PAC Finite-Time Safety Guarantees for Stochastic Systems with Unknown Disturbance Distributions

TL;DR

This paper develops a data-driven framework for providing finite-time safety guarantees in stochastic systems with unknown disturbances, using PAC bounds and barrier certificates based on finite samples.

Contribution

It introduces a novel safety certification method that relies on finite disturbance samples and PAC theory to ensure probabilistic safety over finite horizons.

Findings

Derived PAC generalization bounds for safety certification

Established trade-offs between sample size, model complexity, and safety levels

Provided practical guidelines for data-driven safety guarantees

Abstract

We investigate the problem of establishing finite-time probabilistic safety guarantees for discrete-time stochastic dynamical systems subject to unknown disturbance distributions, using barrier certificate methods. Our approach develops a data-driven safety certification framework that relies only on a finite collection of independent and identically distributed (i.i.d.) disturbance samples. Within this framework, we propose a certification procedure such that, with confidence at least $1-\delta$ over the sampled disturbances, if the output of the certification procedure is accepted, the probability that the system remains within a prescribed safe set over a finite horizon is at least $1-\epsilon$. A key challenge lies in formally characterizing the probably approximately correct (PAC) generalization behavior induced by finite samples. To address this, we derive PAC generalization bounds using tools from VC dimension, scenario optimization, and Rademacher complexity. These results illuminate the fundamental trade-offs between sample size, model complexity, and safety tolerance, providing both theoretical insight and practical guidance for designing reliable, data-driven safety certificates in discrete-time stochastic systems.