Accelerated Zeroth-Order and First-Order Momentum Methods from Mini to Minimax Optimization

In the paper, we propose a class of accelerated zeroth-order and first-order momentum methods for both nonconvex mini-optimization and minimax-optimization. Specifically, we propose a new accelerated zeroth-order momentum (Acc-ZOM) method to solve stochastic mini-optimization problems. We prove that the Acc-ZOM method achieves a lower query complexity of $\tilde{O}(d^{3/4}\epsilon^{-3})$ 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. In particular, the Acc-ZOM does not need large batches that are required in the existing zeroth-order stochastic algorithms. At the same time, we propose an accelerated zeroth-order momentum descent ascent (Acc-ZOMDA) method for black-box minimax-optimization. We prove that the Acc-ZOMDA method reaches the best known query complexity of $\tilde{O}((d_1+d_2)\kappa_y^{3}\epsilon^{-3})$ without large batches for finding an $\epsilon$-stationary point, where $d_1$ and $d_2$ denote dimensions of optimization parameters and $\kappa_y$ is condition number. Moreover, we propose an accelerated first-order momentum descent ascent (Acc-MDA) method for solving white-box minimax problems, and prove that it achieves a lower gradient complexity of $\tilde{O}(\kappa_y^{2.5}\epsilon^{-3})$ given batch size $b=\kappa_y^{4}$ for finding an $\epsilon$-stationary point, which improves the best known result by a factor of $O(\kappa_y^{1/2})$. Extensive experimental results on the black-box adversarial attack to deep neural networks (DNNs) and poisoning attack demonstrate the efficiency of our algorithms.