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Probabilistic Robustness in Deep Learning: A Concise yet Comprehensive Guide

Xingyu Zhao

TL;DR

Probabilistic robustness (PR) addresses robustness in deep learning under stochastic perturbations, offering a practical alternative to traditional adversarial robustness (AR) by focusing on the probability of unaffected predictions within perturbation neighborhoods. The chapter formally defines PR, surveys both black-box and white-box evaluation methods, and introduces AT-PR, a dedicated min-max training framework to enlarge PR-safe regions around inputs. It further discusses how to embed PR evidence into system-level safety assurance, including translating DL-level robustness metrics into reliability claims and propagating uncertainty through reliability models. The work highlights open challenges—benchmarking PR estimators, extending PR to frontier AI tasks, and developing end-to-end case studies for system integration—signaling important directions for safer, more trustworthy AI in safety-critical domains.

Abstract

Deep learning (DL) has demonstrated significant potential across various safety-critical applications, yet ensuring its robustness remains a key challenge. While adversarial robustness has been extensively studied in worst-case scenarios, probabilistic robustness (PR) offers a more practical perspective by quantifying the likelihood of failures under stochastic perturbations. This paper provides a concise yet comprehensive overview of PR, covering its formal definitions, evaluation and enhancement methods. We introduce a reformulated ''min-max'' optimisation framework for adversarial training specifically designed to improve PR. Furthermore, we explore the integration of PR verification evidence into system-level safety assurance, addressing challenges in translating DL model-level robustness to system-level claims. Finally, we highlight open research questions, including benchmarking PR evaluation methods, extending PR to generative AI tasks, and developing rigorous methodologies and case studies for system-level integration.

Probabilistic Robustness in Deep Learning: A Concise yet Comprehensive Guide

TL;DR

Probabilistic robustness (PR) addresses robustness in deep learning under stochastic perturbations, offering a practical alternative to traditional adversarial robustness (AR) by focusing on the probability of unaffected predictions within perturbation neighborhoods. The chapter formally defines PR, surveys both black-box and white-box evaluation methods, and introduces AT-PR, a dedicated min-max training framework to enlarge PR-safe regions around inputs. It further discusses how to embed PR evidence into system-level safety assurance, including translating DL-level robustness metrics into reliability claims and propagating uncertainty through reliability models. The work highlights open challenges—benchmarking PR estimators, extending PR to frontier AI tasks, and developing end-to-end case studies for system integration—signaling important directions for safer, more trustworthy AI in safety-critical domains.

Abstract

Deep learning (DL) has demonstrated significant potential across various safety-critical applications, yet ensuring its robustness remains a key challenge. While adversarial robustness has been extensively studied in worst-case scenarios, probabilistic robustness (PR) offers a more practical perspective by quantifying the likelihood of failures under stochastic perturbations. This paper provides a concise yet comprehensive overview of PR, covering its formal definitions, evaluation and enhancement methods. We introduce a reformulated ''min-max'' optimisation framework for adversarial training specifically designed to improve PR. Furthermore, we explore the integration of PR verification evidence into system-level safety assurance, addressing challenges in translating DL model-level robustness to system-level claims. Finally, we highlight open research questions, including benchmarking PR evaluation methods, extending PR to generative AI tasks, and developing rigorous methodologies and case studies for system-level integration.

Paper Structure

This paper contains 15 sections, 4 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definition of Probabilistic Robustness
  3. Evaluation of Probabilistic Robustness
  4. Enhancement of Probabilistic Robustness
  5. System-level Integration of Probabilistic Robustness
  6. What is the most suitable reliability metric for this specific application? And what are the acceptable risk thresholds for the given metric?
  7. How can system-level reliability metrics be translated into corresponding metrics/definitions at the DL model global and local levels?
  8. How can PR estimates and their uncertainties be propagated from local and global levels to support system-level reliability claims with informed confidence?
  9. Challenges and Open Problems
  10. Benchmarking PR estimators
  11. Extending PR to frontier AI tasks
  12. Improving PR by dedicated AT
  13. Theoretical foundation for PR-focused AT
  14. End-to-end case studies of system-level integration
  15. Conclusion

Figures (3)

  • Figure 1: Four types of problem formulations of robustness evaluation problems zhang2024protip.
  • Figure 2: Two cases of local loss landscape and AEs identified by AT-AR and AT-PR.
  • Figure 3: Metrics studied ranging from local norm-ball, DL model input domain to system safety space. Safety analysis and reliability modelling techniques, considering factors like system architecture, redundancy design, operational environment/profiles, discrete/continuous nature, failure modes/criticality, should be applied at the system level. Then map them to application-specific estimator settings at the local and global levels of DL models, and propagate estimation results with informed confidence

Theorems & Definitions (4)

  • definition thmcounterdefinition: Probabilistic Robustness
  • definition thmcounterdefinition: Probabilistic Lipschitzness
  • definition thmcounterdefinition: Total Statistical Robustness
  • definition thmcounterdefinition: AT for PR