PAC One-Step Safety Certification for Black-Box Discrete-Time Stochastic Systems

TL;DR

This paper introduces a data-driven, one-step safety certification framework for black-box stochastic systems using PAC guarantees, enabling recursive safety assurances with limited data.

Contribution

It develops a novel PAC-based approach for safety certification that relies solely on sampled data and formulates barrier certificates without knowing system dynamics.

Findings

The framework provides theoretical one-step safety guarantees.

Three methods for deriving PAC safety guarantees are proposed and compared.

Numerical examples demonstrate the effectiveness of the methods.

Abstract

This paper investigates the problem of safety certification for black-box discrete-time stochastic systems, where both the system dynamics and disturbance distributions are unknown, and only sampled data are available. Under such limited information, ensuring robust or classical quantitative safety over finite or infinite horizons is generally infeasible. To address this challenge, we propose a data-driven framework that provides theoretical one-step safety guarantees in the Probably Approximately Correct (PAC) sense. This one-step guarantee can be applied recursively at each time step, thereby yielding step-by-step safety assurances over extended horizons. Our approach formulates barrier certificate conditions based solely on sampled data and establishes PAC safety guarantees by leveraging the VC dimension, scenario approaches, Markov's inequality, and Hoeffding's inequality. Two sampling procedures are proposed, and three methods are proposed to derive PAC safety guarantees. The properties and comparative advantages of these three methods are thoroughly discussed. Finally, the effectiveness of the proposed methods are demonstrated through several numerical examples.