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Controlling Multiple Errors Simultaneously with a PAC-Bayes Bound

Reuben Adams, John Shawe-Taylor, Benjamin Guedj

TL;DR

The paper extends PAC-Bayes by introducing an error-type partition and deriving a bound on the KL-divergence between the empirical and true distributions over these error types. This bound holds uniformly for all posteriors in the hypothesis space and enables simultaneous control of all linear combinations of per-type risks, including those weighted by arbitrary loss vectors. A differentiable training objective is constructed via a closed-form kl-inverse formulation, enabling optimization of neural networks under the PAC-Bayes bound. Empirical results on binarised MNIST and HAM10000 illustrate the practicality of the approach, showing non-vacuous bounds and informative reallocation of empirical error probabilities, with discussions on extensions and limitations. The framework promises richer certificates for multiclass, structured, and time-varying error regimes, potentially inspiring robust, risk-aware learning in complex domains.

Abstract

Current PAC-Bayes generalisation bounds are restricted to scalar metrics of performance, such as the loss or error rate. However, one ideally wants more information-rich certificates that control the entire distribution of possible outcomes, such as the distribution of the test loss in regression, or the probabilities of different mis-classifications. We provide the first PAC-Bayes bound capable of providing such rich information by bounding the Kullback-Leibler divergence between the empirical and true probabilities of a set of $M$ error types, which can either be discretized loss values for regression, or the elements of the confusion matrix (or a partition thereof) for classification. We transform our bound into a differentiable training objective. Our bound is especially useful in cases where the severity of different mis-classifications may change over time; existing PAC-Bayes bounds can only bound a particular pre-decided weighting of the error types. In contrast our bound implicitly controls all uncountably many weightings simultaneously.

Controlling Multiple Errors Simultaneously with a PAC-Bayes Bound

TL;DR

The paper extends PAC-Bayes by introducing an error-type partition and deriving a bound on the KL-divergence between the empirical and true distributions over these error types. This bound holds uniformly for all posteriors in the hypothesis space and enables simultaneous control of all linear combinations of per-type risks, including those weighted by arbitrary loss vectors. A differentiable training objective is constructed via a closed-form kl-inverse formulation, enabling optimization of neural networks under the PAC-Bayes bound. Empirical results on binarised MNIST and HAM10000 illustrate the practicality of the approach, showing non-vacuous bounds and informative reallocation of empirical error probabilities, with discussions on extensions and limitations. The framework promises richer certificates for multiclass, structured, and time-varying error regimes, potentially inspiring robust, risk-aware learning in complex domains.

Abstract

Current PAC-Bayes generalisation bounds are restricted to scalar metrics of performance, such as the loss or error rate. However, one ideally wants more information-rich certificates that control the entire distribution of possible outcomes, such as the distribution of the test loss in regression, or the probabilities of different mis-classifications. We provide the first PAC-Bayes bound capable of providing such rich information by bounding the Kullback-Leibler divergence between the empirical and true probabilities of a set of error types, which can either be discretized loss values for regression, or the elements of the confusion matrix (or a partition thereof) for classification. We transform our bound into a differentiable training objective. Our bound is especially useful in cases where the severity of different mis-classifications may change over time; existing PAC-Bayes bounds can only bound a particular pre-decided weighting of the error types. In contrast our bound implicitly controls all uncountably many weightings simultaneously.
Paper Structure (19 sections, 13 theorems, 82 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 19 sections, 13 theorems, 82 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A new framework of errors types
  3. Relation to previous results.
  4. Outline.
  5. Notation
  6. Main result
  7. Construction of a Differentiable Training Objective
  8. Proof of the main bound
  9. Numerical experiments
  10. Perspectives
  11. Recipe for implementing Theorems \ref{['Generalised Corollary']} and \ref{['Proposition: Approximating kl^-1 and its derivatives']}
  12. Additional Experimental Details
  13. Proofs
  14. Proof of Proposition \ref{['prop: kl sorta triangle inequality']}
  15. Proof of Lemma \ref{["Lemma: Generalisation of Maurer's Lemma 3"]}
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\mathcal{X}$ and $\mathcal{Y}$ be arbitrary sets and $\bigcup_{j=1}^M E_j$ be a disjoint partition of $\mathcal{Y}^2$ into $M$ error types. Let $D \in \mathcal{M}(\mathcal{X} \times \mathcal{Y})$ be a data-generating distribution and $\mathcal{H}$ be a simple ($\mathcal{H} \subseteq \mathcal{Y} $\xi(m, M) := \sqrt{\pi} e^{1/(12m)}\left(\frac{m}{2}\right)^{\frac{M-1}{2}} \sum_{z=0}^{M-1} \bino

Figures (2)

  • Figure 1: Experimental results for binarised MNIST. (a) The PAC-Bayes bound on the total risk decreases when tuning the posterior via Theorem \ref{['Proposition: Approximating kl^-1 and its derivatives']}. (b) This is achieved by a shift in the empirical error probabilities. (c) The bound on $\textup{kl}(\bm{R}_S(Q)\|\bm{R}_D(Q))$ is not substantially increased, meaning we still retain good control of $\bm{R}_D(Q)$ after optimizing $Q$ for this particular choice of $\bm{\ell}$.
  • Figure 2: MNIST (first column) and HAM10000 (second column) experiments.

Theorems & Definitions (25)

  • Theorem 1
  • Proposition 1
  • Definition 1
  • Theorem 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • Lemma 1
  • ...and 15 more