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Generalization of the Gibbs algorithm with high probability at low temperatures

Andreas Maurer

TL;DR

This paper addresses generalization bounds for individual hypotheses drawn from the Gibbs posterior, including the challenging low-temperature regime where generalization depends on data-dependent loss landscapes. It develops a disintegrated PAC-Bayesian bound (Theorem Main) that links the generalization gap to the total prior volume of near-minimizers via the complexity term $\Lambda_{\beta}(h,\mathbf{x})$ and the empirical loss CDF $\hat{\varphi}$, and interprets its behavior across temperature regimes, margins, and the zero-temperature limit. The work further extends the framework to beyond-Gibbs stochastic algorithms with similar density-based bounds, and provides corollaries for sub-Gaussian losses and margin-based binary classification. The results offer theoretical justification for why flat minima and data-dependent loss landscapes can enhance generalization, and they illuminate how high- and low-temperature analyses relate to stochastic optimization methods like SGLD/SGD in deep learning.

Abstract

The paper gives a bound on the generalization error of the Gibbs algorithm, which recovers known data-independent bounds for the high temperature range and extends to the low-temperature range, where generalization depends critically on the data-dependent loss-landscape. It is shown, that with high probability the generalization error of a single hypothesis drawn from the Gibbs posterior decreases with the total prior volume of all hypotheses with similar or smaller empirical error. This gives theoretical support to the belief in the benefit of flat minima. The zero temperature limit is discussed and the bound is extended to a class of similar stochastic algorithms.

Generalization of the Gibbs algorithm with high probability at low temperatures

TL;DR

This paper addresses generalization bounds for individual hypotheses drawn from the Gibbs posterior, including the challenging low-temperature regime where generalization depends on data-dependent loss landscapes. It develops a disintegrated PAC-Bayesian bound (Theorem Main) that links the generalization gap to the total prior volume of near-minimizers via the complexity term and the empirical loss CDF , and interprets its behavior across temperature regimes, margins, and the zero-temperature limit. The work further extends the framework to beyond-Gibbs stochastic algorithms with similar density-based bounds, and provides corollaries for sub-Gaussian losses and margin-based binary classification. The results offer theoretical justification for why flat minima and data-dependent loss landscapes can enhance generalization, and they illuminate how high- and low-temperature analyses relate to stochastic optimization methods like SGLD/SGD in deep learning.

Abstract

The paper gives a bound on the generalization error of the Gibbs algorithm, which recovers known data-independent bounds for the high temperature range and extends to the low-temperature range, where generalization depends critically on the data-dependent loss-landscape. It is shown, that with high probability the generalization error of a single hypothesis drawn from the Gibbs posterior decreases with the total prior volume of all hypotheses with similar or smaller empirical error. This gives theoretical support to the belief in the benefit of flat minima. The zero temperature limit is discussed and the bound is extended to a class of similar stochastic algorithms.

Paper Structure

This paper contains 19 sections, 15 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A generic generalization bound
  4. Interpretation of Theorem \ref{['Theorem Main']}
  5. Wide minima
  6. High temperatures
  7. Low temperatures
  8. Margins in binary classification
  9. The zero-temperature limit
  10. Distribution-dependence and reproducibility
  11. Other corollaries of Theorem \ref{['Theorem Main']}
  12. Beyond the Gibbs algorithm
  13. Related work
  14. Conclusion and future directions
  15. Table of notation
  16. ...and 4 more sections

Key Result

Theorem 3.1

Let $F:\mathcal{H}\times \mathcal{X}^{n}\rightarrow \mathbb{R}$ be some measurable function and $\delta >0$. Then with probability at least $1-\delta$ in $\mathbf{x}\sim \mu ^{n}$ and $h\sim \hat{G}_{\beta }\left( \mathbf{x}\right)$

Figures (2)

  • Figure 1: Schematic representation of the loss landscape, with the prior being the length of horizontal intervals. $h$ is drawn from the Gibbs-posterior and the total length of the thick green lines contributes to the prior volume and thus to generalization. Notice that for large $\beta$ and large ${\hat{L}}(h,\bf X)$ the optimal $r$ can also be negative.
  • Figure 2: Schematic and compactified phase diagram of the bounds when $\pi (\widehat{\mathcal{H}}_{\min }\left( \mathbf{x}\right))>0$ with $n$ fixed. The diagonal represents the data-independent bounds of Corollary \ref{['Corollary high temperature']}. The data-dependent bounds have to lie in the shaded region by Corollary \ref{['Corollary general']} and converge to $\ln \left( 1/\pi \left( \widehat{\mathcal{H}}_{\min }\left( \mathbf{x}\right) \right) \right)/n$ by Proposition \ref{['Proposition limit']}, ignoring smaller logarithmic terms.

Theorems & Definitions (23)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 4.1
  • Corollary 4.2
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • proof
  • Proposition 4.5
  • ...and 13 more