On the irreducibility and convergence of a class of nonsmooth nonlinear state-space models on manifolds and their applications to zeroth-order optimization

Abstract

In this paper, we analyze a large class of general nonlinear state-space models on a state-space X, defined by the recursion ${\varphi }_{k+1}=F({\varphi }_k,\alpha ({\varphi }_k,U_{k+1}))$, $k\in \mathbb{N}$, where $F,\alpha$ are some functions and $\{U_{k+1}{\}}_{k\in \mathbb{N}}$ is a sequence of i.i.d. random variables. More precisely, we extend conditions under which this class of Markov chains is irreducible, aperiodic and satisfies important continuity properties, relaxing two key assumptions from prior works. First, the state-space X is supposed to be a smooth manifold instead of an open subset of a Euclidean space. Second, we only suppose that $F$ is locally Lipschitz continuous. We demonstrate the significance of our results through their application to Markov chains underlying optimization algorithms. These schemes belong to the class of evolution strategies with covariance matrix adaptation and step-size adaptation.