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. Moreover, we prove that our 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 variable dimension. In particular, the Acc-ZOM does not require large batches required in the existing zeroth-order stochastic algorithms. Meanwhile, we propose an accelerated \textbf{zeroth-order} momentum descent ascent (Acc-ZOMDA) method for \textbf{black-box} minimax-optimization, which obtains a query complexity of $\tilde{O}((d_1+d_2)^{3/4}\kappa_y^{4.5}\epsilon^{-3})$ without 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 \textbf{first-order} momentum descent ascent (Acc-MDA) method for \textbf{white-box} minimax optimization, which has a gradient complexity of $\tilde{O}(\kappa_y^{4.5}\epsilon^{-3})$ without large batches for finding an $\epsilon$-stationary point. In particular, our Acc-MDA can obtain a lower gradient complexity of $\tilde{O}(\kappa_y^{2.5}\epsilon^{-3})$ with a batch size $O(\kappa_y^4)$. Extensive experimental results on black-box adversarial attack to deep neural networks and poisoning attack to logistic regression demonstrate efficiency of our algorithms.

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