Information-Theoretic Foundations for Machine Learning

2024-07-17

arXiv.org

AI Summary

Machine learning practice has long lacked a unified, rigorous theoretical foundation; existing analyses rely heavily on empirical observations and fail to provide principled guidance for critical challenges such as generalization, robustness, and few-shot learning. Method: This paper establishes the first general learning theory framework unifying Bayesian statistics and Shannon information theory, characterizing the dynamic performance evolution of optimal Bayesian learners under streaming empirical data—rigorously accommodating i.i.d., sequential, hierarchical, and model-misspecified settings. Contribution/Results: The framework achieves the first cross-paradigm theoretical unification in learning theory, overcoming the degradation of classical asymptotic analysis with increasing data complexity. It yields tight, interpretable bounds on generalization error, data efficiency, and belief updating. These results provide computationally tractable, theoretically grounded foundations for distribution shift correction, few-shot learning, and robust algorithm design.

Technology Category

• Machine Learning: Learning Theory
• Game Theory and Economic Paradigms: Adversarial Learning
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Machine learning and data science for the Web

Abstract

The progress of machine learning over the past decade is undeniable. In retrospect, it is both remarkable and unsettling that this progress was achievable with little to no rigorous theory to guide experimentation. Despite this fact, practitioners have been able to guide their future experimentation via observations from previous large-scale empirical investigations. In this work, we propose a theoretical framework which attempts to provide rigor to existing practices in machine learning. To the theorist, we provide a framework which is mathematically rigorous and leaves open many interesting ideas for future exploration. To the practitioner, we provide a framework whose results are simple, and provide intuition to guide future investigations across a wide range of learning paradigms. Concretely, we provide a theoretical framework rooted in Bayesian statistics and Shannon's information theory which is general enough to unify the analysis of many phenomena in machine learning. Our framework characterizes the performance of an optimal Bayesian learner as it learns from a stream of experience. Unlike existing analyses that weaken with increasing data complexity, our theoretical tools provide accurate insights across diverse machine learning settings. Throughout this work, we derive theoretical results and demonstrate their generality by apply them to derive insights specific to settings. These settings range from learning from data which is independently and identically distributed under an unknown distribution, to data which is sequential, to data which exhibits hierarchical structure amenable to meta-learning, and finally to data which is not fully explainable under the learner's beliefs (misspecification). These results are particularly relevant as we strive to understand and overcome increasingly difficult machine learning challenges in this endlessly complex world.

Problem

• Developing a rigorous theoretical framework for machine learning practices
• Unifying analysis of diverse learning paradigms using information theory
• Characterizing optimal Bayesian learner performance across complex data scenarios

Innovation

• Theoretical framework based on Bayesian statistics
• Unifies analysis using Shannon's information theory
• Accurate insights across diverse learning settings