Moment-based Invariants for Probabilistic Loops with Non-polynomial Assignments

Abstract

. We present a method to automatically approximate moment-based invariants of probabilistic programs with non-polynomial updates of continuous state variables to accommodate more complex dynamics. Our approach leverages polynomial chaos expansion to approximate nonlinear functional updates as sums of orthogonal polynomials. We exploit this result to automatically estimate state-variable moments of all orders in Prob-solvable loops with non-polynomial updates. We showcase the accuracy of our estimation approach in several examples, such as the turning vehicle model and the Taylor rule in monetary policy. expansion to approximate non-polynomial general functional assignments. The approximations produced by our technique have optimal exponential convergence when the parameters of the general non-polynomial functions have distributions that are stable across all iterations. We derived an upper bound on the approximation error for the case of un-stable parameter distributions. Our methods can accommodate non-linear, non-polynomial updates in classes of probabilistic loops amenable to automated moment computation, such as the class of Prob-solvable loops. Moreover, our tech-niques can be used for moment approximation for uncertainty quantification in more general probabilistic loops. Our experiments demonstrate the ability of our methods to characterize non-polynomial behavior in stochastic models from various domains via their moments, with high accuracy and in a fraction of the time required by other state-of-the-art tools.