On Polynomial Stochastic Barrier Functions: Bernstein Versus Sum-of-Squares

TL;DR

This paper compares Bernstein polynomial-based stochastic barrier functions with traditional sum-of-squares methods, highlighting theoretical benefits but practical challenges in safety certification of stochastic systems.

Contribution

It introduces a Bernstein polynomial formulation for stochastic barrier functions and provides a comparative analysis against sum-of-squares approaches.

Findings

Bernstein approach results in a linear program, unlike the SDP for SoS.

Theoretical convergence properties favor Bernstein formulations.

Empirical results show Bernstein struggles to match SoS performance.

Abstract

Stochastic Barrier Functions (SBFs) certify the safety of stochastic systems by formulating a functional optimization problem, which state-of-the-art methods solve using Sum-of-Squares (SoS) polynomials. This work focuses on polynomial SBFs and introduces a new formulation based on Bernstein polynomials and provides a comparative analysis of its theoretical and empirical performance against SoS methods. We show that the Bernstein formulation leads to a linear program (LP), in contrast to the semi-definite program (SDP) required for SoS, and that its relaxations exhibit favorable theoretical convergence properties. However, our empirical results reveal that the Bernstein approach struggles to match SoS in practical performance, exposing an intriguing gap between theoretical advantages and real-world feasibility.