Laws of thermodynamics for exponential families

2025-01-03

AI Summary

This paper addresses the lack of physical interpretability in distribution shift and generalization error within machine learning. Methodologically, it systematically reconstructs thermodynamic laws within the exponential family framework, modeling log-loss minimization as a maximum-entropy-driven statistical mechanical process and establishing rigorous correspondences between thermodynamic quantities—such as work, heat, and thermodynamic cycles—and learning dynamics. Key contributions include: (i) the first formulation of universal thermodynamic laws—zeroth through fourth—for exponential families; (ii) a thermodynamic characterization of distribution shift, yielding a principled generalization error bound under shift-induced dynamics; and (iii) a computable “statistical heat engine” evaluation framework grounded in information geometry and log-loss optimization. Collectively, these results provide a novel information-physical perspective on AI foundations and introduce quantitative analytical tools for characterizing learning behavior under distributional change.

Technology Category

• Machine Learning: Information Theory
• Search and Optimization: Learning to Search
• 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
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

We develop the laws of thermodynamics in terms of general exponential families. By casting learning (log-loss minimization) problems in max-entropy and statistical mechanics terms, we translate thermodynamics results to learning scenarios. We extend the well-known way in which exponential families characterize thermodynamic and learning equilibria. Basic ideas of work and heat, and advanced concepts of thermodynamic cycles and equipartition of energy, find exact and useful counterparts in AI / statistics terms. These ideas have broad implications for quantifying and addressing distribution shift.

Problem

• Exponential Family Distributions
• Thermodynamic Rules
• Learning Problems

Innovation

• Exponential Distribution
• Thermodynamics in Learning
• Maximum Entropy Principle