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Entropy, concentration, and learning: a statistical mechanics primer

Akshay Balsubramani

TL;DR

This work reframes loss minimization in AI as a statistical-mechanics problem, deriving learning from first principles via concentration of empirical measures and Boltzmann-type probability calculations. It centers on the information projection onto constraint sets and the exponential-family model class, showing that log-loss minimization, robust Bayes, and probability maximization are equivalent perspectives that converge to maximum entropy under the right priors. Through Sanov's theorem, Gibbs conditioning, and a detailed treatment of energy, free energy, and entropy, the paper connects fundamental information-theoretic quantities to learning dynamics and generalization. The framework provides a principled, axiomatic basis for why KL divergence and cross-entropy are natural loss functions, and it extends to both discrete and continuous spaces, offering a unified lens for understanding regularization, model evaluation, and mean-field approximations in modern AI. Overall, it positions exponential families as a central tool for principled learning, with deep ties to physics-inspired concepts of energy, temperature, and duality.

Abstract

Artificial intelligence models trained through loss minimization have demonstrated significant success, grounded in principles from fields like information theory and statistical physics. This work explores these established connections through the lens of statistical mechanics, starting from first-principles sample concentration behaviors that underpin AI and machine learning. Our development of statistical mechanics for modeling highlights the key role of exponential families, and quantities of statistics, physics, and information theory.

Entropy, concentration, and learning: a statistical mechanics primer

TL;DR

This work reframes loss minimization in AI as a statistical-mechanics problem, deriving learning from first principles via concentration of empirical measures and Boltzmann-type probability calculations. It centers on the information projection onto constraint sets and the exponential-family model class, showing that log-loss minimization, robust Bayes, and probability maximization are equivalent perspectives that converge to maximum entropy under the right priors. Through Sanov's theorem, Gibbs conditioning, and a detailed treatment of energy, free energy, and entropy, the paper connects fundamental information-theoretic quantities to learning dynamics and generalization. The framework provides a principled, axiomatic basis for why KL divergence and cross-entropy are natural loss functions, and it extends to both discrete and continuous spaces, offering a unified lens for understanding regularization, model evaluation, and mean-field approximations in modern AI. Overall, it positions exponential families as a central tool for principled learning, with deep ties to physics-inspired concepts of energy, temperature, and duality.

Abstract

Artificial intelligence models trained through loss minimization have demonstrated significant success, grounded in principles from fields like information theory and statistical physics. This work explores these established connections through the lens of statistical mechanics, starting from first-principles sample concentration behaviors that underpin AI and machine learning. Our development of statistical mechanics for modeling highlights the key role of exponential families, and quantities of statistics, physics, and information theory.
Paper Structure (57 sections, 1 theorem, 69 equations, 1 figure)

This paper contains 57 sections, 1 theorem, 69 equations, 1 figure.

Table of Contents

  1. Foundations: entropy, concentration, and learning
  2. Context among related work
  3. Scope
  4. Concentration: Boltzmann's "probability calculation"
  5. Boltzmann's reasoning
  6. Consequences
  7. Enter entropy
  8. Learning: generalizing Boltzmann's scenario
  9. Concentration: general "probability calculations"
  10. The general calculation
  11. Information projection
  12. The result
  13. Discussion: the nature of concentration in $\mathcal{A}$
  14. Gibbs conditioning principle
  15. The impact of further information
  16. ...and 42 more sections

Key Result

Theorem 1

Figures (1)

  • Figure 1: Approximating entropy for different distributions, varying $n$, for $D = 50000$. At left is for a near-uniform distribution $P$ (parameters sampled according to a Dirichlet(1)). At right is for $P$ sampled uniformly over the positive orthant, for which much more information is contained per sample. Note that even for moderate $n$ ($n / D \geq ~5$) -- the relative error is just a small constant. Meanwhile, the first-order correction $\frac{1}{2} \log \left( \frac{2 \pi n}{ \prod_{i=1}^{D} (2 \pi n P_i) } \right)$ makes the estimate significantly more accurate.

Theorems & Definitions (1)

  • Theorem 1: Sanov's Theorem