Accelerated Zeroth-Order Momentum Methods from Mini to Minimax Optimization

In the paper, we propose a new accelerated zeroth-order momentum (Acc-ZOM) method to solve the non-convex stochastic mini-optimization problems. We prove that the Acc-ZOM method achieves a lower query complexity of $O(d^{3/4}\epsilon^{-3})$ for finding an $\epsilon$-stationary point, which improves the best known result by a factor of $O(d^{1/4})$ where $d$ denotes the parameter dimension. The Acc-ZOM does not require any batches compared to the large batches required in the existing zeroth-order stochastic algorithms. Further, we extend the Acc-ZOM method to solve the non-convex stochastic minimax-optimization problems and propose an accelerated zeroth-order momentum descent ascent (Acc-ZOMDA) method. We prove that the Acc-ZOMDA method reaches the best know query complexity of $\tilde{O}(\kappa_y^3(d_1+d_2)^{3/2}\epsilon^{-3})$ for finding an $\epsilon$-stationary point, where $d_1$ and $d_2$ denote dimensions of the mini and max optimization parameters respectively and $\kappa_y$ is condition number. In particular, our theoretical result does not rely on large batches required in the existing methods. Moreover, we propose a momentum-based accelerated framework for the minimax-optimization problems. At the same time, we present an accelerated momentum descent ascent (Acc-MDA) method for solving the white-box minimax problems, and prove that it achieves the best known gradient complexity of $\tilde{O}(\kappa_y^3\epsilon^{-3})$ without large batches. Extensive experimental results on the black-box adversarial attack to deep neural networks (DNNs) and poisoning attack demonstrate the efficiency of our algorithms.