[论文解读] Generalization Bounds: Perspectives from Information Theory and PAC-Bayes

Fredrik Hellström, Giuseppe Durisi+2 more|arXiv (Cornell University)|Sep 8, 2023

Bayesian Modeling and Causal Inference被引用 6

一句话总结

本专著统一了机器学习中泛化界的信息论与PAC-贝叶斯方法，展示了它们共享的模块化结构和共同的底层原理。该研究证明，条件互信息（CMI）为分析泛化提供了强有力的框架，适用于深度学习和迭代算法，其界比以往方法更紧致且更具可解释性。

ABSTRACT

A fundamental question in theoretical machine learning is generalization. Over the past decades, the PAC-Bayesian approach has been established as a flexible framework to address the generalization capabilities of machine learning algorithms, and design new ones. Recently, it has garnered increased interest due to its potential applicability for a variety of learning algorithms, including deep neural networks. In parallel, an information-theoretic view of generalization has developed, wherein the relation between generalization and various information measures has been established. This framework is intimately connected to the PAC-Bayesian approach, and a number of results have been independently discovered in both strands. In this monograph, we highlight this strong connection and present a unified treatment of PAC-Bayesian and information-theoretic generalization bounds. We present techniques and results that the two perspectives have in common, and discuss the approaches and interpretations that differ. In particular, we demonstrate how many proofs in the area share a modular structure, through which the underlying ideas can be intuited. We pay special attention to the conditional mutual information (CMI) framework; analytical studies of the information complexity of learning algorithms; and the application of the proposed methods to deep learning. This monograph is intended to provide a comprehensive introduction to information-theoretic generalization bounds and their connection to PAC-Bayes, serving as a foundation from which the most recent developments are accessible. It is aimed broadly towards researchers with an interest in generalization and theoretical machine learning.

研究动机与目标

• 统一机器学习中信息论与PAC-贝叶斯泛化界的框架。
• 证明两种方法在信息度量基础上共享模块化的证明结构。
• 利用条件互信息（CMI）分析学习算法的信息复杂度。
• 将泛化界的适用范围扩展至深度神经网络和迭代学习方法。
• 阐明信息论界最优或次优的条件，并识别该领域中的开放问题。

提出的方法

• 通过一种测度变换技术推导泛化界，将信息度量与泛化误差联系起来。
• 将条件互信息（CMI）作为核心工具，量化训练数据与学习假设之间的依赖关系。
• 通过统一的模块化框架重构并重新解释现有的PAC-贝叶斯与信息论界。
• 以在线学习的遗憾界为来源，通过转换方法推导新的泛化界。
• 使用分析与数值技术分析特定学习算法，包括深度神经网络。
• 提出一种系统性方法，根据推导结构与问题设定选择合适的信息度量。

实验结果

研究问题

• RQ1 在底层原理与证明结构方面，信息论与PAC-贝叶斯泛化界之间有何关联？
• RQ2 在何种场景下，信息论界能实现最优或次优性能？
• RQ3 条件互信息（CMI）框架能否用于推导更紧致且更具可解释性的泛化界？
• RQ4 测度变换技术在统一这两种方法中起到什么作用？
• RQ5 泛化界如何适应迭代与深度学习算法？

主要发现

• 信息论与PAC-贝叶斯方法共享一种共同的模块化证明结构，使得泛化界的统一处理成为可能。
• 条件互信息（CMI）为分析学习算法中的泛化提供了强大且可解释的度量。
• 本专著表明，信息论界在深度神经网络中可实现数值上的准确性，并与损失曲面平坦性及模型可压缩性相关联。
• 通过将在线学习的遗憾界转换，该框架恢复了已知的泛化界并推导出新的界，显示出在线学习与统计学习之间的紧密联系。
• 在某些场景下，如高斯位置模型中，信息论界可实现对泛化差距的最优表征。
• 本专著指出了若干开放问题，包括某些界中log√n依赖关系是否可被消除，以及在给定场景下哪种信息度量为最优。