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

TL;DR

This work addresses safety verification for black-box discrete-time stochastic systems lacking knowledge of both dynamics and disturbance distributions. It develops a data-driven framework that provides formal one-step safety guarantees in the PAC sense by leveraging barrier certificates and sampled data, allowing recursive application for longer horizons. Three methods—robust and stochastic barrier certificates combined with scenario approaches, VC-dimension theory, and concentration inequalities—are proposed to obtain probabilistic safety guarantees under different sampling schemes. The results include theoretical guarantees, comparative analyses of sampling strategies, and numerical demonstrations, complemented by a prototype tool and discussions of scalability and future extensions.

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.