGD", 1.0, false), - "SGD Δ0.5" => Experiment("SGD", 0.5, false), - "SGD Δ0.5 + SD" => Experiment("SGD", 0.5, true), - - "WD Δ1.0" => Experiment("WD", 1.0, false), - "WD Δ0.5" => Experiment("WD", 0.5, false), - "WD Δ0.5 + SD" => Experiment("WD", 0.5, true), - - "ADAM Δ1.0" => Experiment("ADAM", 1.0, false), - "ADAM Δ0.5" => Experiment("ADAM", 0.5, false), - "ADAM Δ0.5 + SD" => Experiment("ADAM", 0.5, true), -) - -Experiments.run_experiments(experiments) +# figure 1 +Figures.create_figure_1("medium") + +# figures 2-13 +# experiments = OrderedDict( +# "GD Δ0.5" => Experiment("GD", 0.5, false), +# "GD Δ1.0" => Experiment("GD", 1.0, false), +# "GD Δ0.5 + SD" => Experiment("GD", 0.5, true), +# "GD Δ1.0 + SD" => Experiment("GD", 1.0, true), + +# "SGD Δ0.5" => Experiment("SGD", 0.5, false), +# "SGD Δ1.0" => Experiment("SGD", 1.0, false), +# "SGD Δ0.5 + SD" => Experiment("SGD", 0.5, true), +# "SGD Δ1.0 + SD" => Experiment("SGD", 1.0, true), + +# "WD Δ0.5" => Experiment("WD", 0.5, false), +# "WD Δ1.0" => Experiment("WD", 1.0, false), +# "WD Δ0.5 + SD" => Experiment("WD", 0.5, true), +# "WD Δ1.0 + SD" => Experiment("WD", 1.0, true), + +# "ADAM Δ1.0" => Experiment("ADAM", 1.0, false), +# "ADAM Δ0.5" => Experiment("ADAM", 0.5, false), +# "ADAM Δ0.5 + SD" => Experiment("ADAM", 0.5, true), +# "ADAM Δ1.0 + SD" => Experiment("ADAM", 1.0, true), +# ) +# Experiments.run_experiments(experiments, show_plots = false, random_moons = false) +# experiments_2 = OrderedDict( +# "GD Δ0.5 + SD 10000" => Experiment("GD", 0.5, true), +# ) +# Experiments.run_experiments(experiments_2, show_plots = false, random_moons = false, epochs = 10000) + +# experiments_3 = OrderedDict( +# "ADAM Δ0.5 + SD" => Experiment("ADAM", 0.5, true), +# ) +# Experiments.run_experiments(experiments_3, show_plots = false, random_moons = false, learning_rate = 1e-4, prefix = "1e-4") +# Experiments.run_experiments(experiments_3, show_plots = false, random_moons = false, learning_rate = 1e-3, prefix = "1e-3") end From 95060424da1cb0c6e1ae37fb32ef100658ee90bb Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 22:14:10 +0100 Subject: [PATCH 36/49] manually edit figure 1 --- plots/figure_1.png | Bin 54746 -> 44030 bytes plots/figure_1_edited.png | Bin 0 -> 44030 bytes src/plotlib.jl | 1 + 3 files changed, 1 insertion(+) create mode 100644 plots/figure_1_edited.png diff --git a/plots/figure_1.png b/plots/figure_1.png index 80d8cd4d566a3455d51417509e75c601177e7997..1e17b27d9420f56fad62f7029f8ff58526d5a07c 100644 GIT binary patch literal 44030 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z7VFDw;l^9m{nFXyU;@Z;9hZCk3q&BBY+uCjuU^k0Ze!T5+OM${NyDl2h};@;CTPw} z$J|NFwd)s-wmAVm?Hl*GUe}sWCsX~?>1lbg21;f n&ZS%R;{W>t`Tz5&oKmMO`C(l}7u8P%^m){jwUjCpEJFVWn-^@* literal 0 HcmV?d00001 diff --git a/src/plotlib.jl b/src/plotlib.jl index 5ec1e27a..42e5f65f 100644 --- a/src/plotlib.jl +++ b/src/plotlib.jl @@ -40,6 +40,7 @@ function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loade ), layout) # adjust common layout with individual titles + # does not work for some reason :/ PlotlyJS.relayout!(p_train, title = attr(text = train_title, font = attr(size = title_font_size))) PlotlyJS.relayout!(p_test, title = attr(text = test_title, font = attr(size = title_font_size))) From 3bf644b1870bdc0ca87e78e1c7cebec00fe8320b Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 23:09:27 +0100 Subject: [PATCH 37/49] update notebook --- notebook/gradientstarvation.jl | 120 +++++++++++++++++++-------------- 1 file changed, 71 insertions(+), 49 deletions(-) diff --git a/notebook/gradientstarvation.jl b/notebook/gradientstarvation.jl index c7111da9..e2c2fe2f 100644 --- a/notebook/gradientstarvation.jl +++ b/notebook/gradientstarvation.jl @@ -95,6 +95,20 @@ Despite its popularity, gradient descent learning in NNs can come with some unwa Another not yet published but currently in preprint described observation is the effect of [**_Gradient Starvation_ (GS)**](https://arxiv.org/abs/2011.09468). Pezeshki et al. coined and formalized this phenomenon of over-parameterized NNs in their work. According to them, _Gradient Starvation_ is a predisposition of NNs learned by cross-entropy, which occurs when the loss minimization is driven by only a feature subset regardless of the presence of other meaningful features. Thus, the NN becomes biased towards superficial features. Beyond demonstrating the effects of GS, Pezeshki et al. introduce **_Spectral Decoupling_ (SD)** regularization as a means to counteract gradient starvation. """ +# ╔═╡ 60905936-9373-44de-96e6-cb5d6dd72d33 +md""" +### Spectral Decoupling +Proceeding from the popular "ridge-regularized cross entropy": + +$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot log(1 + exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{θ}||^2$ + +with $\boldsymbol{Y}$ the diagonal matrix of $y$ and $\boldsymbol{θ}$ the concatenation of all layer's weights, SD is a variation where the L2 regularization term is replaced by a penalty on the raw predictions $\boldsymbol{ŷ}$: + +$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot log(1 + exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{ŷ}||^2$ + +This, as stated by the Pezeshki et al., uncouples the learning of multiple features. According to their work, a feature trained close to its optimum inhibits (starves) the training of other features which is thus prevented by SD. +""" + # ╔═╡ 00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 md""" ### Objective @@ -111,29 +125,28 @@ The first topology allows to draw a line by a very small margin between the two # ╔═╡ 2f766a9c-de6e-4f95-aa73-aacb486aaf61 md""" -$(LocalResource("../plots/figure_1.png")) +$(LocalResource("../plots/figure_1_edited.png")) Figure 1: Two moons topologies; one linear separable (offset Δ1.0), the other separable by a curved boundary (offset Δ0.5). """ # ╔═╡ 7f7ed965-fa08-4d6f-8908-dfc8876bdd02 md""" -The idea of Pezeshki et al. was that in the linear separable case the NN is not encouraged to learn a curved decision boundary because the loss becomes neglibile by only discriminating the two moons along one axis, therefore learning only one feature and neglecting the other. In contrast to that, the interleaved moons can only be separated by learning both features which better accommodates the data structure in general. -They observed that a different choice of regularization and optimization methods like Weight Decay (WD) and Adaptive Moment Estimation (ADAM) among others are not able to -stimulate the NN to learn a curved decision boundary. -Moreover, Pezeshki et al. explain that circumstance by cross-entropy loss learning which starves the gradients from the other feature, only reinforcing the superficial one. +The reasoning of the researchers was that in the linear separable case the NN is not encouraged to learn a curved decision boundary because the loss becomes negligible by only discriminating the two moons along one axis, therefore learning only one feature and neglecting the other. In contrast to that, the interleaved moons can only be separated by learning both features which better accommodates the curvatures of the data structure in general. +They observed that a different choice of regularization and optimization methods like weight decay (WD) and _Adaptive Moment Estimation_ (ADAM), among others, are not able to stimulate the NN to learn a curved decision boundary. +Moreover, Pezeshki et al. explained that circumstance by cross-entropy loss learning which starves the gradients from the other feature, only reinforcing the superficial one. #### Repercussions of Gradient Starvation -Why does the shape of the decision boundary even matter as long as it discriminates the data points confidently? As stated by the authors, GS results in a very small distance between the data points and the boundary which translates into a lack of robustness when generalizing to new data. Contrarily, GS could also have the positive effect of the prevention of overfitting by not learning non-dominant features. Pezeshki et al. make the comparison to the aformentioned Ockham's razor: GS leads to simpler decision boundaries and hence NNs that generalize better, thus being a form of implicit regularization. +Why does the shape of the decision boundary even matter as long as it discriminates the data points confidently? As stated by the authors, GS results in a very small distance between the data points and the boundary which translates into a lack of robustness when generalizing to new data. Contrarily, GS could also have a positive effect in terms of preventing overfitting by not learning non-dominant features. The authors made the comparison to the aformentioned Ockham's razor: GS leads to simpler decision boundaries and hence NNs that generalize better, thus GS being a form of implicit regularization. #### Implementation -The implementation is based on [Flux](https://fluxml.ai/), a ML framework for Julia. The NN consists of two hidden layers with 500 nodes each. Rectified Linear Unit (ReLu) is used as an activation function and learning is facilitated with cross-entropy loss for 1000 epochs on a data set with 300 observations (150 blue and 150 red points). In contrast to Pezeshki et al., I opted to train in batches of 50 (6 batches in total) since this is a common practice for training NNs. +The implementation is based on [Flux](https://fluxml.ai/), a ML framework for Julia. The NN consists of two hidden layers with 500 nodes each. Rectified Linear Unit (ReLu) is used as an activation function and learning is facilitated by cross-entropy loss for 1000 epochs on a data set with 300 observations (150 blue and 150 red points). In contrast to Pezeshki et al., I've opted to train in batches of 50 (6 batches in total) since this is a common practice for training NNs. All networks are trained with gradient descent but with different regularization and optimization approaches. This yields the following different setups: -- Gradient descent (GD; no regularization) +- Gradient descent (GD; no regularization/optimization) - Gradient descent with momentum (stochastic gradient descent (SGD); no regularization) -- Gradient descent with weight decay (WD; λ⋅L2 regularization) +- Gradient descent with weight decay (WD; L2 regularization) - Gradient descent with _Adaptive Moment Estimation_ (ADAM; no regularization) -Different sets of hyper parameters were evaluated. However, since the main objective is to reproduce the research conducted by Pezeshki et al., their set of hyperparameters were used: +Different sets of hyperparameters were evaluated while tinkering with the NNs. However, since the main objective is to reproduce the research conducted by Pezeshki et al., their set of hyperparameters were used: - learning rate = 0.01 - weight decay coefficient = 0.01 - spectral decoupling coefficient = 0.003 @@ -148,21 +161,21 @@ md""" md""" ##### Linear separable case $(LocalResource("../plots/GD_Δ1.0.png")) -Figure 2: Gradient descent (GD) on the moons with offset = Δ1.0. The decision boundary is almost linear and the loss continuously falls while training. Accuracy reaches 100%. +Figure 2: GD, moon offset = Δ1.0. The decision boundary is almost linear and the loss continuously falls while training. Accuracy reaches 100%. $(LocalResource("../plots/SGD_Δ1.0.png")) -Figure 3: Stochastic gradient descent (SGD) on the moons with offset = Δ1.0. The decision boundary is similar to the one learned with GD but equally distant from both classes. Loss declines quicker and reaches an overall lower level compared to GD. Accuracy reaches 100% after less training. +Figure 3: SGD, moon offset = Δ1.0. The decision boundary is similar to the one learned with GD but equally distant from both classes. Loss declines quicker and reaches an overall lower level compared to GD. Accuracy reaches 100% after less training. $(LocalResource("../plots/WD_Δ1.0.png")) -Figure 4: Gradient descent with weight decay (WD) on the moons with offset = Δ1.0. Decay coefficient = 0.01. The boundary is similar to GD without WD. +Figure 4: WD, moon offset = Δ1.0, WD coefficient = 0.01. The boundary is similar to GD without WD. $(LocalResource("../plots/ADAM_Δ1.0.png")) -Figure 5: Gradient descent with Adaptive Moment Estimation (ADAM) on the moons with offset = Δ1.0. The decision boundary is slightly curved around the data points closest to the boundary. Loss converges to zero almost immediately while full accuracy is attained. +Figure 5: ADAM, moon offset = Δ1.0. The decision boundary is slightly curved around the data points closest to the boundary. Loss converges to zero almost immediately while full accuracy is attained. """ # ╔═╡ ac453412-0f71-45ee-aefb-0cc519467c6c md""" -As we can see from figures 2-5, the decision boundaries of all networks exhibit some specificites. The GD network learned an almost linear decision boundary which appears to be closer to the blue-labeled moon. The boundary looks similar with SGD, but the distance to both moons is equal. Only the network with ADAM optimization learns a slight curvature around the points closest to the boundary. Furthermore, the ADAM classification learned the fastest as can be seen from the rapid decrease in loss and increase in accuracy. At the same time, the contour lines all closely condense around the decision boundary which indicates that the ADAM network is most confident in discriminating both classes. +As we can see from figures 2 to 5, the decision boundaries of all networks exhibit some specificities. The GD network learned an almost linear decision boundary which appears to be closer to the blue-labeled moon. The boundary looks similar with SGD, but its distance to both moons is equal. Only the network with ADAM optimization learns a slight curvature around the points closest to the boundary. Furthermore, the ADAM classification learned the fastest as can be seen from the rapid decrease in loss and increase in accuracy. At the same time, the contour lines all closely condense around the decision boundary which indicates that the ADAM network is most confident in discriminating both classes. """ # ╔═╡ 5e282e1a-62fc-4911-9b7e-f2631eeb2669 @@ -170,21 +183,21 @@ md""" ##### Linear inseparable case $(LocalResource("../plots/GD_Δ0.5.png")) -Figure 6: GD on the moons with offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes. +Figure 6: GD, moon offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes within the 1000 training iterations. $(LocalResource("../plots/SGD_Δ0.5.png")) -Figure 7: SGD on the moons with offset = Δ0.5. The network learns a curved decision boundary which is very close to the interleaving data points. Loss becomes negligible during learning. +Figure 7: SGD, moon offset = Δ0.5. The network learns a curved decision boundary which is very close to the interleaving data points. Loss becomes negligible during learning. $(LocalResource("../plots/WD_Δ0.5.png")) -Figure 8: WD on the moons with offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes. Again, similar result compared to GD. +Figure 8: WD, moon offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes. Again, similar result compared to GD. $(LocalResource("../plots/ADAM_Δ0.5.png")) -Figure 9: ADAM on the moons with offset = Δ0.5. The network learns a curved decision boundary. Loss fluctuates especially during the early iterations of training. +Figure 9: ADAM, moon offset = Δ0.5. The network learns a curved decision boundary. Loss fluctuates especially during the early iterations of training. """ # ╔═╡ f7bae71f-885a-404d-985b-92d612e1f3d6 md""" -SGD and ADAM are able to learn a curved decision boundary (fig. 7 and 9) while GD and WD fail to do so (fig. 6 and 8). Moreover, the decision boundary learned by ADAM better resembles the data structure. +SGD and ADAM are able to learn a curved decision boundary (fig. 7 and 9) while GD and WD fail to do so (fig. 6 and 8). Moreover, the decision boundary learned by ADAM better resembles the data structure. The contours of the ADAM network show a very strong confidence on the networks classification. This contrasts the results from the GS paper, where ADAM was not able to learn a decision boundary as curved and well-fitting the data structure as shown in fig. 9. """ # ╔═╡ 2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f @@ -192,21 +205,21 @@ md""" #### Spectral Decoupling $(LocalResource("../plots/GD_Δ0.5_+_SD.png")) -Figure 10: GD with SD on the moons with offset = Δ0.5. No curved decision boundary is learned but the loss also did not converge. +Figure 10: GD with SD, moon offset = Δ0.5. No curved decision boundary is learned but the loss also did not converge. $(LocalResource("../plots/SGD_Δ0.5_+_SD.png")) -Figure 11: SGD with SD on the moons with offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the moon data structure. Loss converges to ~15%. +Figure 11: SGD with SD, moon offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the moon data structure and shows invaginations at the data points of both moons closest to the boundary. Loss converges to ~15%. $(LocalResource("../plots/WD_Δ0.5_+_SD.png")) -Figure 12: WD with SD on the moons with offset = Δ0.5 fails to separate the two data classes. +Figure 12: WD with SD, moon offset = Δ0.5. NN fails to separate the two data classes. Similar to GD with SD, loss could still be lowered with further training. $(LocalResource("../plots/ADAM_Δ0.5_+_SD.png")) -Figure 13: ADAM with SD on the moons with offset = Δ0.5 learns a curved decision boundary which greatly resembles the data structure. Loss heavily oscillates and steadily increases with longer training. +Figure 13: ADAM with SD, moon offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the data structure. Loss heavily oscillates and steadily increases with longer training. Interestingly, all data points of each respective class belong to the same contour. """ # ╔═╡ 7460f609-c32b-4704-a361-71c98bb626cd md""" -Spectral decoupling, originally applied as a regularization by Pezeshki et al. only with SGD, is able to yield curved decision boundaries which greatly resemble the data structure of the two moons (SGD and ADAM). While SGD learns a more angular boundary with low loss, ADAM learns a smoother one but exhibits a high loss with strong oscillations. Interestingly, ADAM learns to group all data points of a class within the same contour (fig. 13). GD and WD do not profit from SD regularization and don't learn a curved boundary. +SD, originally applied as a regularization by Pezeshki et al. only with SGD, is able to yield curved decision boundaries which greatly resemble the data structure of the two moons in combination with SGD and ADAM optimization. While SGD learns a more angular boundary with low loss, ADAM learns a smoother one but exhibits a high loss with strong oscillations. Interestingly, ADAM learns to group all data points of a class within the same contour (fig. 13). GD and WD do not profit from SD regularization and don't learn a curved boundary. However, their loss profiles suggest improvement with further learning. """ # ╔═╡ 87cc8172-d5cc-49ff-9a0e-fa734e1010c1 @@ -215,38 +228,45 @@ md""" Given that the loss did not converge for GD + SD after 1000 epochs, the network was trained for 10000 iterations. $(LocalResource("../plots/GD_Δ0.5_+_SD_10000.png")) -Figure 14: GD with SD on the moons with offset = Δ0.5 trained 10 times longer (10000 iterations). The learned curved decision boundary resembles the result obtained with SGD + SD. +Figure 14: GD with SD, moon offset = Δ0.5. NN was trained 10 times longer (10000 iterations). The learned curved decision boundary resembles the result obtained with SGD + SD. To conquer the overfitting observed with ADAM + SD lower learning rates were evaluated. $(LocalResource("../plots/1e-3_ADAM_Δ0.5_+_SD.png")) -Figure 15: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.001 still suffers from severe overfitting. +Figure 15: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.001 still suffers from severe overfitting. Nonetheless, oscillations in loss are dampened. $(LocalResource("../plots/1e-4_ADAM_Δ0.5_+_SD.png")) -Figure 16: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.0001 shows lower loss but a decision boundary with a smaller margin. +Figure 16: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.0001 shows lower loss compared to 0.001 but a decision boundary with a smaller margin. + +The additional experiments show that GD with SD performs equally well compared to SGD with SD but needs much more training for that. +The overfitting problem with ADAM + SD can be successfully conquered with lower learning rates. However, the shape of the decision boundary suffers with learning rates as low as 0.0001. From all three ADAM + SD experiments, the one learning at a rate of 0.001 performed best. """ # ╔═╡ bd4f515c-1e41-48bc-8c36-233535a3022c md""" -### Summary +### Plot Summary This section summarizes the decision boundaries of all figures from above for easier visual comparison in one plot. -Figure 14: +Figure 17: """ # ╔═╡ 2c740919-1fd4-45c4-91d1-ef3d3e081139 md""" ## Discussion -Pezeshki et al. described gradient starvation as a learning proclivity that can exhibit both negative and positive effects on neural network learning. Considering the 2D classification exmample discussed in this notebook, the effect of learning a decision boundary with a small margin, thus hardly robust, could be exemplified for stochastic gradient descent. Combining SGD with spectral decoupling regularization was greatly able to enlarge the margin, therefore yielding a more robust network that not only focussed on a single dominant feature. On the other hand, the potential negative effect of overfitting by precisely learning less dominant features, as well became evident through higher losses (SGD: 8.61% without vs. 13.37% with SD; ADAM: 2.00% without vs. 68.16%). -Contrarily to the results from Pezeshki et al., the here used neural network was perfectly able to learn curved decision boundaries featuring a large margin between the data points using ADAM without SD. This result could potentially be explained by the differing learning rates. While Pezeshki et al. trained their ADAM network with 0.0001 and 0.001, I trained all networks under the same conditions, hence both SGD and ADAM with a learning rate of 0.01. Despite the higher learning rate, ADAM showed no sign of overfitting without SD. However, with SD the overfitting was extreme but ADAM with SD was also a combination of methods not assessed by the researchers. -Regarding the same matter, my approach to apply SD not only to SGD but all other optimizers simultaneously offered only limited insight. First of all, the combination of WD (L2 regularization) and SD was expected to not work well since both methods are somewhat conflicting. ADAM performed well with SD but suffered from overfitting which can be conquered by a lower learning rate as demonstrated in fig. 16. Nontheless, the network with purportedly lower loss learns a boundary with a smaller margin. +Pezeshki et al. described gradient starvation as a learning proclivity that can exhibit both negative and positive effects on neural network learning. Considering the 2D classification task discussed in this notebook, the effect of learning a decision boundary with a small margin, thus supposedly hardly robust, could be exemplified for stochastic gradient descent. Combining SGD with spectral decoupling regularization was greatly able to enlarge the margin, therefore yielding a more robust network that not only focussed on a single dominant feature. On the other hand, the potential negative effect of overfitting by precisely learning less dominant features as well became evident through higher losses (SGD: 8.61% without vs. 13.37% with SD; ADAM: 2.00% without vs. 68.16%). +Contrarily to the results from Pezeshki et al., the here used neural network was perfectly able to learn curved decision boundaries featuring a large margin between the data points using ADAM without SD. While Pezeshki et al. trained their ADAM network with rates of 0.0001 and 0.001, I trained all networks under the same conditions, hence both SGD and ADAM with a learning rate of 0.01. Despite the higher learning rate, ADAM showed no sign of overfitting without SD. However, with SD the overfitting was extreme but ADAM coupled with SD regularization was also a combination of methods not assessed by the researchers. +Regarding the same matter, my approach to apply SD not only to SGD but all other optimizers simultaneously offered limited insight. First of all, the combination of WD (L2 regularization) and SD was expected to not work well since both methods are somewhat conflicting. ADAM performed well with SD but suffered from overfitting which can be conquered by a lower learning rate as demonstrated. Nonetheless, the network with purportedly lower loss learns a boundary with a smaller margin. + +All in all, the results of Pezeshki et al. could be reproduced with the exception of the network using ADAM, which was able to learn decision boundaries with similar characteristics as the ones learned by SGD with SD. + +Finally, it must be mentioned that Pezeshki et al. not only exemplified GS on the rather synthetical moon classification task but showed GS ramifications and the potential of SD as a regularization technique on more complex, well-established datasets like CIFAR, colored MNIST and CelebA which was sadly out of scope of this project. """ -# ╔═╡ 92cca2d8-c2b6-4046-9b86-26c5c9d64ee3 +# ╔═╡ 21854d7b-13ce-477c-bc61-1cb15536e24f md""" -## Conclusion - +### Conclusion +Gradient starvation is another relevant side effect of gradient descent-learned neural networks that may be worth the attention in the optimization process of NNs. However, if GS is a blessing or a curse depends on the use case. Starved models, i.e. models concentrating on superficial features during learning, can be less prone to overfitting while adressing GS promises to increase robustness. """ # ╔═╡ c95db58c-9d47-4ef0-9ede-234d7a693f41 @@ -266,6 +286,7 @@ Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXi # ╔═╡ 9441b599-cb0d-4892-8151-4c40125214be md""" ### Appendix +To foster the understanding of interested reader on the mentioned regularization and optimization approaches, """ # ╔═╡ c7331867-28b5-4a3f-b95c-926a3639dad9 @@ -492,24 +513,25 @@ uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" # ╟─8f0cdbf1-0174-4d18-9072-2a8c5f015177 # ╟─6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b # ╟─512e6240-76e7-11ec-2511-07ffc4b74d27 -# ╠═93bc80a8-673d-457b-96d8-e1bb0b09682e -# ╠═66a3b5cb-a465-42fb-a8c8-404d574312d2 -# ╠═4a14831d-7736-4af5-875f-05ae6e3cf7c4 -# ╠═00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 -# ╠═ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 -# ╠═2f766a9c-de6e-4f95-aa73-aacb486aaf61 -# ╠═7f7ed965-fa08-4d6f-8908-dfc8876bdd02 +# ╟─93bc80a8-673d-457b-96d8-e1bb0b09682e +# ╟─66a3b5cb-a465-42fb-a8c8-404d574312d2 +# ╟─4a14831d-7736-4af5-875f-05ae6e3cf7c4 +# ╟─60905936-9373-44de-96e6-cb5d6dd72d33 +# ╟─00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 +# ╟─ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 +# ╟─2f766a9c-de6e-4f95-aa73-aacb486aaf61 +# ╟─7f7ed965-fa08-4d6f-8908-dfc8876bdd02 # ╟─88ddf822-622b-4d46-a0a4-a7d749c8bc48 # ╟─884bc572-9daa-40ad-8d76-56ec58728f63 # ╟─ac453412-0f71-45ee-aefb-0cc519467c6c # ╟─5e282e1a-62fc-4911-9b7e-f2631eeb2669 -# ╠═f7bae71f-885a-404d-985b-92d612e1f3d6 -# ╠═2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f -# ╠═7460f609-c32b-4704-a361-71c98bb626cd -# ╠═87cc8172-d5cc-49ff-9a0e-fa734e1010c1 +# ╟─f7bae71f-885a-404d-985b-92d612e1f3d6 +# ╟─2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f +# ╟─7460f609-c32b-4704-a361-71c98bb626cd +# ╟─87cc8172-d5cc-49ff-9a0e-fa734e1010c1 # ╠═bd4f515c-1e41-48bc-8c36-233535a3022c -# ╠═2c740919-1fd4-45c4-91d1-ef3d3e081139 -# ╠═92cca2d8-c2b6-4046-9b86-26c5c9d64ee3 +# ╟─2c740919-1fd4-45c4-91d1-ef3d3e081139 +# ╟─21854d7b-13ce-477c-bc61-1cb15536e24f # ╟─c95db58c-9d47-4ef0-9ede-234d7a693f41 # ╠═9441b599-cb0d-4892-8151-4c40125214be # ╠═c7331867-28b5-4a3f-b95c-926a3639dad9 From c34df546b1d7c7fc9a159c23960fbe107c88ddc8 Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 23:11:02 +0100 Subject: [PATCH 38/49] fix 10000 epochs graph --- "plots/GD_\316\2240.5_+_SD_10000.png" | Bin 178474 -> 180181 bytes src/experiments.jl | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git "a/plots/GD_\316\2240.5_+_SD_10000.png" "b/plots/GD_\316\2240.5_+_SD_10000.png" index fb2cf7cb027abf5e2174753efa677711999f11a8..1b2e6bf4298e293568b1645f258de02fe0221ab9 100644 GIT binary patch literal 180181 zcmeFYWmjBX(=FV%1=kSVp>YWA65Iog)3^i;8r(Iw1b5fQ8~0!d1P>P6Ex5z!>$>k} zjPsuJ2i_0o)9&4C^!`@EZ%Wml zSO5SjfP#$VM=#^!E|hfbxwhAyn*-~KE3vBr5FDI_EfHdBAX{{>#IGikD~8=a+oiU( zTy|!>CzS6QzK0XyG6xrG7BkDAIWP_ImfrG|oxNVxcFl_&9WR#se2WwAnHv$2!PDV? 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b/src/experiments.jl @@ -31,7 +31,7 @@ function run_experiments(experiments; prefix = "", random_moons = true, show_plo # create plots p1 = PlotLib.plot_decision_boundary(train_loader, model, title = name) - p2 = PlotLib.plot_loss_and_accuracy(stats["loss"], stats["accuracy"]) + p2 = PlotLib.plot_loss_and_accuracy(stats["loss"], stats["accuracy"]; args...) if plot_type == "both" # combine subplots From a4d8351c1d847a376dae6e3640092c851e149e0e Mon Sep 17 00:00:00 2001 From: Peter Date: Sun, 30 Jan 2022 23:50:48 +0100 Subject: [PATCH 39/49] update readme --- README.md | 11 +++++++++-- notebook/gradientstarvation.jl | 21 --------------------- 2 files changed, 9 insertions(+), 23 deletions(-) diff --git a/README.md b/README.md index bbf84e3e..b1e1bbfd 100644 --- a/README.md +++ b/README.md @@ -1,3 +1,10 @@ -# Gradient Starvation +# Gradient Starvation 🍴💀 -This project is still under construction. \ No newline at end of file +Welcome to my STMO project! + +I wanted to explore the findings of Pezeshki et al. presented in their publication [Gradient Starvation: A Learning Proclivity in Neural Networks](https://arxiv.org/abs/2011.09468). +The goal was to explain the problem of gradient starvation and to investigate its effects and a proposed remedy on a simple 2D classification task. + +The notebook and the code of my project are separated! You can find them in their respective subfolders `notebook` and `src`. + +All experiments conducted, including the related figures, can be simply repeated and re-created by running `include("src\\gradientstarvation.jl")` from a Julia REPL. \ No newline at end of file diff --git a/notebook/gradientstarvation.jl b/notebook/gradientstarvation.jl index e2c2fe2f..7728c762 100644 --- a/notebook/gradientstarvation.jl +++ b/notebook/gradientstarvation.jl @@ -242,15 +242,6 @@ The additional experiments show that GD with SD performs equally well compared t The overfitting problem with ADAM + SD can be successfully conquered with lower learning rates. However, the shape of the decision boundary suffers with learning rates as low as 0.0001. From all three ADAM + SD experiments, the one learning at a rate of 0.001 performed best. """ -# ╔═╡ bd4f515c-1e41-48bc-8c36-233535a3022c -md""" -### Plot Summary -This section summarizes the decision boundaries of all figures from above for easier visual comparison in one plot. - - -Figure 17: -""" - # ╔═╡ 2c740919-1fd4-45c4-91d1-ef3d3e081139 md""" ## Discussion @@ -283,15 +274,6 @@ Russell, S., & Norvig, P. (2002). Artificial intelligence: a modern approach. Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. """ -# ╔═╡ 9441b599-cb0d-4892-8151-4c40125214be -md""" -### Appendix -To foster the understanding of interested reader on the mentioned regularization and optimization approaches, -""" - -# ╔═╡ c7331867-28b5-4a3f-b95c-926a3639dad9 - - # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] @@ -529,11 +511,8 @@ uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" # ╟─2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f # ╟─7460f609-c32b-4704-a361-71c98bb626cd # ╟─87cc8172-d5cc-49ff-9a0e-fa734e1010c1 -# ╠═bd4f515c-1e41-48bc-8c36-233535a3022c # ╟─2c740919-1fd4-45c4-91d1-ef3d3e081139 # ╟─21854d7b-13ce-477c-bc61-1cb15536e24f # ╟─c95db58c-9d47-4ef0-9ede-234d7a693f41 -# ╠═9441b599-cb0d-4892-8151-4c40125214be -# ╠═c7331867-28b5-4a3f-b95c-926a3639dad9 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002 From edc9504301075413f8a5050c61835ce7df1e088d Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 00:03:06 +0100 Subject: [PATCH 40/49] final version of main module --- src/gradientstarvation.jl | 66 ++++++++++++++++++++------------------- 1 file changed, 34 insertions(+), 32 deletions(-) diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index f3db9b31..f4d09213 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -17,37 +17,39 @@ include("structs.jl") Figures.create_figure_1("medium") # figures 2-13 -# experiments = OrderedDict( -# "GD Δ0.5" => Experiment("GD", 0.5, false), -# "GD Δ1.0" => Experiment("GD", 1.0, false), -# "GD Δ0.5 + SD" => Experiment("GD", 0.5, true), -# "GD Δ1.0 + SD" => Experiment("GD", 1.0, true), - -# "SGD Δ0.5" => Experiment("SGD", 0.5, false), -# "SGD Δ1.0" => Experiment("SGD", 1.0, false), -# "SGD Δ0.5 + SD" => Experiment("SGD", 0.5, true), -# "SGD Δ1.0 + SD" => Experiment("SGD", 1.0, true), - -# "WD Δ0.5" => Experiment("WD", 0.5, false), -# "WD Δ1.0" => Experiment("WD", 1.0, false), -# "WD Δ0.5 + SD" => Experiment("WD", 0.5, true), -# "WD Δ1.0 + SD" => Experiment("WD", 1.0, true), - -# "ADAM Δ1.0" => Experiment("ADAM", 1.0, false), -# "ADAM Δ0.5" => Experiment("ADAM", 0.5, false), -# "ADAM Δ0.5 + SD" => Experiment("ADAM", 0.5, true), -# "ADAM Δ1.0 + SD" => Experiment("ADAM", 1.0, true), -# ) -# Experiments.run_experiments(experiments, show_plots = false, random_moons = false) -# experiments_2 = OrderedDict( -# "GD Δ0.5 + SD 10000" => Experiment("GD", 0.5, true), -# ) -# Experiments.run_experiments(experiments_2, show_plots = false, random_moons = false, epochs = 10000) - -# experiments_3 = OrderedDict( -# "ADAM Δ0.5 + SD" => Experiment("ADAM", 0.5, true), -# ) -# Experiments.run_experiments(experiments_3, show_plots = false, random_moons = false, learning_rate = 1e-4, prefix = "1e-4") -# Experiments.run_experiments(experiments_3, show_plots = false, random_moons = false, learning_rate = 1e-3, prefix = "1e-3") +experiments = OrderedDict( + "GD Δ0.5" => Experiment("GD", 0.5, false), + "GD Δ1.0" => Experiment("GD", 1.0, false), + "GD Δ0.5 + SD" => Experiment("GD", 0.5, true), + "GD Δ1.0 + SD" => Experiment("GD", 1.0, true), + + "SGD Δ0.5" => Experiment("SGD", 0.5, false), + "SGD Δ1.0" => Experiment("SGD", 1.0, false), + "SGD Δ0.5 + SD" => Experiment("SGD", 0.5, true), + "SGD Δ1.0 + SD" => Experiment("SGD", 1.0, true), + + "WD Δ0.5" => Experiment("WD", 0.5, false), + "WD Δ1.0" => Experiment("WD", 1.0, false), + "WD Δ0.5 + SD" => Experiment("WD", 0.5, true), + "WD Δ1.0 + SD" => Experiment("WD", 1.0, true), + + "ADAM Δ1.0" => Experiment("ADAM", 1.0, false), + "ADAM Δ0.5" => Experiment("ADAM", 0.5, false), + "ADAM Δ0.5 + SD" => Experiment("ADAM", 0.5, true), + "ADAM Δ1.0 + SD" => Experiment("ADAM", 1.0, true), +) +Experiments.run_experiments(experiments, show_plots = false, random_moons = false) + +# additional experiments +# figure 14 +Experiments.run_experiments( + OrderedDict("GD Δ0.5 + SD 10000" => Experiment("GD", 0.5, true)), + show_plots = false, random_moons = false, epochs = 10000 +) + +# figure 15, 16 +add_exp_adam = OrderedDict("ADAM Δ0.5 + SD" => Experiment("ADAM", 0.5, true)) +Experiments.run_experiments(add_exp_adam, show_plots = false, random_moons = false, learning_rate = 1e-4, prefix = "1e-4") +Experiments.run_experiments(add_exp_adam, show_plots = false, random_moons = false, learning_rate = 1e-3, prefix = "1e-3") end From 255010173d902201eba84ba5001347cc62516779 Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 00:31:46 +0100 Subject: [PATCH 41/49] update documentation --- src/data.jl | 52 +++++++++++++++++++++++++++++++++++++++----- src/experiments.jl | 11 +++++++++- src/figures.jl | 7 +++++- src/neuralnetwork.jl | 16 ++++++-------- src/plotlib.jl | 26 +++++++++++++++++++--- src/plotutils.jl | 18 ++++++++++++++- 6 files changed, 109 insertions(+), 21 deletions(-) diff --git a/src/data.jl b/src/data.jl index b1c5c412..37f7e75b 100644 --- a/src/data.jl +++ b/src/data.jl @@ -11,7 +11,13 @@ using ScikitLearn export get_moon_data_loader, get_nmist_data_loader -function get_moon_data_loader(; n = 300, offset = 0.5, shuffle = true, seed = rand((1, 2^31)), args...) +""" + get_moon_data_loader(n = 300, offset = 0.5, shuffle = true, seed = 1337) + +Returns a Flux.DataLoader with moon data for the specified parameters. +Supports creation of random moons and shuffling of the data points. +""" +function get_moon_data_loader(; n::Int64 = 300, offset::Float64 = 0.5, shuffle::Bool = true, seed::Int64 = rand((1, 2^31)), args...) args = Args(; args...) # offset: linearly separable => 1.0; not lin. sep. => 0.5 @@ -22,7 +28,12 @@ function get_moon_data_loader(; n = 300, offset = 0.5, shuffle = true, seed = ra return DataLoader((x, y), batchsize = args.batchsize, shuffle = shuffle) end -function get_nmist_data_loader(; shuffle = true, args...) +""" + get_nmist_data_loader(shuffle = true) + +Returns a Flux.DataLoader with MNIST data for the specified parameters. +""" +function get_nmist_data_loader(; shuffle::Bool = true, args...) args = Args(; args...) x, y = MLDatasets.MNIST.traindata(Float32) @@ -35,7 +46,13 @@ function get_nmist_data_loader(; shuffle = true, args...) return DataLoader((x, y), batchsize = args.batchsize, shuffle = shuffle) end -function generate_moons_from_publication(; offset = 0.0, coord_down_scale = 1, show_plot = false) +""" + generate_moons_from_publication(offset = 0.5, coord_down_scale = 1.0, show_plot = false) + +Returns the exact same moons that were used on the demo website +of the publication authors at https://mohammadpz.github.io/GradientStarvation.html +""" +function generate_moons_from_publication(; offset::Float64 = 0.0, coord_down_scale::Float64 = 1.0, show_plot::Bool = false) moons = hcat(moon_class_1(offset, coord_down_scale), moon_class_2(offset, coord_down_scale)) labels = [repeat([0], 150); repeat([1], 150)] @@ -46,7 +63,13 @@ function generate_moons_from_publication(; offset = 0.0, coord_down_scale = 1, s return moons, labels end -function generate_moons(n; noise = 0.09, offset = 0.0, rotation = 90, seed = rand((1, 2^31)), show_plot = false) +""" + generate_moons(n = 300, noise = 0.42, offset = 0.5, rotation = 90, seed = 42, show_plot = true) + +Wrapper function for scikitlearn make_moons function. Additionally features the options to rotate and offset the +moons or plot the generated data. +""" +function generate_moons(n::Int64; noise::Float64 = 0.09, offset::Float64 = 0.0, rotation::Int64 = 90, seed::Int64 = rand((1, 2^31)), show_plot::Bool = false) X, y = make_moons(n_samples = n, noise = noise, random_state = seed) p1 = scatter(X[:,1], X[:,2], c = y, title = "generated state") @@ -73,6 +96,11 @@ function generate_moons(n; noise = 0.09, offset = 0.0, rotation = 90, seed = ran return X, y end +""" + apply_moon_offset(X, y, 0.5) + +Separates both moons by the specified offset. +""" function apply_moon_offset(X, y, offset) # @assert direction in ["x", "y"] "Error: direction must be either x or y" # use offset as total offset, thus devide by number of labels; assume 2 labels @@ -97,7 +125,13 @@ function apply_moon_offset(X, y, offset) return X end -function moon_class_1(offset = 0.0, coord_down_scale = 1) +""" + moon_class_1(0.0, 1.0) + +Returns the hardcoded coordinates of the first moon from the author's demo website +at https://mohammadpz.github.io/GradientStarvation.html +""" +function moon_class_1(offset::Float64 = 0.0, coord_down_scale::Float64 = 1.0) x = [ -2.717835893494339 - offset, -2.0025756608302556 - offset, @@ -406,7 +440,13 @@ function moon_class_1(offset = 0.0, coord_down_scale = 1) return vcat(transpose.((x, y))...) / coord_down_scale end -function moon_class_2(offset = 0.0, coord_down_scale = 1) +""" + moon_class_2(0.0, 1.0) + +Returns the hardcoded coordinates of the second moon from the author's demo website +at https://mohammadpz.github.io/GradientStarvation.html +""" +function moon_class_2(offset::Float64 = 0.0, coord_down_scale::Float64 = 1.0) x = [ 1.2605697631874289 + offset, 0.4428841761287988 + offset, diff --git a/src/experiments.jl b/src/experiments.jl index 37ebbea4..21051221 100644 --- a/src/experiments.jl +++ b/src/experiments.jl @@ -6,13 +6,21 @@ include("plotlib.jl") include("plotutils.jl") include("structs.jl") +using DataStructures using PlotlyJS export run_experiments plot_dir = "plots" -function run_experiments(experiments; prefix = "", random_moons = true, show_plots = true, save_plots = true, plot_size = "medium", plot_type = "both", args...) +""" + run_experiments(experiments, prefix = "exp_1", random_moons = true, show_plots = false, save_plots = true, plot_size = "medium", plot_type = "both", epochs = 500, batchsize = 42) + +Runs all experiments as specified by an DataStructures.OrderedDict. +This includes the creation of moon data, the network, the training and the plotting of the results. +This function is the work horse to conduct any analyses for the notebook. +""" +function run_experiments(experiments::DataStructures.OrderedDict; prefix::String = "", random_moons::Bool = true, show_plots::Bool = true, save_plots::Bool = true, plot_size::String = "medium", plot_type::String = "both", args...) plot_type = lowercase(plot_type) @assert plot_type in ["boundary", "stats", "both"] "Requested plot type '$plot_type' is not supported." @@ -54,6 +62,7 @@ function run_experiments(experiments; prefix = "", random_moons = true, show_plo display(p) end end + @info "Finished $(length(experiments)) experiments." end diff --git a/src/figures.jl b/src/figures.jl index 9de0c916..916ef36f 100644 --- a/src/figures.jl +++ b/src/figures.jl @@ -10,7 +10,12 @@ export create_figure_1 plot_dir = "plots" -function create_figure_1(plot_size) +""" + create_figure_1("medium") + +Convenience function to create figure 1 of the notebook. +""" +function create_figure_1(plot_size::String) width, height = PlotUtils.get_plot_resolution(plot_size) loader_1 = Data.get_moon_data_loader(offset = 1.0, seed = 1337) loader_2 = Data.get_moon_data_loader(offset = 0.5, seed = 1337) diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 3d261e67..71894e4a 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -52,9 +52,9 @@ end Creates a Flux neural network according to the topology used by Pezeshki et al. That is a NN with 2 hidden layers, 500 units each and ReLU activation. The hidden -layer dimensions can be adjusted. +layer dimensions can be adjusted by the first and only parameter. """ -function neural_network(hidden_dim = 500) +function neural_network(hidden_dim::Int64 = 500) return Chain( Dense(2, hidden_dim, relu), Dense(hidden_dim, 2) @@ -71,12 +71,10 @@ the specified optimizer and additionally spectral decoupling as a loss regulariz ```julia-repl julia> train(train_loader, test_loader, "SGD", false) [ Info: opt = SGD, sd = false, lr = 0.01, bs = 50 -Training... 100%|███████████████████████████████████████████████████████████████████████████| Time: 0:00:42 - epoch: 1000 - train_loss: 0.1337 - test_loss: 0.1337 - train_accuracy: 1.0 - test_accuracy: 1.0 +Training... 100%|███████████████████████████████████████████████████████████████████████████| Time: 0:00:05 + epoch: 1000 + loss: 0.1337 + accuracy: 1.0 ``` """ function train(data_loader::Flux.Data.DataLoader, optimizer::String = "SGD", spectral_decoupling::Bool = false; args...) @@ -147,7 +145,7 @@ for the given learning rate. Supported optimizers are: - Stochastic Gradient Descent (SGD) - Weight Decay (WD) """ -function get_optimizer(learning_rate, optimizer = "SGD"; args...) +function get_optimizer(learning_rate::Float64, optimizer::String = "SGD"; args...) args = Args(; args...) optimizer = uppercase(optimizer) @assert optimizer in ["ADAM", "GD", "SGD", "WD"] "Requested optimizer '$optimizer' is not supported." diff --git a/src/plotlib.jl b/src/plotlib.jl index 42e5f65f..6790c3b1 100644 --- a/src/plotlib.jl +++ b/src/plotlib.jl @@ -12,6 +12,11 @@ export plot_train_and_test_data, plot_loss_and_accuracy, plot_features, plot_dec title_font_size = 18 tick_font_size = 16 +""" + plot_train_and_test_data(train_loader, test_loader, "Train", "Test") + +Returns a PlotlyJS subplot with one row and two columns visualizing the two provided datasets. +""" function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loader::Flux.Data.DataLoader; train_title = "", test_title = "") # define common layout layout = Layout( @@ -50,7 +55,12 @@ function plot_train_and_test_data(train_loader::Flux.Data.DataLoader, test_loade p end -function plot_loss_and_accuracy(loss, accuracy; args...) +""" + plot_loss_and_accuracy(loss, accuracy) + +Returns a PlotlyJS line plot with the data of the provided loss and accuracy vectors. +""" +function plot_loss_and_accuracy(loss::Vector{Any}, accuracy::Vector{Any}; args...) args = Args(; args...) PlotlyJS.plot([ PlotlyJS.scatter(y = loss, x = 1:args.epochs, mode = "lines", name = "loss"), @@ -76,7 +86,12 @@ function plot_loss_and_accuracy(loss, accuracy; args...) ) end -function plot_features(z1, z2; args...) +""" + plot_features(z1, z2) + +Returns a PlotlyJS line plot with the data of both features z1 and z2. +""" +function plot_features(z1::Vector{Float64}, z2::Vector{Float64}; args...) args = Args(; args...) PlotlyJS.plot([ PlotlyJS.scatter(y = z1, x = 1:args.epochs, mode = "lines", name = "z1"), @@ -99,7 +114,12 @@ function plot_features(z1, z2; args...) ) end -function plot_decision_boundary(loader, model; title = "") +""" + plot_decision_boundary(train_loader, model; title = "SGD Δ0.5") + +Returns a PlotlyJS contour plot of the provided dataset according to the specified model. +""" +function plot_decision_boundary(loader::Flux.DataLoader, model; title = "") # grid and range step size n = 100 # determine limits of given data diff --git a/src/plotutils.jl b/src/plotutils.jl index ee0bed7b..7b87a3b0 100644 --- a/src/plotutils.jl +++ b/src/plotutils.jl @@ -2,6 +2,12 @@ module PlotUtils export get_custom_rdbu_scale, get_plot_resolution, map_bool_to_color +""" + get_custom_rdbu_scale(0.75) + +Returns a customized red blue scale adjusted for a very thin white colorband +in the middle of the interval. +""" function get_custom_rdbu_scale(opacity::Union{Float32, Float64}) @assert opacity >= 0 || opacity <= 1 "Opacity must be in [0, 1]" return [ @@ -13,7 +19,12 @@ function get_custom_rdbu_scale(opacity::Union{Float32, Float64}) ] end -function get_plot_resolution(size) +""" + get_plot_resolution("large") + +Returns a tuple of a resolution with 16:9 aspect ratio. +""" +function get_plot_resolution(size::String) size = lowercase(size) @assert size in ["small", "medium", "large", "xlarge"] "Requested plot size '$size' is not supported." @@ -32,6 +43,11 @@ function get_plot_resolution(size) end end +""" + map_bool_to_color([false, true, true, false], "red", "blue") + +Maps a boolean vector to the two specified colors. +""" function map_bool_to_color(bool_vec::Union{BitVector, Vector{Bool}}, color_true::String, color_false::String) return map(el -> el == true ? color_true : color_false, bool_vec) end From e9c80ec584e2d8e3677db99e208efa477996ee63 Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 00:36:11 +0100 Subject: [PATCH 42/49] add static version of notebook --- notebook/gradientstarvation.jl.html | 62 +++++++++++++++++++++++++++++ 1 file changed, 62 insertions(+) create mode 100644 notebook/gradientstarvation.jl.html diff --git a/notebook/gradientstarvation.jl.html b/notebook/gradientstarvation.jl.html new file mode 100644 index 00000000..56e2faec --- /dev/null +++ b/notebook/gradientstarvation.jl.html @@ -0,0 +1,62 @@ + + + + + ⚡ Pluto.jl ⚡ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + From f1ef6802dfa503612a36a67290dab89ecf553038 Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 09:25:03 +0100 Subject: [PATCH 43/49] test Resource instead of LocalResource --- notebook/gradientstarvation_test.jl | 518 ++++++++++++++++++++++++++++ 1 file changed, 518 insertions(+) create mode 100644 notebook/gradientstarvation_test.jl diff --git a/notebook/gradientstarvation_test.jl b/notebook/gradientstarvation_test.jl new file mode 100644 index 00000000..84fdee8c --- /dev/null +++ b/notebook/gradientstarvation_test.jl @@ -0,0 +1,518 @@ +### A Pluto.jl notebook ### +# v0.17.5 + +using Markdown +using InteractiveUtils + +# ╔═╡ 5c563230-df6e-4f65-a106-a707d2454fe7 +using PlutoUI + +# ╔═╡ 8bd774c7-1acf-4beb-bc99-4c303882d262 +md""" +# Starving Neurons - Neural Networks and Gradient Starvation + +## Introduction + +Artificial intelligence (AI) and machine learning are disciplines whose methods have become ubiquitously applied in a broad range of scientific domains, enabling the implicit learning of features inherent to a dataset. A common AI tool for self-learning of data characteristics by examples are neural networks (NNs) which attempt to mimic biological neurons. NNs are broadly applicable, e.g. for tasks such as image and speech recognition, weather forecasting, stock market prediction, robotics and data mining among many others. But how do these artificial neural networks learn? +There is a myriad of neural network subtypes and topologies, but in this notebook, I want to focus on the multi-layer perceptron (MLP). + +Basically speaking, a NN consists of multiple layers: an input layer, one or more hidden layers and an output layer (see fig. 1). + +""" + +# ╔═╡ 8f0cdbf1-0174-4d18-9072-2a8c5f015177 +html""" +An image visualizing a basic NN topology with an input layer, a single hidden layer and an output layer + +""" + + +# ╔═╡ 6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b +md""" +Figure 1: Basic NN/MLP topology with input, a single hidden and output layer. +CC-BY-SA 3.0. Source: [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Artificial_neural_network.svg) +""" + +# ╔═╡ 512e6240-76e7-11ec-2511-07ffc4b74d27 +md""" +The input layer encodes the data presented to the network and feeds it weighted into the first hidden layer; all other layers consist of artificial neurons with non-linear activation functions. MLPs are fully connected, meaning that every node of one layer connects to every node of the next layer and each connection has a weight assigned to it. The weights can be understood as the actual learned parameters being adjusted as they get passed from layer to layer. The MLP learns supervised, i.e. by presentation of labelled training data. Without going into too much detail, I want to briefly formalize the learning process by the example of a classification task, that is deciding to which discrete class a given input belongs. This is exactly the use case we will evaluate later on. +""" + +# ╔═╡ 93bc80a8-673d-457b-96d8-e1bb0b09682e +md""" +### Supervised Learning – Learning from Examples + +Supervised learning tries to learn a function which maps input to output values based on known input-output pairs (i.e. the pair ($x$, $y$); a correct classification $y$ for a given input $x$). The observed outputs $y$ originate from an unknown function $f(x) = y$. Thus, the goal of the learning process is to approximate $f(x)$ by a hypothesis $h(x) \approx f(x)$ while minimizing the error of the approximation, refered to as loss. More formally, the loss is defined as the utility lost by substituting the correct classification $f(x) = y$ by the hypothesis $h(x) = \hat{y}$: + +$L(x, y, ŷ) = +Utility\\(result\ of\ using\ y\ given\ x\\) +- Utility\\(result\ of\ using\ ŷ\ given\ x\\)$ + +Commonly applied loss functions are the mean absolute error and mean squared error loss. Especially popular for classification tasks is the cross-entropy loss, also called logarithmic loss: + +$L(y, ŷ) = -\sum_{i}y_i\cdot log(ŷ_i)$ + +The learning process maximizes the expected utility through choosing the optimal hypothesis by minimizing the loss function. In practice, this is estimated by the empirical loss observed on the set $X$ of $N$ training examples. + +$EmpiricalLoss(h) = \frac{1}{N}\cdot\sum_{(x, y) \in X} L(y, h(x))$ + +But what about the weights I’ve introduced earlier as the “actual learned parameters”? Keep in mind that a hypothesis $h$ is a function and the weights simply represent the function coefficients. For the linear case that would mean $h(x) = w_1 x + w_0$. +In the multivariate case there is no closed-form solution for the minimum loss. Therefore, we are confronted with an optimization problem which can be solved by gradient descent. The weight update with learning rate $\alpha$ and the vector of weights $\bf{w}$ looks like this: + +$w_i = w_i - \alpha\cdot \frac{\partial}{\partial w_i}\cdot Loss(\bf{w})$ + +""" + +# ╔═╡ 66a3b5cb-a465-42fb-a8c8-404d574312d2 +md""" +### Regularization + +> "Entities should not be multiplied beyond necessity." – William of Ockham +This quote, known as Ockham’s razor and usually paraphrased as “simple solutions are better solutions” captures the core idea of regularization. The aim of regularization is to find the hypothesis which minimizes the total cost, defined as the sum of empirical loss and its complexity: + +$Cost(h) = EmpiricalLoss(h) + \lambda\cdot Complexity(h)$ + +Regularization penalizes complex hypotheses (mind the added regularization term in the formula $\lambda\cdot Complexity(h)$) and favors more regular functions. It limits the values that the weights can take during the learning process. Therefore, regularization avoids overfitting of the data. +The estimated best hypothesis $ĥ^*$ from the hypothesis space $\mathcal{H}$ now becomes: + +$ĥ^* = \arg\min_{h \in \mathcal{H}} Cost(h)$ + +### Optimization +Given the major workings of supervised learning, it should have become clear that learning is a mathematical optimization problem. A classic way to achieve the minimization of the cost is the application of gradient descent (GD). However, the path the gradient take on the surface of the solution space can limit its ability to find the global optimum. To conquer that, the steps can be modified by additional optimization methods. Common choices for that purpose are the addition of momentum, resulting in stochastic gradient descent (SGD); an adaptive learning rate, different forms of parameter initialization as well as normalization techniques. +""" + +# ╔═╡ 4a14831d-7736-4af5-875f-05ae6e3cf7c4 +md""" +### Side Effects of Gradient-based Learning +Despite its popularity, gradient descent learning in NNs can come with some unwanted side effects such as the vanishing gradient problem, which condenses down to the gradient being too small to effectively update a weight and thus preventing learning. This phenomenon is complemented in the other direction by exploding gradients. + +Another not yet published but currently in preprint described observation is the effect of [**_Gradient Starvation_ (GS)**](https://arxiv.org/abs/2011.09468). Pezeshki et al. coined and formalized this phenomenon of over-parameterized NNs in their work. According to them, _Gradient Starvation_ is a predisposition of NNs learned by cross-entropy, which occurs when the loss minimization is driven by only a feature subset regardless of the presence of other meaningful features. Thus, the NN becomes biased towards superficial features. Beyond demonstrating the effects of GS, Pezeshki et al. introduce **_Spectral Decoupling_ (SD)** regularization as a means to counteract gradient starvation. +""" + +# ╔═╡ 60905936-9373-44de-96e6-cb5d6dd72d33 +md""" +### Spectral Decoupling +Proceeding from the popular "ridge-regularized cross entropy": + +$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot log(1 + exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{θ}||^2$ + +with $\boldsymbol{Y}$ the diagonal matrix of $y$ and $\boldsymbol{θ}$ the concatenation of all layer's weights, SD is a variation where the L2 regularization term is replaced by a penalty on the raw predictions $\boldsymbol{ŷ}$: + +$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot log(1 + exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{ŷ}||^2$ + +This, as stated by the Pezeshki et al., uncouples the learning of multiple features. According to their work, a feature trained close to its optimum inhibits (starves) the training of other features which is thus prevented by SD. +""" + +# ╔═╡ 00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 +md""" +### Objective + +My project and thereby this notebook aims to reproduce and demonstrate the effects of GS on a simple 2D classification task. Furthermore, the proposed SD regularization method is implemented and compared to other methods. On top of that, spectral decoupling is not only applied in the context of SGD but all other investigated optimization approaches. +""" + +# ╔═╡ ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 +md""" +### Waning or Waxing Moon? A 2D Classification Experiment +A modified version of the [Moons dataset](https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_moons.html) was used with two different topologies as exemplified in figure 1. +The first topology allows to draw a line by a very small margin between the two moons. Therefore, it represents the fully linearly separable case (fig. 1, left). The second one applies a smaller offset to interleave both moons which only allows to separate them by learning a curved classification boundary (fig. 1, right). The task of the NN is to discriminate if a given data point belongs to the waning (blue) or waxing moon (red). +""" + +# ╔═╡ 2f766a9c-de6e-4f95-aa73-aacb486aaf61 +md""" +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/figure_1_edited.png")) +Figure 1: Two moons topologies; one linear separable (offset Δ1.0), the other separable by a curved boundary (offset Δ0.5). +""" + +# ╔═╡ 7f7ed965-fa08-4d6f-8908-dfc8876bdd02 +md""" +The reasoning of the researchers was that in the linear separable case the NN is not encouraged to learn a curved decision boundary because the loss becomes negligible by only discriminating the two moons along one axis, therefore learning only one feature and neglecting the other. In contrast to that, the interleaved moons can only be separated by learning both features which better accommodates the curvatures of the data structure in general. +They observed that a different choice of regularization and optimization methods like weight decay (WD) and _Adaptive Moment Estimation_ (ADAM), among others, are not able to stimulate the NN to learn a curved decision boundary. +Moreover, Pezeshki et al. explained that circumstance by cross-entropy loss learning which starves the gradients from the other feature, only reinforcing the superficial one. + +#### Repercussions of Gradient Starvation +Why does the shape of the decision boundary even matter as long as it discriminates the data points confidently? As stated by the authors, GS results in a very small distance between the data points and the boundary which translates into a lack of robustness when generalizing to new data. Contrarily, GS could also have a positive effect in terms of preventing overfitting by not learning non-dominant features. The authors made the comparison to the aformentioned Ockham's razor: GS leads to simpler decision boundaries and hence NNs that generalize better, thus GS being a form of implicit regularization. + +#### Implementation +The implementation is based on [Flux](https://fluxml.ai/), a ML framework for Julia. The NN consists of two hidden layers with 500 nodes each. Rectified Linear Unit (ReLu) is used as an activation function and learning is facilitated by cross-entropy loss for 1000 epochs on a data set with 300 observations (150 blue and 150 red points). In contrast to Pezeshki et al., I've opted to train in batches of 50 (6 batches in total) since this is a common practice for training NNs. +All networks are trained with gradient descent but with different regularization and optimization approaches. This yields the following different setups: +- Gradient descent (GD; no regularization/optimization) +- Gradient descent with momentum (stochastic gradient descent (SGD); no regularization) +- Gradient descent with weight decay (WD; L2 regularization) +- Gradient descent with _Adaptive Moment Estimation_ (ADAM; no regularization) + +Different sets of hyperparameters were evaluated while tinkering with the NNs. However, since the main objective is to reproduce the research conducted by Pezeshki et al., their set of hyperparameters were used: +- learning rate = 0.01 +- weight decay coefficient = 0.01 +- spectral decoupling coefficient = 0.003 +""" + +# ╔═╡ 88ddf822-622b-4d46-a0a4-a7d749c8bc48 +md""" +#### Results +""" + +# ╔═╡ 884bc572-9daa-40ad-8d76-56ec58728f63 +md""" +##### Linear separable case +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/GD_%CE%941.0.png")) +Figure 2: GD, moon offset = Δ1.0. The decision boundary is almost linear and the loss continuously falls while training. Accuracy reaches 100%. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/SGD_%CE%941.0.png")) +Figure 3: SGD, moon offset = Δ1.0. The decision boundary is similar to the one learned with GD but equally distant from both classes. Loss declines quicker and reaches an overall lower level compared to GD. Accuracy reaches 100% after less training. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/WD_%CE%941.0.png")) +Figure 4: WD, moon offset = Δ1.0, WD coefficient = 0.01. The boundary is similar to GD without WD. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/ADAM_%CE%941.0.png")) +Figure 5: ADAM, moon offset = Δ1.0. The decision boundary is slightly curved around the data points closest to the boundary. Loss converges to zero almost immediately while full accuracy is attained. +""" + +# ╔═╡ ac453412-0f71-45ee-aefb-0cc519467c6c +md""" +As we can see from figures 2 to 5, the decision boundaries of all networks exhibit some specificities. The GD network learned an almost linear decision boundary which appears to be closer to the blue-labeled moon. The boundary looks similar with SGD, but its distance to both moons is equal. Only the network with ADAM optimization learns a slight curvature around the points closest to the boundary. Furthermore, the ADAM classification learned the fastest as can be seen from the rapid decrease in loss and increase in accuracy. At the same time, the contour lines all closely condense around the decision boundary which indicates that the ADAM network is most confident in discriminating both classes. +""" + +# ╔═╡ 5e282e1a-62fc-4911-9b7e-f2631eeb2669 +md""" +##### Linear inseparable case + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/WD_%CE%940.5.png")) +Figure 6: GD, moon offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes within the 1000 training iterations. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/SGD_%CE%940.5.png")) +Figure 7: SGD, moon offset = Δ0.5. The network learns a curved decision boundary which is very close to the interleaving data points. Loss becomes negligible during learning. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/WD_%CE%940.5.png")) +Figure 8: WD, moon offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes. Again, similar result compared to GD. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/ADAM_%CE%940.5.png")) +Figure 9: ADAM, moon offset = Δ0.5. The network learns a curved decision boundary. Loss fluctuates especially during the early iterations of training. +""" + +# ╔═╡ f7bae71f-885a-404d-985b-92d612e1f3d6 +md""" +SGD and ADAM are able to learn a curved decision boundary (fig. 7 and 9) while GD and WD fail to do so (fig. 6 and 8). Moreover, the decision boundary learned by ADAM better resembles the data structure. The contours of the ADAM network show a very strong confidence on the networks classification. This contrasts the results from the GS paper, where ADAM was not able to learn a decision boundary as curved and well-fitting the data structure as shown in fig. 9. +""" + +# ╔═╡ 2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f +md""" +#### Spectral Decoupling + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/GD_%CE%940.5_%2B_SD.png")) +Figure 10: GD with SD, moon offset = Δ0.5. No curved decision boundary is learned but the loss also did not converge. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/SGD_%CE%940.5_%2B_SD.png")) +Figure 11: SGD with SD, moon offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the moon data structure and shows invaginations at the data points of both moons closest to the boundary. Loss converges to ~15%. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/WD_%CE%940.5_%2B_SD.png")) +Figure 12: WD with SD, moon offset = Δ0.5. NN fails to separate the two data classes. Similar to GD with SD, loss could still be lowered with further training. + +$(Resource(".https://github.com/justinsane1337/GradientStarvation/blob/master/plots/ADAM_%CE%940.5_%2B_SD.png")) +Figure 13: ADAM with SD, moon offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the data structure. Loss heavily oscillates and steadily increases with longer training. Interestingly, all data points of each respective class belong to the same contour. +""" + +# ╔═╡ 7460f609-c32b-4704-a361-71c98bb626cd +md""" +SD, originally applied as a regularization by Pezeshki et al. only with SGD, is able to yield curved decision boundaries which greatly resemble the data structure of the two moons in combination with SGD and ADAM optimization. While SGD learns a more angular boundary with low loss, ADAM learns a smoother one but exhibits a high loss with strong oscillations. Interestingly, ADAM learns to group all data points of a class within the same contour (fig. 13). GD and WD do not profit from SD regularization and don't learn a curved boundary. However, their loss profiles suggest improvement with further learning. +""" + +# ╔═╡ 87cc8172-d5cc-49ff-9a0e-fa734e1010c1 +md""" +#### Additional experiments +Given that the loss did not converge for GD + SD after 1000 epochs, the network was trained for 10000 iterations. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/GD_%CE%940.5_%2B_SD_10000.png")) +Figure 14: GD with SD, moon offset = Δ0.5. NN was trained 10 times longer (10000 iterations). The learned curved decision boundary resembles the result obtained with SGD + SD. + +To conquer the overfitting observed with ADAM + SD lower learning rates were evaluated. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/1e-3_ADAM_%CE%940.5_%2B_SD.png")) +Figure 15: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.001 still suffers from severe overfitting. Nonetheless, oscillations in loss are dampened. + +$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/1e-4_ADAM_%CE%940.5_%2B_SD.png")) +Figure 16: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.0001 shows lower loss compared to 0.001 but a decision boundary with a smaller margin. + +The additional experiments show that GD with SD performs equally well compared to SGD with SD but needs much more training for that. +The overfitting problem with ADAM + SD can be successfully conquered with lower learning rates. However, the shape of the decision boundary suffers with learning rates as low as 0.0001. From all three ADAM + SD experiments, the one learning at a rate of 0.001 performed best. +""" + +# ╔═╡ 2c740919-1fd4-45c4-91d1-ef3d3e081139 +md""" +## Discussion +Pezeshki et al. described gradient starvation as a learning proclivity that can exhibit both negative and positive effects on neural network learning. Considering the 2D classification task discussed in this notebook, the effect of learning a decision boundary with a small margin, thus supposedly hardly robust, could be exemplified for stochastic gradient descent. Combining SGD with spectral decoupling regularization was greatly able to enlarge the margin, therefore yielding a more robust network that not only focussed on a single dominant feature. On the other hand, the potential negative effect of overfitting by precisely learning less dominant features as well became evident through higher losses (SGD: 8.61% without vs. 13.37% with SD; ADAM: 2.00% without vs. 68.16%). +Contrarily to the results from Pezeshki et al., the here used neural network was perfectly able to learn curved decision boundaries featuring a large margin between the data points using ADAM without SD. While Pezeshki et al. trained their ADAM network with rates of 0.0001 and 0.001, I trained all networks under the same conditions, hence both SGD and ADAM with a learning rate of 0.01. Despite the higher learning rate, ADAM showed no sign of overfitting without SD. However, with SD the overfitting was extreme but ADAM coupled with SD regularization was also a combination of methods not assessed by the researchers. +Regarding the same matter, my approach to apply SD not only to SGD but all other optimizers simultaneously offered limited insight. First of all, the combination of WD (L2 regularization) and SD was expected to not work well since both methods are somewhat conflicting. ADAM performed well with SD but suffered from overfitting which can be conquered by a lower learning rate as demonstrated. Nonetheless, the network with purportedly lower loss learns a boundary with a smaller margin. + +All in all, the results of Pezeshki et al. could be reproduced with the exception of the network using ADAM, which was able to learn decision boundaries with similar characteristics as the ones learned by SGD with SD. + +Finally, it must be mentioned that Pezeshki et al. not only exemplified GS on the rather synthetical moon classification task but showed GS ramifications and the potential of SD as a regularization technique on more complex, well-established datasets like CIFAR, colored MNIST and CelebA which was sadly out of scope of this project. +""" + +# ╔═╡ 21854d7b-13ce-477c-bc61-1cb15536e24f +md""" +### Conclusion +Gradient starvation is another relevant side effect of gradient descent-learned neural networks that may be worth the attention in the optimization process of NNs. However, if GS is a blessing or a curse depends on the use case. Starved models, i.e. models concentrating on superficial features during learning, can be less prone to overfitting while adressing GS promises to increase robustness. +""" + +# ╔═╡ c95db58c-9d47-4ef0-9ede-234d7a693f41 +md""" +### References +Pezeshki, M., Kaba, S. O., Bengio, Y., Courville, A., Precup, D., & Lajoie, G. (2020). Gradient starvation: A learning proclivity in neural networks. arXiv preprint arXiv:2011.09468. + +Pizurica, A. (2020). E016330: Artificial Intelligence – Learning from examples [PDF course notes]. Ufora. + +Pizurica, A. (2020). E016330: Artificial Intelligence - Neural networks [PDF course notes]. Ufora. + +Russell, S., & Norvig, P. (2002). Artificial intelligence: a modern approach. + +Kingma, D. P., & Ba, J. (2014). 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["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" + +[[deps.p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" +""" + +# ╔═╡ Cell order: +# ╟─5c563230-df6e-4f65-a106-a707d2454fe7 +# ╟─8bd774c7-1acf-4beb-bc99-4c303882d262 +# ╟─8f0cdbf1-0174-4d18-9072-2a8c5f015177 +# ╟─6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b +# ╟─512e6240-76e7-11ec-2511-07ffc4b74d27 +# ╟─93bc80a8-673d-457b-96d8-e1bb0b09682e +# ╟─66a3b5cb-a465-42fb-a8c8-404d574312d2 +# ╟─4a14831d-7736-4af5-875f-05ae6e3cf7c4 +# ╟─60905936-9373-44de-96e6-cb5d6dd72d33 +# ╟─00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 +# ╟─ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 +# ╟─2f766a9c-de6e-4f95-aa73-aacb486aaf61 +# ╟─7f7ed965-fa08-4d6f-8908-dfc8876bdd02 +# ╟─88ddf822-622b-4d46-a0a4-a7d749c8bc48 +# ╟─884bc572-9daa-40ad-8d76-56ec58728f63 +# ╟─ac453412-0f71-45ee-aefb-0cc519467c6c +# ╟─5e282e1a-62fc-4911-9b7e-f2631eeb2669 +# ╟─f7bae71f-885a-404d-985b-92d612e1f3d6 +# ╟─2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f +# ╟─7460f609-c32b-4704-a361-71c98bb626cd +# ╟─87cc8172-d5cc-49ff-9a0e-fa734e1010c1 +# ╟─2c740919-1fd4-45c4-91d1-ef3d3e081139 +# ╟─21854d7b-13ce-477c-bc61-1cb15536e24f +# ╟─c95db58c-9d47-4ef0-9ede-234d7a693f41 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 From 8754891e1351d50a459dcd1091aee53336414882 Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 09:46:51 +0100 Subject: [PATCH 44/49] add author to notebook, fix usage of local resources --- notebook/gradientstarvation.jl | 34 +- notebook/gradientstarvation_test.jl | 518 ---------------------------- 2 files changed, 18 insertions(+), 534 deletions(-) delete mode 100644 notebook/gradientstarvation_test.jl diff --git a/notebook/gradientstarvation.jl b/notebook/gradientstarvation.jl index 7728c762..b29c88fb 100644 --- a/notebook/gradientstarvation.jl +++ b/notebook/gradientstarvation.jl @@ -11,6 +11,8 @@ using PlutoUI md""" # Starving Neurons - Neural Networks and Gradient Starvation +Author: Peter Merseburger + ## Introduction Artificial intelligence (AI) and machine learning are disciplines whose methods have become ubiquitously applied in a broad range of scientific domains, enabling the implicit learning of features inherent to a dataset. A common AI tool for self-learning of data characteristics by examples are neural networks (NNs) which attempt to mimic biological neurons. NNs are broadly applicable, e.g. for tasks such as image and speech recognition, weather forecasting, stock market prediction, robotics and data mining among many others. But how do these artificial neural networks learn? @@ -125,7 +127,7 @@ The first topology allows to draw a line by a very small margin between the two # ╔═╡ 2f766a9c-de6e-4f95-aa73-aacb486aaf61 md""" -$(LocalResource("../plots/figure_1_edited.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/figure_1_edited.png")) Figure 1: Two moons topologies; one linear separable (offset Δ1.0), the other separable by a curved boundary (offset Δ0.5). """ @@ -160,16 +162,16 @@ md""" # ╔═╡ 884bc572-9daa-40ad-8d76-56ec58728f63 md""" ##### Linear separable case -$(LocalResource("../plots/GD_Δ1.0.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/GD_%CE%941.0.png")) Figure 2: GD, moon offset = Δ1.0. The decision boundary is almost linear and the loss continuously falls while training. Accuracy reaches 100%. -$(LocalResource("../plots/SGD_Δ1.0.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/SGD_%CE%941.0.png")) Figure 3: SGD, moon offset = Δ1.0. The decision boundary is similar to the one learned with GD but equally distant from both classes. Loss declines quicker and reaches an overall lower level compared to GD. Accuracy reaches 100% after less training. -$(LocalResource("../plots/WD_Δ1.0.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/WD_%CE%941.0.png")) Figure 4: WD, moon offset = Δ1.0, WD coefficient = 0.01. The boundary is similar to GD without WD. -$(LocalResource("../plots/ADAM_Δ1.0.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/ADAM_%CE%941.0.png")) Figure 5: ADAM, moon offset = Δ1.0. The decision boundary is slightly curved around the data points closest to the boundary. Loss converges to zero almost immediately while full accuracy is attained. """ @@ -182,16 +184,16 @@ As we can see from figures 2 to 5, the decision boundaries of all networks exhib md""" ##### Linear inseparable case -$(LocalResource("../plots/GD_Δ0.5.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/GD_%CE%940.5.png")) Figure 6: GD, moon offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes within the 1000 training iterations. -$(LocalResource("../plots/SGD_Δ0.5.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/SGD_%CE%940.5.png")) Figure 7: SGD, moon offset = Δ0.5. The network learns a curved decision boundary which is very close to the interleaving data points. Loss becomes negligible during learning. -$(LocalResource("../plots/WD_Δ0.5.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/WD_%CE%940.5.png")) Figure 8: WD, moon offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes. Again, similar result compared to GD. -$(LocalResource("../plots/ADAM_Δ0.5.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/ADAM_%CE%940.5.png")) Figure 9: ADAM, moon offset = Δ0.5. The network learns a curved decision boundary. Loss fluctuates especially during the early iterations of training. """ @@ -204,16 +206,16 @@ SGD and ADAM are able to learn a curved decision boundary (fig. 7 and 9) while G md""" #### Spectral Decoupling -$(LocalResource("../plots/GD_Δ0.5_+_SD.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/GD_%CE%940.5_%2B_SD.png")) Figure 10: GD with SD, moon offset = Δ0.5. No curved decision boundary is learned but the loss also did not converge. -$(LocalResource("../plots/SGD_Δ0.5_+_SD.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/SGD_%CE%940.5_%2B_SD.png")) Figure 11: SGD with SD, moon offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the moon data structure and shows invaginations at the data points of both moons closest to the boundary. Loss converges to ~15%. -$(LocalResource("../plots/WD_Δ0.5_+_SD.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/WD_%CE%940.5_%2B_SD.png")) Figure 12: WD with SD, moon offset = Δ0.5. NN fails to separate the two data classes. Similar to GD with SD, loss could still be lowered with further training. -$(LocalResource("../plots/ADAM_Δ0.5_+_SD.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/ADAM_%CE%940.5_%2B_SD.png")) Figure 13: ADAM with SD, moon offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the data structure. Loss heavily oscillates and steadily increases with longer training. Interestingly, all data points of each respective class belong to the same contour. """ @@ -227,15 +229,15 @@ md""" #### Additional experiments Given that the loss did not converge for GD + SD after 1000 epochs, the network was trained for 10000 iterations. -$(LocalResource("../plots/GD_Δ0.5_+_SD_10000.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/GD_%CE%940.5_%2B_SD_10000.png")) Figure 14: GD with SD, moon offset = Δ0.5. NN was trained 10 times longer (10000 iterations). The learned curved decision boundary resembles the result obtained with SGD + SD. To conquer the overfitting observed with ADAM + SD lower learning rates were evaluated. -$(LocalResource("../plots/1e-3_ADAM_Δ0.5_+_SD.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/1e-3_ADAM_%CE%940.5_%2B_SD.png")) Figure 15: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.001 still suffers from severe overfitting. Nonetheless, oscillations in loss are dampened. -$(LocalResource("../plots/1e-4_ADAM_Δ0.5_+_SD.png")) +$(Resource("https://raw.githubusercontent.com/justinsane1337/GradientStarvation/master/plots/1e-4_ADAM_%CE%940.5_%2B_SD.png")) Figure 16: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.0001 shows lower loss compared to 0.001 but a decision boundary with a smaller margin. The additional experiments show that GD with SD performs equally well compared to SGD with SD but needs much more training for that. diff --git a/notebook/gradientstarvation_test.jl b/notebook/gradientstarvation_test.jl deleted file mode 100644 index 84fdee8c..00000000 --- a/notebook/gradientstarvation_test.jl +++ /dev/null @@ -1,518 +0,0 @@ -### A Pluto.jl notebook ### -# v0.17.5 - -using Markdown -using InteractiveUtils - -# ╔═╡ 5c563230-df6e-4f65-a106-a707d2454fe7 -using PlutoUI - -# ╔═╡ 8bd774c7-1acf-4beb-bc99-4c303882d262 -md""" -# Starving Neurons - Neural Networks and Gradient Starvation - -## Introduction - -Artificial intelligence (AI) and machine learning are disciplines whose methods have become ubiquitously applied in a broad range of scientific domains, enabling the implicit learning of features inherent to a dataset. A common AI tool for self-learning of data characteristics by examples are neural networks (NNs) which attempt to mimic biological neurons. NNs are broadly applicable, e.g. for tasks such as image and speech recognition, weather forecasting, stock market prediction, robotics and data mining among many others. But how do these artificial neural networks learn? -There is a myriad of neural network subtypes and topologies, but in this notebook, I want to focus on the multi-layer perceptron (MLP). - -Basically speaking, a NN consists of multiple layers: an input layer, one or more hidden layers and an output layer (see fig. 1). - -""" - -# ╔═╡ 8f0cdbf1-0174-4d18-9072-2a8c5f015177 -html""" -An image visualizing a basic NN topology with an input layer, a single hidden layer and an output layer - -""" - - -# ╔═╡ 6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b -md""" -Figure 1: Basic NN/MLP topology with input, a single hidden and output layer. -CC-BY-SA 3.0. Source: [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Artificial_neural_network.svg) -""" - -# ╔═╡ 512e6240-76e7-11ec-2511-07ffc4b74d27 -md""" -The input layer encodes the data presented to the network and feeds it weighted into the first hidden layer; all other layers consist of artificial neurons with non-linear activation functions. MLPs are fully connected, meaning that every node of one layer connects to every node of the next layer and each connection has a weight assigned to it. The weights can be understood as the actual learned parameters being adjusted as they get passed from layer to layer. The MLP learns supervised, i.e. by presentation of labelled training data. Without going into too much detail, I want to briefly formalize the learning process by the example of a classification task, that is deciding to which discrete class a given input belongs. This is exactly the use case we will evaluate later on. -""" - -# ╔═╡ 93bc80a8-673d-457b-96d8-e1bb0b09682e -md""" -### Supervised Learning – Learning from Examples - -Supervised learning tries to learn a function which maps input to output values based on known input-output pairs (i.e. the pair ($x$, $y$); a correct classification $y$ for a given input $x$). The observed outputs $y$ originate from an unknown function $f(x) = y$. Thus, the goal of the learning process is to approximate $f(x)$ by a hypothesis $h(x) \approx f(x)$ while minimizing the error of the approximation, refered to as loss. More formally, the loss is defined as the utility lost by substituting the correct classification $f(x) = y$ by the hypothesis $h(x) = \hat{y}$: - -$L(x, y, ŷ) = -Utility\\(result\ of\ using\ y\ given\ x\\) -- Utility\\(result\ of\ using\ ŷ\ given\ x\\)$ - -Commonly applied loss functions are the mean absolute error and mean squared error loss. Especially popular for classification tasks is the cross-entropy loss, also called logarithmic loss: - -$L(y, ŷ) = -\sum_{i}y_i\cdot log(ŷ_i)$ - -The learning process maximizes the expected utility through choosing the optimal hypothesis by minimizing the loss function. In practice, this is estimated by the empirical loss observed on the set $X$ of $N$ training examples. - -$EmpiricalLoss(h) = \frac{1}{N}\cdot\sum_{(x, y) \in X} L(y, h(x))$ - -But what about the weights I’ve introduced earlier as the “actual learned parameters”? Keep in mind that a hypothesis $h$ is a function and the weights simply represent the function coefficients. For the linear case that would mean $h(x) = w_1 x + w_0$. -In the multivariate case there is no closed-form solution for the minimum loss. Therefore, we are confronted with an optimization problem which can be solved by gradient descent. The weight update with learning rate $\alpha$ and the vector of weights $\bf{w}$ looks like this: - -$w_i = w_i - \alpha\cdot \frac{\partial}{\partial w_i}\cdot Loss(\bf{w})$ - -""" - -# ╔═╡ 66a3b5cb-a465-42fb-a8c8-404d574312d2 -md""" -### Regularization - -> "Entities should not be multiplied beyond necessity." – William of Ockham -This quote, known as Ockham’s razor and usually paraphrased as “simple solutions are better solutions” captures the core idea of regularization. The aim of regularization is to find the hypothesis which minimizes the total cost, defined as the sum of empirical loss and its complexity: - -$Cost(h) = EmpiricalLoss(h) + \lambda\cdot Complexity(h)$ - -Regularization penalizes complex hypotheses (mind the added regularization term in the formula $\lambda\cdot Complexity(h)$) and favors more regular functions. It limits the values that the weights can take during the learning process. Therefore, regularization avoids overfitting of the data. -The estimated best hypothesis $ĥ^*$ from the hypothesis space $\mathcal{H}$ now becomes: - -$ĥ^* = \arg\min_{h \in \mathcal{H}} Cost(h)$ - -### Optimization -Given the major workings of supervised learning, it should have become clear that learning is a mathematical optimization problem. A classic way to achieve the minimization of the cost is the application of gradient descent (GD). However, the path the gradient take on the surface of the solution space can limit its ability to find the global optimum. To conquer that, the steps can be modified by additional optimization methods. Common choices for that purpose are the addition of momentum, resulting in stochastic gradient descent (SGD); an adaptive learning rate, different forms of parameter initialization as well as normalization techniques. -""" - -# ╔═╡ 4a14831d-7736-4af5-875f-05ae6e3cf7c4 -md""" -### Side Effects of Gradient-based Learning -Despite its popularity, gradient descent learning in NNs can come with some unwanted side effects such as the vanishing gradient problem, which condenses down to the gradient being too small to effectively update a weight and thus preventing learning. This phenomenon is complemented in the other direction by exploding gradients. - -Another not yet published but currently in preprint described observation is the effect of [**_Gradient Starvation_ (GS)**](https://arxiv.org/abs/2011.09468). Pezeshki et al. coined and formalized this phenomenon of over-parameterized NNs in their work. According to them, _Gradient Starvation_ is a predisposition of NNs learned by cross-entropy, which occurs when the loss minimization is driven by only a feature subset regardless of the presence of other meaningful features. Thus, the NN becomes biased towards superficial features. Beyond demonstrating the effects of GS, Pezeshki et al. introduce **_Spectral Decoupling_ (SD)** regularization as a means to counteract gradient starvation. -""" - -# ╔═╡ 60905936-9373-44de-96e6-cb5d6dd72d33 -md""" -### Spectral Decoupling -Proceeding from the popular "ridge-regularized cross entropy": - -$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot log(1 + exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{θ}||^2$ - -with $\boldsymbol{Y}$ the diagonal matrix of $y$ and $\boldsymbol{θ}$ the concatenation of all layer's weights, SD is a variation where the L2 regularization term is replaced by a penalty on the raw predictions $\boldsymbol{ŷ}$: - -$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot log(1 + exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{ŷ}||^2$ - -This, as stated by the Pezeshki et al., uncouples the learning of multiple features. According to their work, a feature trained close to its optimum inhibits (starves) the training of other features which is thus prevented by SD. -""" - -# ╔═╡ 00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 -md""" -### Objective - -My project and thereby this notebook aims to reproduce and demonstrate the effects of GS on a simple 2D classification task. Furthermore, the proposed SD regularization method is implemented and compared to other methods. On top of that, spectral decoupling is not only applied in the context of SGD but all other investigated optimization approaches. -""" - -# ╔═╡ ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 -md""" -### Waning or Waxing Moon? A 2D Classification Experiment -A modified version of the [Moons dataset](https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_moons.html) was used with two different topologies as exemplified in figure 1. -The first topology allows to draw a line by a very small margin between the two moons. Therefore, it represents the fully linearly separable case (fig. 1, left). The second one applies a smaller offset to interleave both moons which only allows to separate them by learning a curved classification boundary (fig. 1, right). The task of the NN is to discriminate if a given data point belongs to the waning (blue) or waxing moon (red). -""" - -# ╔═╡ 2f766a9c-de6e-4f95-aa73-aacb486aaf61 -md""" -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/figure_1_edited.png")) -Figure 1: Two moons topologies; one linear separable (offset Δ1.0), the other separable by a curved boundary (offset Δ0.5). -""" - -# ╔═╡ 7f7ed965-fa08-4d6f-8908-dfc8876bdd02 -md""" -The reasoning of the researchers was that in the linear separable case the NN is not encouraged to learn a curved decision boundary because the loss becomes negligible by only discriminating the two moons along one axis, therefore learning only one feature and neglecting the other. In contrast to that, the interleaved moons can only be separated by learning both features which better accommodates the curvatures of the data structure in general. -They observed that a different choice of regularization and optimization methods like weight decay (WD) and _Adaptive Moment Estimation_ (ADAM), among others, are not able to stimulate the NN to learn a curved decision boundary. -Moreover, Pezeshki et al. explained that circumstance by cross-entropy loss learning which starves the gradients from the other feature, only reinforcing the superficial one. - -#### Repercussions of Gradient Starvation -Why does the shape of the decision boundary even matter as long as it discriminates the data points confidently? As stated by the authors, GS results in a very small distance between the data points and the boundary which translates into a lack of robustness when generalizing to new data. Contrarily, GS could also have a positive effect in terms of preventing overfitting by not learning non-dominant features. The authors made the comparison to the aformentioned Ockham's razor: GS leads to simpler decision boundaries and hence NNs that generalize better, thus GS being a form of implicit regularization. - -#### Implementation -The implementation is based on [Flux](https://fluxml.ai/), a ML framework for Julia. The NN consists of two hidden layers with 500 nodes each. Rectified Linear Unit (ReLu) is used as an activation function and learning is facilitated by cross-entropy loss for 1000 epochs on a data set with 300 observations (150 blue and 150 red points). In contrast to Pezeshki et al., I've opted to train in batches of 50 (6 batches in total) since this is a common practice for training NNs. -All networks are trained with gradient descent but with different regularization and optimization approaches. This yields the following different setups: -- Gradient descent (GD; no regularization/optimization) -- Gradient descent with momentum (stochastic gradient descent (SGD); no regularization) -- Gradient descent with weight decay (WD; L2 regularization) -- Gradient descent with _Adaptive Moment Estimation_ (ADAM; no regularization) - -Different sets of hyperparameters were evaluated while tinkering with the NNs. However, since the main objective is to reproduce the research conducted by Pezeshki et al., their set of hyperparameters were used: -- learning rate = 0.01 -- weight decay coefficient = 0.01 -- spectral decoupling coefficient = 0.003 -""" - -# ╔═╡ 88ddf822-622b-4d46-a0a4-a7d749c8bc48 -md""" -#### Results -""" - -# ╔═╡ 884bc572-9daa-40ad-8d76-56ec58728f63 -md""" -##### Linear separable case -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/GD_%CE%941.0.png")) -Figure 2: GD, moon offset = Δ1.0. The decision boundary is almost linear and the loss continuously falls while training. Accuracy reaches 100%. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/SGD_%CE%941.0.png")) -Figure 3: SGD, moon offset = Δ1.0. The decision boundary is similar to the one learned with GD but equally distant from both classes. Loss declines quicker and reaches an overall lower level compared to GD. Accuracy reaches 100% after less training. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/WD_%CE%941.0.png")) -Figure 4: WD, moon offset = Δ1.0, WD coefficient = 0.01. The boundary is similar to GD without WD. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/ADAM_%CE%941.0.png")) -Figure 5: ADAM, moon offset = Δ1.0. The decision boundary is slightly curved around the data points closest to the boundary. Loss converges to zero almost immediately while full accuracy is attained. -""" - -# ╔═╡ ac453412-0f71-45ee-aefb-0cc519467c6c -md""" -As we can see from figures 2 to 5, the decision boundaries of all networks exhibit some specificities. The GD network learned an almost linear decision boundary which appears to be closer to the blue-labeled moon. The boundary looks similar with SGD, but its distance to both moons is equal. Only the network with ADAM optimization learns a slight curvature around the points closest to the boundary. Furthermore, the ADAM classification learned the fastest as can be seen from the rapid decrease in loss and increase in accuracy. At the same time, the contour lines all closely condense around the decision boundary which indicates that the ADAM network is most confident in discriminating both classes. -""" - -# ╔═╡ 5e282e1a-62fc-4911-9b7e-f2631eeb2669 -md""" -##### Linear inseparable case - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/WD_%CE%940.5.png")) -Figure 6: GD, moon offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes within the 1000 training iterations. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/SGD_%CE%940.5.png")) -Figure 7: SGD, moon offset = Δ0.5. The network learns a curved decision boundary which is very close to the interleaving data points. Loss becomes negligible during learning. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/WD_%CE%940.5.png")) -Figure 8: WD, moon offset = Δ0.5. The network fails to learn a decision boundary to separate the two classes. Again, similar result compared to GD. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/ADAM_%CE%940.5.png")) -Figure 9: ADAM, moon offset = Δ0.5. The network learns a curved decision boundary. Loss fluctuates especially during the early iterations of training. -""" - -# ╔═╡ f7bae71f-885a-404d-985b-92d612e1f3d6 -md""" -SGD and ADAM are able to learn a curved decision boundary (fig. 7 and 9) while GD and WD fail to do so (fig. 6 and 8). Moreover, the decision boundary learned by ADAM better resembles the data structure. The contours of the ADAM network show a very strong confidence on the networks classification. This contrasts the results from the GS paper, where ADAM was not able to learn a decision boundary as curved and well-fitting the data structure as shown in fig. 9. -""" - -# ╔═╡ 2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f -md""" -#### Spectral Decoupling - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/GD_%CE%940.5_%2B_SD.png")) -Figure 10: GD with SD, moon offset = Δ0.5. No curved decision boundary is learned but the loss also did not converge. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/SGD_%CE%940.5_%2B_SD.png")) -Figure 11: SGD with SD, moon offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the moon data structure and shows invaginations at the data points of both moons closest to the boundary. Loss converges to ~15%. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/WD_%CE%940.5_%2B_SD.png")) -Figure 12: WD with SD, moon offset = Δ0.5. NN fails to separate the two data classes. Similar to GD with SD, loss could still be lowered with further training. - -$(Resource(".https://github.com/justinsane1337/GradientStarvation/blob/master/plots/ADAM_%CE%940.5_%2B_SD.png")) -Figure 13: ADAM with SD, moon offset = Δ0.5. The network learns a curved decision boundary which greatly resembles the data structure. Loss heavily oscillates and steadily increases with longer training. Interestingly, all data points of each respective class belong to the same contour. -""" - -# ╔═╡ 7460f609-c32b-4704-a361-71c98bb626cd -md""" -SD, originally applied as a regularization by Pezeshki et al. only with SGD, is able to yield curved decision boundaries which greatly resemble the data structure of the two moons in combination with SGD and ADAM optimization. While SGD learns a more angular boundary with low loss, ADAM learns a smoother one but exhibits a high loss with strong oscillations. Interestingly, ADAM learns to group all data points of a class within the same contour (fig. 13). GD and WD do not profit from SD regularization and don't learn a curved boundary. However, their loss profiles suggest improvement with further learning. -""" - -# ╔═╡ 87cc8172-d5cc-49ff-9a0e-fa734e1010c1 -md""" -#### Additional experiments -Given that the loss did not converge for GD + SD after 1000 epochs, the network was trained for 10000 iterations. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/GD_%CE%940.5_%2B_SD_10000.png")) -Figure 14: GD with SD, moon offset = Δ0.5. NN was trained 10 times longer (10000 iterations). The learned curved decision boundary resembles the result obtained with SGD + SD. - -To conquer the overfitting observed with ADAM + SD lower learning rates were evaluated. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/1e-3_ADAM_%CE%940.5_%2B_SD.png")) -Figure 15: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.001 still suffers from severe overfitting. Nonetheless, oscillations in loss are dampened. - -$(Resource("https://github.com/justinsane1337/GradientStarvation/blob/master/plots/1e-4_ADAM_%CE%940.5_%2B_SD.png")) -Figure 16: ADAM with SD on the moons with offset = Δ0.5 and a learning rate of 0.0001 shows lower loss compared to 0.001 but a decision boundary with a smaller margin. - -The additional experiments show that GD with SD performs equally well compared to SGD with SD but needs much more training for that. -The overfitting problem with ADAM + SD can be successfully conquered with lower learning rates. However, the shape of the decision boundary suffers with learning rates as low as 0.0001. From all three ADAM + SD experiments, the one learning at a rate of 0.001 performed best. -""" - -# ╔═╡ 2c740919-1fd4-45c4-91d1-ef3d3e081139 -md""" -## Discussion -Pezeshki et al. described gradient starvation as a learning proclivity that can exhibit both negative and positive effects on neural network learning. Considering the 2D classification task discussed in this notebook, the effect of learning a decision boundary with a small margin, thus supposedly hardly robust, could be exemplified for stochastic gradient descent. Combining SGD with spectral decoupling regularization was greatly able to enlarge the margin, therefore yielding a more robust network that not only focussed on a single dominant feature. On the other hand, the potential negative effect of overfitting by precisely learning less dominant features as well became evident through higher losses (SGD: 8.61% without vs. 13.37% with SD; ADAM: 2.00% without vs. 68.16%). -Contrarily to the results from Pezeshki et al., the here used neural network was perfectly able to learn curved decision boundaries featuring a large margin between the data points using ADAM without SD. While Pezeshki et al. trained their ADAM network with rates of 0.0001 and 0.001, I trained all networks under the same conditions, hence both SGD and ADAM with a learning rate of 0.01. Despite the higher learning rate, ADAM showed no sign of overfitting without SD. However, with SD the overfitting was extreme but ADAM coupled with SD regularization was also a combination of methods not assessed by the researchers. -Regarding the same matter, my approach to apply SD not only to SGD but all other optimizers simultaneously offered limited insight. First of all, the combination of WD (L2 regularization) and SD was expected to not work well since both methods are somewhat conflicting. ADAM performed well with SD but suffered from overfitting which can be conquered by a lower learning rate as demonstrated. Nonetheless, the network with purportedly lower loss learns a boundary with a smaller margin. - -All in all, the results of Pezeshki et al. could be reproduced with the exception of the network using ADAM, which was able to learn decision boundaries with similar characteristics as the ones learned by SGD with SD. - -Finally, it must be mentioned that Pezeshki et al. not only exemplified GS on the rather synthetical moon classification task but showed GS ramifications and the potential of SD as a regularization technique on more complex, well-established datasets like CIFAR, colored MNIST and CelebA which was sadly out of scope of this project. -""" - -# ╔═╡ 21854d7b-13ce-477c-bc61-1cb15536e24f -md""" -### Conclusion -Gradient starvation is another relevant side effect of gradient descent-learned neural networks that may be worth the attention in the optimization process of NNs. However, if GS is a blessing or a curse depends on the use case. Starved models, i.e. models concentrating on superficial features during learning, can be less prone to overfitting while adressing GS promises to increase robustness. -""" - -# ╔═╡ c95db58c-9d47-4ef0-9ede-234d7a693f41 -md""" -### References -Pezeshki, M., Kaba, S. O., Bengio, Y., Courville, A., Precup, D., & Lajoie, G. (2020). Gradient starvation: A learning proclivity in neural networks. arXiv preprint arXiv:2011.09468. - -Pizurica, A. (2020). E016330: Artificial Intelligence – Learning from examples [PDF course notes]. Ufora. - -Pizurica, A. (2020). E016330: Artificial Intelligence - Neural networks [PDF course notes]. Ufora. - -Russell, S., & Norvig, P. (2002). Artificial intelligence: a modern approach. - -Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. -""" - -# ╔═╡ 00000000-0000-0000-0000-000000000001 -PLUTO_PROJECT_TOML_CONTENTS = """ -[deps] -PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" - -[compat] -PlutoUI = "~0.7.32" -""" - -# ╔═╡ 00000000-0000-0000-0000-000000000002 -PLUTO_MANIFEST_TOML_CONTENTS = """ -# This file is machine-generated - editing it directly is not advised - -julia_version = "1.7.0" -manifest_format = "2.0" - -[[deps.AbstractPlutoDingetjes]] -deps = ["Pkg"] -git-tree-sha1 = "8eaf9f1b4921132a4cff3f36a1d9ba923b14a481" -uuid = "6e696c72-6542-2067-7265-42206c756150" -version = "1.1.4" - -[[deps.ArgTools]] -uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" - -[[deps.Artifacts]] -uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" - -[[deps.Base64]] -uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" - -[[deps.ColorTypes]] -deps = ["FixedPointNumbers", "Random"] -git-tree-sha1 = "024fe24d83e4a5bf5fc80501a314ce0d1aa35597" -uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f" -version = "0.11.0" - 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-[[deps.Test]] -deps = ["InteractiveUtils", "Logging", "Random", "Serialization"] -uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40" - -[[deps.UUIDs]] -deps = ["Random", "SHA"] -uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4" - -[[deps.Unicode]] -uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5" - -[[deps.Zlib_jll]] -deps = ["Libdl"] -uuid = "83775a58-1f1d-513f-b197-d71354ab007a" - -[[deps.libblastrampoline_jll]] -deps = ["Artifacts", "Libdl", "OpenBLAS_jll"] -uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" - -[[deps.nghttp2_jll]] -deps = ["Artifacts", "Libdl"] -uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" - -[[deps.p7zip_jll]] -deps = ["Artifacts", "Libdl"] -uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" -""" - -# ╔═╡ Cell order: -# ╟─5c563230-df6e-4f65-a106-a707d2454fe7 -# ╟─8bd774c7-1acf-4beb-bc99-4c303882d262 -# ╟─8f0cdbf1-0174-4d18-9072-2a8c5f015177 -# ╟─6a79cfe1-12f9-4ae0-a917-14b4f3a1b31b -# ╟─512e6240-76e7-11ec-2511-07ffc4b74d27 -# ╟─93bc80a8-673d-457b-96d8-e1bb0b09682e -# ╟─66a3b5cb-a465-42fb-a8c8-404d574312d2 -# ╟─4a14831d-7736-4af5-875f-05ae6e3cf7c4 -# ╟─60905936-9373-44de-96e6-cb5d6dd72d33 -# ╟─00d4161d-6f8f-42d6-8f13-7d5aa7e70ab3 -# ╟─ac02597f-a7ca-4ede-ab8a-c0739c60d0d5 -# ╟─2f766a9c-de6e-4f95-aa73-aacb486aaf61 -# ╟─7f7ed965-fa08-4d6f-8908-dfc8876bdd02 -# ╟─88ddf822-622b-4d46-a0a4-a7d749c8bc48 -# ╟─884bc572-9daa-40ad-8d76-56ec58728f63 -# ╟─ac453412-0f71-45ee-aefb-0cc519467c6c -# ╟─5e282e1a-62fc-4911-9b7e-f2631eeb2669 -# ╟─f7bae71f-885a-404d-985b-92d612e1f3d6 -# ╟─2bfb5f10-cb87-41a4-9dca-f3f0b1157c6f -# ╟─7460f609-c32b-4704-a361-71c98bb626cd -# ╟─87cc8172-d5cc-49ff-9a0e-fa734e1010c1 -# ╟─2c740919-1fd4-45c4-91d1-ef3d3e081139 -# ╟─21854d7b-13ce-477c-bc61-1cb15536e24f -# ╟─c95db58c-9d47-4ef0-9ede-234d7a693f41 -# ╟─00000000-0000-0000-0000-000000000001 -# ╟─00000000-0000-0000-0000-000000000002 From 150123fd505f8be7bc0c3142a057951f9ebc7dd4 Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 09:49:36 +0100 Subject: [PATCH 45/49] fix typo in readme --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index b1e1bbfd..85194304 100644 --- a/README.md +++ b/README.md @@ -5,6 +5,6 @@ Welcome to my STMO project! I wanted to explore the findings of Pezeshki et al. presented in their publication [Gradient Starvation: A Learning Proclivity in Neural Networks](https://arxiv.org/abs/2011.09468). The goal was to explain the problem of gradient starvation and to investigate its effects and a proposed remedy on a simple 2D classification task. -The notebook and the code of my project are separated! You can find them in their respective subfolders `notebook` and `src`. +The notebook and the code of my project are separate! You can find them in their respective subfolders `notebook` and `src`. All experiments conducted, including the related figures, can be simply repeated and re-created by running `include("src\\gradientstarvation.jl")` from a Julia REPL. \ No newline at end of file From 1eac1f5ea5a3fa13c6095eabadcae1ea8fc85022 Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 09:50:36 +0100 Subject: [PATCH 46/49] make instructions more clear --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 85194304..bd7c4105 100644 --- a/README.md +++ b/README.md @@ -7,4 +7,4 @@ The goal was to explain the problem of gradient starvation and to investigate it The notebook and the code of my project are separate! You can find them in their respective subfolders `notebook` and `src`. -All experiments conducted, including the related figures, can be simply repeated and re-created by running `include("src\\gradientstarvation.jl")` from a Julia REPL. \ No newline at end of file +All experiments conducted, including the related figures, can be simply repeated and re-created by running `include("src\\gradientstarvation.jl")` from a Julia REPL after checking out the project on your local machine. \ No newline at end of file From 0f855a5e7da425cb5d666158befe6f2af0720392 Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 09:51:37 +0100 Subject: [PATCH 47/49] remove static html version --- notebook/gradientstarvation.jl.html | 62 ----------------------------- 1 file changed, 62 deletions(-) delete mode 100644 notebook/gradientstarvation.jl.html diff --git a/notebook/gradientstarvation.jl.html b/notebook/gradientstarvation.jl.html deleted file mode 100644 index 56e2faec..00000000 --- a/notebook/gradientstarvation.jl.html +++ /dev/null @@ -1,62 +0,0 @@ - - - - - ⚡ Pluto.jl ⚡ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - From 5881d1c2f29d6e4989ce0c549bc3ad16c4e16b25 Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 12:27:20 +0100 Subject: [PATCH 48/49] fixes suggested by review --- notebook/gradientstarvation.jl | 23 +++++++++++------------ 1 file changed, 11 insertions(+), 12 deletions(-) diff --git a/notebook/gradientstarvation.jl b/notebook/gradientstarvation.jl index b29c88fb..1d6cfb6e 100644 --- a/notebook/gradientstarvation.jl +++ b/notebook/gradientstarvation.jl @@ -50,24 +50,23 @@ The input layer encodes the data presented to the network and feeds it weighted md""" ### Supervised Learning – Learning from Examples -Supervised learning tries to learn a function which maps input to output values based on known input-output pairs (i.e. the pair ($x$, $y$); a correct classification $y$ for a given input $x$). The observed outputs $y$ originate from an unknown function $f(x) = y$. Thus, the goal of the learning process is to approximate $f(x)$ by a hypothesis $h(x) \approx f(x)$ while minimizing the error of the approximation, refered to as loss. More formally, the loss is defined as the utility lost by substituting the correct classification $f(x) = y$ by the hypothesis $h(x) = \hat{y}$: +Supervised learning tries to learn a function which maps input to output values based on known input-output pairs (i.e. the pair ($x$, $y$); a correct classification $y$ for a given input $x$). The observed outputs $y$ originate from an unknown function $f(x) = y$. Thus, the goal of the learning process is to approximate $f(x)$ by a hypothesis $h(x) \approx f(x)$ while minimizing the error of the approximation, referred to as loss. More formally, the loss is defined as the utility lost by substituting the correct classification $f(x) = y$ by the hypothesis $h(x) = \hat{y}$: -$L(x, y, ŷ) = -Utility\\(result\ of\ using\ y\ given\ x\\) -- Utility\\(result\ of\ using\ ŷ\ given\ x\\)$ +$L(x, y, ŷ) = \text{Utility(result of using}\ y\ \text{given}\ x\\) +- \text{Utility(result of using}\ ŷ\ \text{given}\ x\\)$ Commonly applied loss functions are the mean absolute error and mean squared error loss. Especially popular for classification tasks is the cross-entropy loss, also called logarithmic loss: -$L(y, ŷ) = -\sum_{i}y_i\cdot log(ŷ_i)$ +$L(y, ŷ) = -\sum_{i}y_i\cdot \log(ŷ_i)$ The learning process maximizes the expected utility through choosing the optimal hypothesis by minimizing the loss function. In practice, this is estimated by the empirical loss observed on the set $X$ of $N$ training examples. -$EmpiricalLoss(h) = \frac{1}{N}\cdot\sum_{(x, y) \in X} L(y, h(x))$ +$\text{EmpiricalLoss}(h) = \frac{1}{N}\cdot\sum_{(x, y) \in X} L(y, h(x))$ But what about the weights I’ve introduced earlier as the “actual learned parameters”? Keep in mind that a hypothesis $h$ is a function and the weights simply represent the function coefficients. For the linear case that would mean $h(x) = w_1 x + w_0$. In the multivariate case there is no closed-form solution for the minimum loss. Therefore, we are confronted with an optimization problem which can be solved by gradient descent. The weight update with learning rate $\alpha$ and the vector of weights $\bf{w}$ looks like this: -$w_i = w_i - \alpha\cdot \frac{\partial}{\partial w_i}\cdot Loss(\bf{w})$ +$w_i = w_i - \alpha\cdot \frac{\partial}{\partial w_i}\cdot \text{Loss}(\bf{w})$ """ @@ -78,12 +77,12 @@ md""" > "Entities should not be multiplied beyond necessity." – William of Ockham This quote, known as Ockham’s razor and usually paraphrased as “simple solutions are better solutions” captures the core idea of regularization. The aim of regularization is to find the hypothesis which minimizes the total cost, defined as the sum of empirical loss and its complexity: -$Cost(h) = EmpiricalLoss(h) + \lambda\cdot Complexity(h)$ +$\text{Cost}(h) = \text{EmpiricalLoss}(h) + \lambda\cdot \text{Complexity}(h)$ -Regularization penalizes complex hypotheses (mind the added regularization term in the formula $\lambda\cdot Complexity(h)$) and favors more regular functions. It limits the values that the weights can take during the learning process. Therefore, regularization avoids overfitting of the data. +Regularization penalizes complex hypotheses (mind the added regularization term in the formula $\lambda\cdot \text{Complexity}(h)$) and favors more regular functions. It limits the values that the weights can take during the learning process. Therefore, regularization avoids overfitting of the data. The estimated best hypothesis $ĥ^*$ from the hypothesis space $\mathcal{H}$ now becomes: -$ĥ^* = \arg\min_{h \in \mathcal{H}} Cost(h)$ +$ĥ^* = \arg\min_{h \in \mathcal{H}} \text{Cost}(h)$ ### Optimization Given the major workings of supervised learning, it should have become clear that learning is a mathematical optimization problem. A classic way to achieve the minimization of the cost is the application of gradient descent (GD). However, the path the gradient take on the surface of the solution space can limit its ability to find the global optimum. To conquer that, the steps can be modified by additional optimization methods. Common choices for that purpose are the addition of momentum, resulting in stochastic gradient descent (SGD); an adaptive learning rate, different forms of parameter initialization as well as normalization techniques. @@ -102,11 +101,11 @@ md""" ### Spectral Decoupling Proceeding from the popular "ridge-regularized cross entropy": -$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot log(1 + exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{θ}||^2$ +$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot \log(1 + \exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{θ}||^2$ with $\boldsymbol{Y}$ the diagonal matrix of $y$ and $\boldsymbol{θ}$ the concatenation of all layer's weights, SD is a variation where the L2 regularization term is replaced by a penalty on the raw predictions $\boldsymbol{ŷ}$: -$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot log(1 + exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{ŷ}||^2$ +$\mathcal{L}(\boldsymbol{θ}) = \boldsymbol{1} \cdot \log(1 + \exp(-\boldsymbol{Y}\boldsymbol{ŷ})) + \frac{λ}{2} ||\boldsymbol{ŷ}||^2$ This, as stated by the Pezeshki et al., uncouples the learning of multiple features. According to their work, a feature trained close to its optimum inhibits (starves) the training of other features which is thus prevented by SD. """ From fcc1a5cc7e0d1cf1214a61bc6c196d92ea46a2fd Mon Sep 17 00:00:00 2001 From: Peter Date: Mon, 31 Jan 2022 17:46:01 +0100 Subject: [PATCH 49/49] update package documentation --- docs/Project.toml | 3 ++- docs/make.jl | 26 +++++++++++++++++++++----- docs/src/index.md | 5 +---- docs/src/man/data.md | 20 ++++++++++++++++++++ docs/src/man/example.md | 14 -------------- docs/src/man/experiments.md | 7 +++++++ docs/src/man/figures.md | 7 +++++++ docs/src/man/neuralnetwork.md | 14 ++++++++++++++ docs/src/man/plotlib.md | 10 ++++++++++ docs/src/man/plotutils.md | 9 +++++++++ docs/src/man/regularization.md | 7 +++++++ src/data.jl | 4 +--- src/gradientstarvation.jl | 14 +++++++++++--- src/neuralnetwork.jl | 2 +- src/regularization.jl | 10 ++++++++-- 15 files changed, 119 insertions(+), 33 deletions(-) create mode 100644 docs/src/man/data.md delete mode 100644 docs/src/man/example.md create mode 100644 docs/src/man/experiments.md create mode 100644 docs/src/man/figures.md create mode 100644 docs/src/man/neuralnetwork.md create mode 100644 docs/src/man/plotlib.md create mode 100644 docs/src/man/plotutils.md create mode 100644 docs/src/man/regularization.md diff --git a/docs/Project.toml b/docs/Project.toml index f1f63797..09530034 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -1,3 +1,4 @@ [deps] +DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8" Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" -GradientStarvation = "8e7dc78f-7190-4116-bf88-3d148ae9f088" +GradientStarvation = "b11d05d9-c3d3-46bc-b9c9-1875ab7cf247" diff --git a/docs/make.jl b/docs/make.jl index 3b1a8a61..5f08d008 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -1,16 +1,32 @@ using Documenter - -using GradientStarvation using GradientStarvation.Data +using GradientStarvation.Figures +using GradientStarvation.Experiments using GradientStarvation.NeuralNetwork +using GradientStarvation.PlotLib +using GradientStarvation.PlotUtils +using GradientStarvation.Regularization makedocs( - sitename = "Gradient Starvation - An STMO ZOO Project", + sitename = "Gradient Starvation", format = Documenter.HTML(), - modules = [Data, NeuralNetwork], + modules = [ + GradientStarvation.Data, + GradientStarvation.Figures, + GradientStarvation.Experiments, + GradientStarvation.NeuralNetwork, + GradientStarvation.PlotLib, + GradientStarvation.PlotUtils, + GradientStarvation.Regularization + ], pages = Any[ "Data" => "man/data.md", - "Neural Network" => "man/neuralnetwork.md" + "Figures" => "man/figures.md", + "Experiments" => "man/experiments.md", + "Neural Network" => "man/neuralnetwork.md", + "Plots" => "man/plotlib.md", + "Plot Utils" => "man/plotutils.md", + "Regularization" => "man/regularization.md" ] ) diff --git a/docs/src/index.md b/docs/src/index.md index ad33d99d..e6bdf681 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -1,7 +1,4 @@ -# STMO ZOO documentation - -This is the documentation page for the STMO ZOO packages (edition 2020-2021). This is the place for the **documentation** -of your function. So explain the main functionality of your module and list the documentation of your funtions. +# Gradient Starvation documentation ```@contents ``` \ No newline at end of file diff --git a/docs/src/man/data.md b/docs/src/man/data.md new file mode 100644 index 00000000..3196a500 --- /dev/null +++ b/docs/src/man/data.md @@ -0,0 +1,20 @@ +## Data + +This module provides basic functionality to obtain training-ready `Flux.DataLoader`s. + +Following methods are exported: + +```@docs +get_moon_data_loader +get_nmist_data_loader +``` + +These helper functions are private to the module: + +```@docs +generate_moons_from_publication +generate_moons +apply_moon_offset +moon_class_1 +moon_class_2 +``` diff --git a/docs/src/man/example.md b/docs/src/man/example.md deleted file mode 100644 index 08857ecb..00000000 --- a/docs/src/man/example.md +++ /dev/null @@ -1,14 +0,0 @@ -## Example - -This is the example code, written by Michiel Stock. It contains some basic functionality -to solve quadratic systems of the form: - -``\min_{\mathbf{x}} \frac{1}{2} \mathbf{x}^\intercal P\mathbf{x} + \mathbf{q} \cdot \mathbf{x} + r\,,`` - -It contains a function to solve this optimization problem and a helper function that generates a quadratic function from -the parameters in their canonical form. - -```@docs -solve_quadratic_system -quadratic_function -``` \ No newline at end of file diff --git a/docs/src/man/experiments.md b/docs/src/man/experiments.md new file mode 100644 index 00000000..f03beeba --- /dev/null +++ b/docs/src/man/experiments.md @@ -0,0 +1,7 @@ +## Experiments + +The Experiments module provides a single function to run experiments as defined by a `DataStructures.OrderedDict` containing experiments according to the `Experiment` struct. + +```@docs +run_experiments +``` \ No newline at end of file diff --git a/docs/src/man/figures.md b/docs/src/man/figures.md new file mode 100644 index 00000000..4b88a197 --- /dev/null +++ b/docs/src/man/figures.md @@ -0,0 +1,7 @@ +## Figures + +This module collects all functions to create any special figures appearing in the notebook. This does not include plots that are produced along during the processing of experiments. + +```@docs +create_figure_1 +``` \ No newline at end of file diff --git a/docs/src/man/neuralnetwork.md b/docs/src/man/neuralnetwork.md new file mode 100644 index 00000000..beeddc4b --- /dev/null +++ b/docs/src/man/neuralnetwork.md @@ -0,0 +1,14 @@ +# Neural Network + +Here, we basically have the main module of the whole project. The neural network architecture is defined in this module, as well as functions to train it, to retrieve an optimization strategy or obtain information about training loss and accuracy. + +```@docs +get_loss_and_accuracy +train +``` + +Private helper methods: +```@docs +neural_network +get_optimizer +``` \ No newline at end of file diff --git a/docs/src/man/plotlib.md b/docs/src/man/plotlib.md new file mode 100644 index 00000000..1d702e20 --- /dev/null +++ b/docs/src/man/plotlib.md @@ -0,0 +1,10 @@ +# Plot Library + +To avoid naming confusion with `Plots`, my plot module was coined `PlotLib` and contains solely functions that are concerned with the creation of plots. Thus, all of these return ready to display `PlotlyJS` plots. In contrast to the functionality provided in `Figures`, functions presented here are more raw or general purpose in the sense that they plot a certain aspect of the neural network and the training process. + +```@docs +plot_train_and_test_data +plot_loss_and_accuracy +plot_features +plot_decision_boundary +``` \ No newline at end of file diff --git a/docs/src/man/plotutils.md b/docs/src/man/plotutils.md new file mode 100644 index 00000000..cf0b9faf --- /dev/null +++ b/docs/src/man/plotutils.md @@ -0,0 +1,9 @@ +# Plot Utils + +Addtional functionality related to plotting. These functions are helper functions that do not directly produce plots. + +```@docs +get_custom_rdbu_scale +get_plot_resolution +map_bool_to_color +``` \ No newline at end of file diff --git a/docs/src/man/regularization.md b/docs/src/man/regularization.md new file mode 100644 index 00000000..f496474e --- /dev/null +++ b/docs/src/man/regularization.md @@ -0,0 +1,7 @@ +## Regularization + +This module contains custom-implemented regularizations like spectral decoupling. + +```@docs +spectral_decoupling +``` \ No newline at end of file diff --git a/src/data.jl b/src/data.jl index 37f7e75b..7842b3ae 100644 --- a/src/data.jl +++ b/src/data.jl @@ -102,12 +102,10 @@ end Separates both moons by the specified offset. """ function apply_moon_offset(X, y, offset) - # @assert direction in ["x", "y"] "Error: direction must be either x or y" # use offset as total offset, thus devide by number of labels; assume 2 labels offset /= 2 - # direction == "x" ? direction = 1 : direction = 2 - moons = [] + for label in unique(y) push!(moons, findall(l -> l == label, y)) end diff --git a/src/gradientstarvation.jl b/src/gradientstarvation.jl index f4d09213..775563a1 100644 --- a/src/gradientstarvation.jl +++ b/src/gradientstarvation.jl @@ -7,16 +7,24 @@ if isfile("Project.toml") && isfile("Manifest.toml") end using DataStructures -using PlotlyJS +include("data.jl") include("experiments.jl") include("figures.jl") +include("neuralnetwork.jl") +include("plotlib.jl") +include("plotutils.jl") +include("regularization.jl") include("structs.jl") +# includes & export needed for documentation +export Data, Figures, Experiments, NeuralNetwork, PlotLib, PlotUtils, Regularization + # figure 1 Figures.create_figure_1("medium") # figures 2-13 +# run different experiments on the 2D classification task (moons) experiments = OrderedDict( "GD Δ0.5" => Experiment("GD", 0.5, false), "GD Δ1.0" => Experiment("GD", 1.0, false), @@ -40,13 +48,13 @@ experiments = OrderedDict( ) Experiments.run_experiments(experiments, show_plots = false, random_moons = false) -# additional experiments +# additional experiments with different parameters + # figure 14 Experiments.run_experiments( OrderedDict("GD Δ0.5 + SD 10000" => Experiment("GD", 0.5, true)), show_plots = false, random_moons = false, epochs = 10000 ) - # figure 15, 16 add_exp_adam = OrderedDict("ADAM Δ0.5 + SD" => Experiment("ADAM", 0.5, true)) Experiments.run_experiments(add_exp_adam, show_plots = false, random_moons = false, learning_rate = 1e-4, prefix = "1e-4") diff --git a/src/neuralnetwork.jl b/src/neuralnetwork.jl index 71894e4a..f7b17fe2 100644 --- a/src/neuralnetwork.jl +++ b/src/neuralnetwork.jl @@ -6,7 +6,7 @@ include("regularization.jl") using Flux using Flux: onecold using Flux: Optimiser -using Flux.Losses: binarycrossentropy, crossentropy, logitcrossentropy, logitbinarycrossentropy +using Flux.Losses: logitcrossentropy using LinearAlgebra using MLDatasets using ProgressMeter diff --git a/src/regularization.jl b/src/regularization.jl index 172db5a4..6b4ce5da 100644 --- a/src/regularization.jl +++ b/src/regularization.jl @@ -1,10 +1,16 @@ module Regularization -using Flux.Losses: crossentropy, logitcrossentropy, logitbinarycrossentropy +using Flux.Losses: logitcrossentropy using Statistics +""" + spectral_decoupling(ŷ, y, λ) + +Logitcrossentropy extended with spectral decoupling regularization as proposed by +[Pezeshki et al.](https://arxiv.org/abs/2011.09468). The implementation here somewhat differs from the formula presented +in the paper but the methodology is consistent with the python code provided by the authors. +""" function spectral_decoupling(ŷ, y, λ) - # return mean(1.0 .* log.(1.0 .+ exp.(-ŷ .* y))) + λ * mean(ŷ .^ 2) return logitcrossentropy(ŷ, y) + λ * mean(ŷ .^ 2) end

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