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 O(d^3/4^-3) for finding an -stationary point, which improves the best known result by a factor of O(d^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 O((d_1+d_2)^3/4_y^4.5^-3) without requiring large batches for finding an -stationary point, where d_1 and d_2 denote variable dimensions and _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 O(_y^4.5^-3) without requiring large batches for finding an -stationary point. In particular, our Acc-MDA can obtain a lower gradient complexity of O(_y^2.5^-3) with a batch size O(_y^4), which improves the best known result by a factor of O(_y^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.

Tasks

Reproductions