Faster Gradient-Free Algorithms for Nonsmooth Nonconvex Stochastic Optimization

We consider the optimization problem of the form $\min{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}{\xi} [F(x; \xi)]$, where the component $F(x;\xi)$ is $L$-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most $\mathcal{O}( L⁴ d^{3/2} \epsilon^{-4} + \Delta L³ d^{3/2} \delta^{-1} \epsilon^{-4})$ stochastic zeroth-order oracle complexity to find a $(\delta,\epsilon)$-Goldstein stationary point of objective function, where $\Delta = f(x0) - \inf{x \in \mathbb{R}^d} f(x)$ 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 $\mathcal{O}(L³ d^{3/2} \epsilon^{-3}+ \Delta L² d^{3/2} \delta^{-1} \epsilon^{-3})$.