Zeroth-order Gradient and Quasi-Newton Methods for Nonsmooth Nonconvex Stochastic Optimization

Abstract

We consider the minimization of a Lipschitz continuous and expectation-valued function, denoted by $f$ and defined as $f(\mathbf{x}) \triangleq \mathbb{E}[\tilde{f}(\mathbf{x}, \mathbf{\xi})]$, over a closed and convex set $\mathcal{X}$. We obtain asymptotics as well as rate and complexity guarantees for computing approximate Clarke-stationary points via zeroth-order schemes. We adopt an approach reliant on minimizing $f_{\eta}$ where $f_{\eta}(\mathbf{x}) \triangleq \mathbb{E}_{\mathbf{u}}\left[\mathbf{x}, f(\mathbf{x}+\eta \mathbf{u})\, \right]$, $\mathbf{u}$ is a random variable defined on a unit sphere, and $\eta > 0$. In fact, it is known that a stationary point of the $\eta$-smoothed problem is an $\eta$-stationary point for the original problem in the Clarke sense. In such a setting, we develop two schemes with promising empirical behavior. (I) We develop a variance-reduced zeroth-order gradient framework (VRG-ZO) for minimizing $f_{\eta}$ over $\mathcal{X}$. In this setting, we make two sets of contributions for the sequence generated by the proposed zeroth-order gradient scheme. (a) The residual function of the smoothed problem tends to zero almost surely along the generated sequence, guaranteeing $\eta$-Clarke stationary solutions of the original problem; (b) To compute an $\mathbf{x}$ such that the expected norm of the residual of the $\eta$-smoothed problem is within $\epsilon$ requires no greater than $\mathcal{O}({n^{1/2}}{(L_0\eta^{-1} +L_0^2)} \epsilon^{-2})$ projection steps and $\mathcal{O}({n^{3/2}(L_0^3\eta^{-2}+L_0^5)} \epsilon^{-4})$ function evaluations. (II) Our second scheme is a zeroth-order stochastic quasi-Newton scheme (VRSQN-ZO) reliant on randomized and Moreau smoothing; the iteration and sample complexities are $\mathcal{O}({L_0^{4}}{n^{2}}{\eta^{-4}}\epsilon^{-2})$ and $\mathcal{O}(L_0^{9} n^{5}\eta^{-5}\epsilon^{-5})$, respectively.