Information-theoretic analysis of generalization capability of learning algorithms (1705.07809v2)

Abstract: We derive upper bounds on the generalization error of a learning algorithm in terms of the mutual information between its input and output. The bounds provide an information-theoretic understanding of generalization in learning problems, and give theoretical guidelines for striking the right balance between data fit and generalization by controlling the input-output mutual information. We propose a number of methods for this purpose, among which are algorithms that regularize the ERM algorithm with relative entropy or with random noise. Our work extends and leads to nontrivial improvements on the recent results of Russo and Zou.

• The paper introduces mutual information bounds that quantify generalization error, offering a novel metric for controlling overfitting.
• It outlines algorithm design guidelines that regularize ERM by managing input-output mutual information to balance empirical risk and generalization.
• The study extends its approach to adaptive algorithm composition, providing insights for robust and theoretically grounded learning systems.

Information-theoretic Analysis of Generalization Capability of Learning Algorithms

This paper presents a detailed investigation into the generalization capabilities of learning algorithms using an information-theoretic approach, emphasizing the role of mutual information between input data and learned hypotheses. Aolin Xu and Maxim Raginsky propose a novel perspective in which generalization error is bounded using mutual information, extending traditional approaches that rely on hypothesis space complexity measures or algorithmic stability.

Key Contributions

1. Mutual Information Bounds: The paper derives upper bounds on the generalization error of learning algorithms in terms of the mutual information between the input dataset and the output hypothesis. This approach offers a new lens to analyze and improve generalization capabilities, especially in cases where the hypothesis space may be uncountably infinite.
2. Algorithm Design Guidelines: The information-theoretic bounds serve as theoretical guidelines for balancing empirical risk minimization (ERM) with generalization. Specifically, by controlling the input-output mutual information, it is possible to mitigate overfitting.
3. New Algorithmic Proposals: The authors propose modified learning algorithms that manage input-output mutual information by regularizing ERM with relative entropy or by introducing random noise. For example, the Gibbs algorithm is retrieved as a natural consequence of this framework, providing a probabilistic method to select hypotheses that is informed by mutual information constraints.
4. Extended Results for Adaptive Composition: Beyond individual algorithms, the paper discusses the generalization error of complex algorithms created through adaptive composition, whereby multiple learning algorithms are executed sequentially on the same dataset. The results extend to show that the overall generalization can be controlled by examining the mutual information at each step.

Implications and Future Directions

Theoretical Implications

The presented bounds and methods contribute to a deeper theoretical understanding of generalization in machine learning. By focusing on mutual information, the authors provide a metric that aligns closely with the properties of the dataset, the hypothesis space, the learning algorithm, and the loss function.

Practical Implications

These insights not only offer a more robust method for analyzing the generalization capabilities of learning algorithms but also help in designing new algorithms that can effectively balance the trade-off between fitting the data and ensuring that the model generalizes well to unseen samples.

Speculative Future Developments

Given the trajectory of this research, future extensions could explore more refined mutual information measures or alternative information-theoretic frameworks, potentially bringing more nuanced insights into modern complex models such as deep neural networks. Additionally, further exploration into the stability of various learning algorithms through an information-theoretic lens might reveal new regularization techniques or novel algorithmic strategies tailored for specific task domains.

Conclusion

The paper by Xu and Raginsky offers a compelling alternative to conventional complexity-based generalization bounds by leveraging mutual information as a key tool. This approach promises to reshape our understanding of learning theory, providing clearer guidance for both the analysis and design of algorithms, aligning with both theoretical and practical objectives in the field of machine learning.