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

Abstract: 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 for black-box mini-optimization where only function values can be obtained. Moreover, we prove that our Acc-ZOM method achieves a lower query complexity of $\tilde{O}(d<sup>{3/4}\epsilon<sup>{-3})$ for finding an $\epsilon$-stationary point, which improves the best known result by a factor of $O(d<sup>{1/4})$ where $d$ denotes the variable dimension. In particular, our Acc-ZOM does not need large batches required in the existing zeroth-order stochastic algorithms. Meanwhile, we propose an accelerated zeroth-order momentum descent ascent (Acc-ZOMDA) method for black-box minimax optimization, where only function values can be obtained. Our Acc-ZOMDA obtains a low query complexity of $$\tilde{O}((d_1+d_2)<sup>{3/4}\kappa_y<sup>{4.5}\epsilon<sup>{-3})$$ without requiring large batches for finding an $\epsilon$-stationary point, where $d_1$ and $d_2$ denote variable dimensions and $\kappa_y$ is condition number. Moreover, we propose an accelerated first-order momentum descent ascent (Acc-MDA) method for minimax optimization, whose explicit gradients are accessible. Our Acc-MDA achieves a low gradient complexity of $\tilde{O}(\kappa_y<sup>{4.5}\epsilon<sup>{-3})$ without requiring large batches for finding an $\epsilon$-stationary point. In particular, our Acc-MDA can obtain a lower gradient complexity of $\tilde{O}(\kappa_y<sup>{2.5}\epsilon<sup>{-3})$ with a batch size $O(\kappa_y<sup>4)$, which improves the best known result by a factor of $O(\kappa_y<sup>{1/2})$. Extensive experimental results on black-box adversarial attack to deep neural networks and poisoning attack to logistic regression demonstrate efficiency of our algorithms.