Stochastic Zeroth-Order Optimization Under Nonstationarity and Nonconvexity

Abstract

Stochastic zeroth-order optimization algorithms have been predominantly analyzed under the assumption that the objective function being optimized is time-invariant. Motivated by dynamic matrix sensing and completion problems, and online reinforcement learning problems, in this work, we propose and analyze stochastic zeroth-order optimization algorithms when the objective being optimized changes with time. Considering general nonconvex functions, we propose nonstationary versions of regret measures based on first-order and second-order optimal solutions, and provide the corresponding regret bounds. For the case of first-order optimal solution based regret measures, we provide regret bounds in both the low- and high-dimensional settings. For the case of second-order optimal solution based regret, we propose zeroth-order versions of the stochastic cubic-regularized Newton's method based on estimating the Hessian matrices in the bandit setting via second-order Gaussian Stein's identity. Our nonstationary regret bounds in terms of second-order optimal solutions have interesting consequences for avoiding saddle points in the nonstationary setting.