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We consider the minimization of a Lipschitz continuous and expectation-valuedfunction defined as $f(\mathbf{x}) \triangleq \mathbb{E}[{\tilde f}(\mathbf{x},\boldsymbol{\xi})]$, over a closed and convex set. Our focus lies on obtainingboth asymptotics as well as rate and complexity guarantees for computing anapproximate stationary point (in a Clarke sense) via zeroth-order schemes. Weadopt a smoothing-based approach reliant on minimizing $f_{\eta}$ where$f_{\eta}(\mathbf{x}) = \mathbb{E}_{\mathbf{u}}[f(\mathbf{x}+\eta\mathbf{u})]$, $\mathbf{u}$ is a random variable defined on a unit sphere, and$\eta > 0$. It has been observed that a stationary point of the $\eta$-smoothedproblem is a $2\eta$-stationary point for the original problem in the Clarkesense. In such a setting, we develop two sets of schemes with promisingempirical behavior. (I) We develop a smoothing-enabled variance-reducedzeroth-order gradient framework (VRG-ZO) and make two sets of contributions forthe sequence generated by the proposed zeroth-order gradient scheme. (a) Theresidual function of the smoothed problem tends to zero almost surely along thegenerated sequence, allowing for making guarantees for $\eta$-Clarke stationarysolutions of the original problem; (b) To compute an $\mathbf{x}$ that ensuresthat the expected norm of the residual of the $\eta$-smoothed problem is within$\epsilon$ requires no greater than $O(\eta^{-1} \epsilon^{-2})$ projectionsteps and $ O\left(\eta^{-2} \epsilon^{-4}\right)$ function evaluations. (II)Our second scheme is a zeroth-order stochastic quasi-Newton scheme (VRSQN-ZO)reliant on a combination of randomized and Moreau smoothing; the correspondingiteration and sample complexities for this scheme are $O\left(\eta^{-5}\epsilon^{-2}\right)$ and $O\left(\eta^{-7}\epsilon^{-4}\right)$, respectively

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