In the paper, we propose a class of accelerated stochastic gradient-free and projection-free (a.k.a., zeroth-order Frank-Wolfe) methods to solve the constrained stochastic and finite-sum nonconvex optimization. Specifically, we propose an accelerated stochastic zeroth-order Frank-Wolfe (Acc-SZOFW) method based on the variance reduced technique of SPIDER/SpiderBoost and a novel momentum accelerated technique. Moreover, under some mild conditions, we prove that the Acc-SZOFW has the function query complexity of $O(d\sqrt{n}\epsilon^{-2})$ for finding an $\epsilon$-stationary point in the finite-sum problem, which improves the exiting best result by a factor of $O(\sqrt{n}\epsilon^{-2})$, and has the function query complexity of $O(d\epsilon^{-3})$ in the stochastic problem, which improves the exiting best result by a factor of $O(\epsilon^{-1})$. To relax the large batches required in the Acc-SZOFW, we further propose a novel accelerated stochastic zeroth-order Frank-Wolfe (Acc-SZOFW*) based on a new variance reduced technique of STORM, which still reaches the function query complexity of $O(d\epsilon^{-3})$ in the stochastic problem without relying on any large batches. In particular, we present an accelerated framework of the Frank-Wolfe methods based on the proposed momentum accelerated technique. The extensive experimental results on black-box adversarial attack and robust black-box classification demonstrate the efficiency of our algorithms.

本文提出了一种加速的随机零阶Frank-Wolfe优化算法，通过使用SPIDER/SpiderBoost技术和一种新的动量加速技术，它可以在非凸优化中实现O(d√nε⁻²)的函数查询复杂度，并改进了现有最佳结果，同时在随机问题中实现了O(dε⁻³)的函数查询复杂度，同时提出了基于STORM的Acc-SZOFW *，它不需要大批量也可以达到与Acc-SZOFW相同的函数查询复杂度。

加速的随机无梯度投影方法