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22 changes: 11 additions & 11 deletions README.md
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### 目录

- 第0章 [绪论](https://datawhalechina.github.io/key-book/#/chapter0/chapter0)
- 第1章 [预备知识](https://datawhalechina.github.io/key-book/#/chapter1/chapter1)
- 第2章 [可学性](https://datawhalechina.github.io/key-book/#/chapter2/chapter2)
- 第3章 [复杂度](https://datawhalechina.github.io/key-book/#/chapter3/chapter3)
- 第4章 [泛化界](https://datawhalechina.github.io/key-book/#/chapter4/chapter4)
- 第5章 [稳定性](https://datawhalechina.github.io/key-book/#/chapter5/chapter5)
- 第6章 [一致性](https://datawhalechina.github.io/key-book/#/chapter6/chapter6)
- 第7章 [收敛率](https://datawhalechina.github.io/key-book/#/chapter7/chapter7)
- 第8章 [遗憾界](https://datawhalechina.github.io/key-book/#/chapter8/chapter8)
- 第0章 [绪论](https://datawhalechina.github.io/key-book/#/chapter0)
- 第1章 [预备知识](https://datawhalechina.github.io/key-book/#/chapter1)
- 第2章 [可学性](https://datawhalechina.github.io/key-book/#/chapter2)
- 第3章 [复杂度](https://datawhalechina.github.io/key-book/#/chapter3)
- 第4章 [泛化界](https://datawhalechina.github.io/key-book/#/chapter4)
- 第5章 [稳定性](https://datawhalechina.github.io/key-book/#/chapter5)
- 第6章 [一致性](https://datawhalechina.github.io/key-book/#/chapter6)
- 第7章 [收敛率](https://datawhalechina.github.io/key-book/#/chapter7)
- 第8章 [遗憾界](https://datawhalechina.github.io/key-book/#/chapter8)

### 选用的《机器学习理论导引》版本

<center><img src="docs/images/mlt.jpg" width="300" height= "300"></center>
<center><img src="docs/images/original_book.jpg" width="300" height= "300"></center>

> 版次:2020年6月第1版<br>
Expand All @@ -61,7 +61,7 @@ https://github.com/datawhalechina/key-book/releases
## 关注我们
<div align=center>
<p>扫描下方二维码,或关注公众号「Datawhale」,然后回复关键词“钥匙书”,即可加入“钥匙书读者交流群”</p>
<img src="docs/images/qr.jpeg" width="300" height= "300">
<img src="docs/images/qr_code.jpg" width="300" height= "300">
<p>或者加入QQ群:704768061</p>
</div>

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### 目录

- 第0章 [绪论](https://datawhalechina.github.io/key-book/#/chapter0/chapter0)
- 第1章 [预备知识](https://datawhalechina.github.io/key-book/#/chapter1/chapter1)
- 第2章 [可学性](https://datawhalechina.github.io/key-book/#/chapter2/chapter2)
- 第3章 [复杂度](https://datawhalechina.github.io/key-book/#/chapter3/chapter3)
- 第4章 [泛化界](https://datawhalechina.github.io/key-book/#/chapter4/chapter4)
- 第5章 [稳定性](https://datawhalechina.github.io/key-book/#/chapter5/chapter5)
- 第6章 [一致性](https://datawhalechina.github.io/key-book/#/chapter6/chapter6)
- 第7章 [收敛率](https://datawhalechina.github.io/key-book/#/chapter7/chapter7)
- 第8章 [遗憾界](https://datawhalechina.github.io/key-book/#/chapter8/chapter8)
- 第0章 [绪论](https://datawhalechina.github.io/key-book/#/chapter0)
- 第1章 [预备知识](https://datawhalechina.github.io/key-book/#/chapter1)
- 第2章 [可学性](https://datawhalechina.github.io/key-book/#/chapter2)
- 第3章 [复杂度](https://datawhalechina.github.io/key-book/#/chapter3)
- 第4章 [泛化界](https://datawhalechina.github.io/key-book/#/chapter4)
- 第5章 [稳定性](https://datawhalechina.github.io/key-book/#/chapter5)
- 第6章 [一致性](https://datawhalechina.github.io/key-book/#/chapter6)
- 第7章 [收敛率](https://datawhalechina.github.io/key-book/#/chapter7)
- 第8章 [遗憾界](https://datawhalechina.github.io/key-book/#/chapter8)

### 选用的《机器学习理论导引》版本

<center><img src="res/mlt.jpg" width="300" height= "300"></center>
<center><img src="res/original_book.jpg" width="300" height= "300"></center>

> 版次:2020年6月第1版<br>
Expand All @@ -61,7 +61,7 @@ https://github.com/datawhalechina/key-book/releases
## 关注我们
<div align=center>
<p>扫描下方二维码,或关注公众号「Datawhale」,然后回复关键词“钥匙书”,即可加入“钥匙书读者交流群”</p>
<img src="res/qr.jpeg" width="300" height= "300">
<img src="res/qr_code.jpg" width="300" height= "300">
<p>或者加入QQ群:704768061</p>
</div>

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- [第0章 序言](chapter0/chapter0.md)
- [第1章 预备知识](chapter1/chapter1.md)
- [第2章 可学性](chapter2/chapter2.md)
- [第3章 复杂度](chapter3/chapter3.md)
- [第4章 泛化界](chapter4/chapter4.md)
- [第5章 稳定性](chapter5/chapter5.md)
- [第6章 一致性](chapter6/chapter6.md)
- [第7章 收敛率](chapter7/chapter7.md)
- [第8章 遗憾界](chapter8/chapter8.md)

- [第0章 序言](chapter0.md)
- [第1章 预备知识](chapter1.md)
- [第2章 可学性](chapter2.md)
- [第3章 复杂度](chapter3.md)
- [第4章 泛化界](chapter4.md)
- [第5章 稳定性](chapter5.md)
- [第6章 一致性](chapter6.md)
- [第7章 收敛率](chapter7.md)
- [第8章 遗憾界](chapter8.md)
- [参考文献](references.md)
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虽然这种理解适用于机器学习,但我们必须注意,对于深度学习,需要进一步的考虑。例如双下降现象与传统机器学习理论相矛盾,后者认为增加模型大小和数据量通常会提高模型的泛化性能。

![double_descent](../images/double_descent.png)
![double_descent](images/double_descent.png)

双下降现象中描绘模型泛化性能的曲线图由三个阶段组成:
1. 第一阶段:当模型规模小且数据量不足时,模型泛化性能较差。
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**示例:**对于二维空间$R^2$中的三个点,线性分类器$sign(wx+b)$可以实现三点的所有对分,但无法实现四点的所有对分,如下图所示:

![shattering](../images/shattering.png)
![shattering](images/shattering.png)

因此,线性分类器在$R^2$中的VC维为3。
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一个典型的例子是我们熟悉的决策树模型:

![decision_tree](../images/decision_tree.png)
![decision_tree](images/decision_tree.png)

每当构造一个决策树的节点时,相当于在样本空间上进行了一次划分(即划分机制)。这种洞察方式同样适用于解释剪枝操作,即通过减少不必要的节点来简化树结构,同时保持或提高模型的性能。

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# 参考文献

Abernethy, Jacob, et al. "Optimal strategies and minimax lower bounds for online convex games." Proceedings of the 21st annual conference on learning theory. 2008.

Auer, Peter. "Using confidence bounds for exploitation-exploration trade-offs." Journal of Machine Learning Research 3.Nov (2002): 397-422.

Bouneffouf, Djallel. "Finite-time analysis of the multi-armed bandit problem with known trend." 2016 IEEE Congress on Evolutionary Computation (CEC). IEEE, 2016.

Bubeck, Sébastien, Ronen Eldan, and Yin Tat Lee. "Kernel-based methods for bandit convex optimization." Journal of the ACM (JACM) 68.4 (2021): 1-35.

Boyd, Stephen, and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.

Cesa-Bianchi, N., Conconi, A., and Gentile, C. "On the Generalization Ability of On-Line Learning Algorithms." IEEE Transactions on Information Theory, vol. 50, no. 9, 2004, pp. 2050-2057, doi:10.1109/TIT.2004.833339.

Devroye, Luc, László Györfi, and Gábor Lugosi. A probabilistic theory of pattern recognition. Vol. 31. Springer Science & Business Media, 2013.

Feller, William. "An introduction to probability theory and its applications." (1971).

Flaxman, Abraham D., Adam Tauman Kalai, and H. Brendan McMahan. "Online convex optimization in the bandit setting: gradient descent without a gradient." arXiv preprint cs/0408007 (2004).

Hazan, Elad, Amit Agarwal, and Satyen Kale. "Logarithmic regret algorithms for online convex optimization." Machine Learning 69.2 (2007): 169-192.

Kakade, Sham M., and Ambuj Tewari. "On the generalization ability of online strongly convex programming algorithms." Advances in neural information processing systems 21 (2008).

Kearns, Michael J., and Umesh Vazirani. An introduction to computational learning theory. MIT press, 1994.

Lai, Tze Leung, and Herbert Robbins. "Asymptotically efficient adaptive allocation rules." Advances in applied mathematics 6.1 (1985): 4-22.

McAllester, David A. "PAC-Bayesian stochastic model selection." Machine Learning 51.1 (2003): 5-21.

Mohri, Mehryar. "Foundations of machine learning." (2018).

Nakkiran, Preetum, et al. "Deep double descent: Where bigger models and more data hurt." Journal of Statistical Mechanics: Theory and Experiment 2021.12 (2021): 124003.

Penot, Jean-Paul. "On regularity conditions in mathematical programming." Optimality and Stability in Mathematical Programming (1982): 167-199.

Robbins, Herbert. "Some aspects of the sequential design of experiments." (1952): 527-535.

Thompson, William R. "On the likelihood that one unknown probability exceeds another in view of the evidence of two samples." Biometrika 25.3-4 (1933): 285-294.

Wainwright, Martin J. High-dimensional statistics: A non-asymptotic viewpoint. Vol. 48. Cambridge university press, 2019.

Wang, Guanghui, Shiyin Lu, and Lijun Zhang. "Adaptivity and optimality: A universal algorithm for online convex optimization." Uncertainty in Artificial Intelligence. PMLR, 2020.

Zhang, Lijun, Shiyin Lu, and Zhi-Hua Zhou. "Adaptive online learning in dynamic environments." Advances in neural information processing systems 31 (2018)

Zhang, Lijun, Tie-Yan Liu, and Zhi-Hua Zhou. "Adaptive regret of convex and smooth functions." International Conference on Machine Learning. PMLR, 2019.

Zinkevich, Martin. "Online convex programming and generalized infinitesimal gradient ascent." Proceedings of the 20th international conference on machine learning (icml-03). 2003.

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