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 $\mathcal{O}( L^4 d^{3/2} \epsilon^{-4} + \Delta L^3 d^{3/2} \delta^{-1} \epsilon^{-4})$ stochastic zeroth-order oracle complexity to find a $(\delta,\epsilon)$-Goldstein stationary point of objective function, where $\Delta = f(x_0) - \inf_{x \in \mathbb{R}^d} f(x)$ 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 $\mathcal{O}(L^3 d^{3/2} \epsilon^{-3}+ \Delta L^2 d^{3/2} \delta^{-1} \epsilon^{-3})$.

翻译：我们考虑形如 $\min_{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}_{\xi} [F(x; \xi)]$ 的优化问题，其中分量 $F(x;\xi)$ 是 $L$-均方Lipschitz连续的，但可能非凸且非光滑。近期提出的无梯度方法需要最多 $\mathcal{O}( L^4 d^{3/2} \epsilon^{-4} + \Delta L^3 d^{3/2} \delta^{-1} \epsilon^{-4})$ 的随机零阶预言复杂度，以找到目标函数的 $(\delta,\epsilon)$-Goldstein平稳点，其中 $\Delta = f(x_0) - \inf_{x \in \mathbb{R}^d} f(x)$，而 $x_0$ 是算法的初始点。本文提出了一种更高效的算法，采用随机递归梯度估计器，将复杂度改进至 $\mathcal{O}(L^3 d^{3/2} \epsilon^{-3}+ \Delta L^2 d^{3/2} \delta^{-1} \epsilon^{-3})$。