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Tightening the Evaluation of PAC Bounds Using Formal Verification Results

Thomas Walker, Alessio Lomuscio

TL;DR

This work addresses the problem of loose PAC bounds for neural networks in safety-critical settings. It proposes a novel approach that tightens generalisation bounds by conditioning on formal verification results obtained from regions around test points, rather than solely relying on pointwise evaluations. By introducing a zero-loss region $\Delta$ with mass $p_{\Delta}$ and applying the law of total probability, the authors derive tightened bounds and improved confidence that depend on $p_{\Delta}$ and the empirical risk $\hat{R}(\mathbf{w})$, with practical estimation strategies for $p_{\Delta}$ discussed in the appendix. The framework shifts the evaluation of generalisation from purely data-driven assessments to a verification-informed, region-based perspective, offering a data-efficient path toward certifiable deployment of neural networks in data-limited, safety-critical domains.

Abstract

Probably Approximately Correct (PAC) bounds are widely used to derive probabilistic guarantees for the generalisation of machine learning models. They highlight the components of the model which contribute to its generalisation capacity. However, current state-of-the-art results are loose in approximating the generalisation capacity of deployed machine learning models. Consequently, while PAC bounds are theoretically useful, their applicability for evaluating a model's generalisation property in a given operational design domain is limited. The underlying classical theory is supported by the idea that bounds can be tightened when the number of test points available to the user to evaluate the model increases. Yet, in the case of neural networks, the number of test points required to obtain bounds of interest is often impractical even for small problems. In this paper, we take the novel approach of using the formal verification of neural systems to inform the evaluation of PAC bounds. Rather than using pointwise information obtained from repeated tests, we use verification results on regions around test points. We show that conditioning existing bounds on verification results leads to a tightening proportional to the underlying probability mass of the verified region.

Tightening the Evaluation of PAC Bounds Using Formal Verification Results

TL;DR

This work addresses the problem of loose PAC bounds for neural networks in safety-critical settings. It proposes a novel approach that tightens generalisation bounds by conditioning on formal verification results obtained from regions around test points, rather than solely relying on pointwise evaluations. By introducing a zero-loss region with mass and applying the law of total probability, the authors derive tightened bounds and improved confidence that depend on and the empirical risk , with practical estimation strategies for discussed in the appendix. The framework shifts the evaluation of generalisation from purely data-driven assessments to a verification-informed, region-based perspective, offering a data-efficient path toward certifiable deployment of neural networks in data-limited, safety-critical domains.

Abstract

Probably Approximately Correct (PAC) bounds are widely used to derive probabilistic guarantees for the generalisation of machine learning models. They highlight the components of the model which contribute to its generalisation capacity. However, current state-of-the-art results are loose in approximating the generalisation capacity of deployed machine learning models. Consequently, while PAC bounds are theoretically useful, their applicability for evaluating a model's generalisation property in a given operational design domain is limited. The underlying classical theory is supported by the idea that bounds can be tightened when the number of test points available to the user to evaluate the model increases. Yet, in the case of neural networks, the number of test points required to obtain bounds of interest is often impractical even for small problems. In this paper, we take the novel approach of using the formal verification of neural systems to inform the evaluation of PAC bounds. Rather than using pointwise information obtained from repeated tests, we use verification results on regions around test points. We show that conditioning existing bounds on verification results leads to a tightening proportional to the underlying probability mass of the verified region.
Paper Structure (15 sections, 4 theorems, 39 equations, 3 tables)

This paper contains 15 sections, 4 theorems, 39 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. PAC Bounds
  5. Verification
  6. Conditioning PAC Bounds on Formal Verification Results
  7. Problem Formalisation
  8. Improving Bound Tightness
  9. Improving Bound Confidence
  10. Evaluation of Results
  11. Conclusion
  12. Future Work
  13. Appendix
  14. Approximating $p_{\Delta}$
  15. Supporting Lemmas

Key Result

Theorem 1

Let $\vert\mathcal{W}\vert=M<\infty$, $\delta\in(0,1)$ and $\mathbf{w}\in\mathcal{W}$. Then

Theorems & Definitions (7)

  • Theorem 1: alquier_user-friendly_2023
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4: scott_hoeffdings_2014
  • proof