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Generalization Bounds: Information-Theoretic and PAC-Bayes Perspectives


Core Concepts
Information-theoretic and PAC-Bayesian perspectives provide insights into generalization bounds in machine learning.
Abstract
The content delves into the foundations of generalization bounds, covering information-theoretic and PAC-Bayesian approaches. It discusses the connection between generalization and information theory, tools for deriving bounds, and applications in various learning models. The structure includes an introduction, foundations, tools, generalization bounds in expectation and probability, the CMI framework, applications, and concluding remarks.
Stats
arXiv:2309.04381v2 [cs.LG] 27 Mar 2024
Quotes
"In this monograph, we highlight this strong connection and present a unified treatment of PAC-Bayesian and information-theoretic generalization bounds." "Information-theoretic generalization bounds make this intuition precise by characterizing the generalization error of (randomized) learning algorithms in terms of information-theoretic metrics."

Key Insights Distilled From

by Fred... at arxiv.org 03-28-2024

https://arxiv.org/pdf/2309.04381.pdf
Generalization Bounds

Deeper Inquiries

How do information-theoretic and PAC-Bayesian approaches differ in deriving generalization bounds

The information-theoretic and PAC-Bayesian approaches differ in deriving generalization bounds primarily in their focus and underlying principles. The information-theoretic approach emphasizes the role of information measures, such as mutual information, relative entropy, and entropy, in quantifying the relationship between the hypothesis and the training data. It aims to capture the amount of information shared between the hypothesis and the data, providing insights into the generalization capabilities of the learning algorithm. On the other hand, the PAC-Bayesian approach incorporates Bayesian techniques by considering distributions over hypotheses and deriving bounds that hold uniformly over these distributions. It focuses on probabilistic guarantees of generalization performance based on Bayesian posterior distributions.

What are the implications of the mutual information term in the information-theoretic generalization bound

The mutual information term in the information-theoretic generalization bound plays a crucial role in quantifying the dependency between the hypothesis and the training data. Mutual information measures the amount of information shared between two random variables, in this case, the hypothesis and the training data. A higher mutual information implies a stronger relationship between the hypothesis and the data, indicating that the hypothesis is more influenced by the training samples. In the context of the generalization bound, a high mutual information value suggests that the hypothesis is overfitting to the training data, leading to potential challenges in generalizing well to unseen data. Therefore, controlling and minimizing the mutual information term is essential for ensuring good generalization performance of the learning algorithm.

How can the insights from this content be applied to real-world machine learning scenarios beyond theoretical bounds

The insights from the content can be applied to real-world machine learning scenarios beyond theoretical bounds by guiding the development and evaluation of learning algorithms. By understanding the role of information measures, such as mutual information and relative entropy, in quantifying generalization capabilities, practitioners can design algorithms that balance model complexity and data fidelity. For instance, in deep learning applications, where overfitting is a common challenge, controlling the mutual information between the model and the training data can help improve generalization performance. Additionally, the principles of information-theoretic generalization bounds can inform regularization strategies, data preprocessing techniques, and model selection processes to enhance the robustness and reliability of machine learning models in practical settings.
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