Inner approximations of stochastic programs for data-driven stochastic barrier function design

Frederik Baymler Mathiesen, Licio Romao, Simeon Craig Calvert, Alessandro Abate, Luca Laurenti

Abstract: This paper studies finite-horizon safety guarantees for discrete-time piece-wise affine systems with stochastic noise of unknown distributions. Our approach is based on a novel approach to synthesise a stochastic barrier function from noise data. In particular, we first build a chance-constraint tightening to obtain an inner approximation of a stochastic program. Then, we apply this methodology for stochastic barrier function design, yielding a robust linear program to which the scenario approach theory applies. In contrast to existing approaches, our method is data efficient as it only requires the number of data to be proportional to the logarithm in the negative inverse of the confidence level and is computationally efficient due to its reduction to linear programming. Furthermore, while state-of-the-art methods assume known statistics on the noise distribution, our approach does not require any information about it. We empirically evaluate the efficacy of our method on various verification benchmarks. Experiments show a significant improvement with respect to state-of-the-art, obtaining tighter certificates with a confidence that is several orders of magnitude higher.