Accelerated Stochastic Gradient-free and Projection-free Methods
This work addresses optimization problems in machine learning where gradient information is unavailable or projections are costly, offering incremental improvements in efficiency for applications like adversarial attacks.
The paper tackles constrained stochastic and finite-sum nonconvex optimization by proposing accelerated stochastic gradient-free and projection-free methods, achieving improved function query complexities, such as O(d√n ε^{-2}) for finite-sum problems, which is a factor of O(√n ε^{-2}) better than prior results.
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 $O(d\sqrt{n}ε^{-2})$ for finding an $ε$-stationary point in the finite-sum problem, which improves the exiting best result by a factor of $O(\sqrt{n}ε^{-2})$, and has the function query complexity of $O(dε^{-3})$ in the stochastic problem, which improves the exiting best result by a factor of $O(ε^{-1})$. 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 $O(dε^{-3})$ 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.