Faster Gradient-Free Algorithms for Nonsmooth Nonconvex Stochastic Optimization

Abstract

We consider the optimization problem of the form $\min_{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}_ξ [F(x; ξ)]$, where the component $F(x;ξ)$ is $L$-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most $\mathcal{O}( L^4 d^{3/2} ε^{-4} + ΔL^3 d^{3/2} δ^{-1} ε^{-4})$ stochastic zeroth-order oracle complexity to find a $(δ,ε)$-Goldstein stationary point of objective function, where $Δ= f(x_0) - \inf_{x \in \mathbb{R}^d} f(x)$ and $x_0$ is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to $\mathcal{O}(L^3 d^{3/2} ε^{-3}+ ΔL^2 d^{3/2} δ^{-1} ε^{-3})$.