Jul 12, 2020
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 problem of 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 and a novel momentum technique. Moreover, under some mild conditions, we prove that the Acc-SZOFW has the function query complexity of O(d√(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(√(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). Further, we propose a novel accelerated stochastic zeroth-order Frank Wolfe (Acc-SZOFW*) to relax the large mini-batch size required in the Acc-SZOFW. In particular, we prove that the Acc-SZOFW* still has the function query complexity of O(dϵ^-3) in the stochastic problem. Finally, we use extensive experiments including black-box adversarial attack and robust black-box classification to verify the efficiency of our algorithms.
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