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摘要: 假设空间复杂性是统计学习理论中用于分析学习模型泛化能力的关键因素.与数据无关的复杂度不同，Rademacher复杂度是与数据分布相关的，因而通常能得到比传统复杂度更紧致的泛化界表达.近年来，Rademacher复杂度在统计学习理论泛化能力分析的应用发展中起到了重要的作用.鉴于其重要性，本文梳理了各种形式的Rademacher复杂度及其与传统复杂度之间的关联性，并探讨了基于Rademacher复杂度进行学习模型泛化能力分析的基本技巧.考虑样本数据的独立同分布和非独立同分布两种产生环境，总结并分析了Rademacher复杂度在泛化能力分析方面的研究现状.展望了当前Rademacher复杂度在非监督框架与非序列环境等方面研究的不足，及其进一步应用与发展.
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关键词:
- 机器学习 /
- 统计学习理论 /
- 泛化界 /
- Rademacher 复杂度
Abstract: Measuring the complexity of a hypothesis space plays a crucial role in statistical learning theory. Unlike those data-independent complexities, Rademacher complexity, which is data-dependent, can attain a much more compact generalization representation. In recent years, Rademacher complexity has attracted more attention and found broad applications in the development of statistical learning theory. Because of its importance, in this paper we review several complexity measures of function classes and their relations with Rademacher complexities. Next, we describe the techniques of Rademacher complexities in generalization analysis. Then, we present the recent researches of Rademacher complexity learning bounds for independent and identical distribution (i.i.d.) and non-independent and identical distribution (non-i.i.d.). Finally, we discuss the potential issues and possible directions of Rademacher complexities in statistical learning theory. -
表 1 Rademacher 复杂度及传统复杂度
Table 1 Rademacher complexities and kinds of complexities of function classes
复杂度类型 本数据集
产生环境复杂度名称 传统复杂度 i.i.d./non-
i.i.d.VC 熵,退火 VC 熵,生长函数,VC 维,
覆盖数,伪维度,Fat-shattering维等
经典 Rademacher 复杂度,局部
Rademacher 复杂度Rade-
macher
复杂度i.i.d. Rademacher chaos 复杂度,单模态
Rademacher 复杂度,
多模态 Rademacher 复杂度,Dropout
Rademacher 复杂度non-i.i.d. 独立不同分布 Rademacher 复杂度,
块Rademacher 复杂度,
序列 Rademacher 复杂度
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表 2 i.i.d.情形的泛化界分析
Table 2 Generalization analysis for i.i.d.
本数据集产生环境 学习策略 假设空间 泛化能力 i.i.d. ERM 1) $\mathcal{X}\rightarrow\{-1,+1\}$ O$O\left( \frac{1}{\sqrt{n}} \right)$ [52],O$\left(\frac{1}{n} \right)$[53-55]等 2) $\mathcal{X}\rightarrow{\bf R}$ O$\left( \frac{1}{n} \right)$[47],$\text{O}{{\left( \frac{{{\log }^{p}}n}{n} \right)}^{1/(2-\alpha )}}$ [28, 32]等 3) ${\bf R}^{d}\rightarrow{\bf R}$ O$O\left( \frac{1}{\sqrt{n}} \right)$ [51] 4) $\mathcal{X}\rightarrow\{-1,+1\}^{L}$ O$O\left( \frac{1}{\sqrt{n}} \right)$ [56],$O\left( \frac{\log n}{n} \right)$[33] 5) 自由样条函数空间 $O\left( {{\left( \frac{\log n}{n} \right)}^{2\alpha /(1+2\alpha )}} \right)$ [31-32]等 6) $\mathcal{X}× R_{s}\rightarrow\mathcal{{\bf R}}$ $O\left( \frac{{{p}^{(k+1)/2}}}{\sqrt{n}}+\frac{1}{\sqrt{n}} \right),O\left( \frac{{{p}^{k+1}}}{\sqrt{n}}+\frac{1}{\sqrt{n}} \right)$[34] 正则化 7) RKHS $O\left(\frac{\sqrt{{\rm{tr}}(G)}}{n}+\frac{1}{\sqrt{n}}\right)$ [48] 8) ${S}^{d}$ $O\left( \frac{1}{\sqrt{n}} \right)$ [36]等 9) ${S}^{d×(md)}$ $O\left( \frac{1}{\sqrt{n}} \right)$ [27, 32]等 SRM 10) 神经网络空间 $O\left(\left(\frac{1}{n}\right)^{\frac{1}{2-\alpha_{p}}}\right)$[30, 32]
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表 3 non-i.i.d.情形的泛化界分析
Table 3 Generalization analysis for non-i.i.d.
样本数据集产生环境 假设空间 泛化能力 独立不同分布 1)$\mathcal{X}\rightarrow\mathcal{Y}$ $O\left(\frac{1}{\sqrt{n}}\right)$[37] 平稳分布且$\beta\textrm{-mixing}$ 2) $\mathcal{X}\rightarrow{\bf R}$ $O\left(\frac{\sqrt{{\rm{tr}}(G)}}{n}+\frac{1}{\sqrt{n}}\right)$[38] 非平稳分布且$\beta\textrm{-mixing}$ 3) $\mathcal{X}\rightarrow\mathcal{Y}$ 非平稳性可能导致
不收敛于0[42]鞅 4) $\mathcal{Z}\rightarrow $[-1,1] 一致收敛[39-41]
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