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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(d3=4ffl3) for finding an -stationary point, which improves the best known result by a factor of O(d1=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((d1 + d2)3=4≤4:5 y ffl3) without requiring large batches for finding an ffl-stationary point, where d1 and d2 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(≤4:5 y ffl3) without requiring large batches for finding an ffl-stationary point. In particular, our Acc-MDA can obtain a lower gradient complexity of O (≤2:5 y ffl3) with a batch size O(≤4 y), which improves the best known result by a factor of O(≤1=2 y). Extensive experimental results on black-box adversarial attack to deep neural networks and poisoning attack to logistic regression demonstrate efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

Zeroth-order (ZO) optimization has been the key technique for various machine learning applications especially for black-box adversarial attack, where models need to be learned in a gradient-free manner. Although many ZO algorithms have been proposed, the high function query complexities hinder their applications seriously. To address this challenging problem, we propose two stagewise black-box reduction frameworks for ZO algorithms under convex and non-convex settings respectively, which lower down the function query complexities of ZO algorithms. Moreover, our frameworks can directly derive the convergence results of ZO algorithms under convex and non-convex settings without extra analyses, as long as convergence results under strongly convex setting are given. To illustrate the advantages, we further study ZO-SVRG, ZO-SAGA and ZO-Varag under strongly-convex setting and use our frameworks to directly derive the convergence results under convex and non-convex settings. The function query complexities of these algorithms derived by our frameworks are lower than that of their vanilla counterparts without frameworks, or even lower than that of state-of-the-art algorithms. Finally we conduct numerical experiments to illustrate the superiority of our frameworks. [ABSTRACT FROM AUTHOR]

Asynchronous stochastic algorithms with various variance reduction techniques (such as SVRG, S2GD, SAGA and q-SAGA) are popular in solving large scale learning problems. Recently, Reddi et al. (2015) proposed an unified variance reduction framework (i.e., HSAG) to analyze the asynchronous stochastic gradient optimization. However, the HSAG framework cannot incorporate the S2GD technique, the analysis of the HSAG framework is limited to the SVRG and SAGA techniques on the smooth convex optimization. They did not analyze other important various variance techniques (e.g., S2GD and q-SAGA) and other important optimization problems (e.g., convex optimization with non-smooth regularization and non-convex optimization with cardinality constraint). In this paper, we bridge this gap by using an unified q-memorization framework for various variance reduction techniques (including SVRG, S2GD, SAGA, q-SAGA) to analyze asynchronous stochastic algorithms for three important optimization problems. Specifically, based on the q-memorization framework, 1) we propose an asynchronous stochastic gradient hard thresholding algorithm with q-memorization (AsySGHT-qM) for the non-convex optimization with cardinality constraint, and prove that the convergence rate of AsySGHT-qM before reaching the inherent error induced by gradient hard thresholding methods is geometric. 2) We propose an asynchronous stochastic proximal gradient algorithm (AsySPG-qM) for the convex optimization with non-smooth regularization, and prove that AsySPG-qM can achieve a linear convergence rate. 3) We propose an asynchronous stochastic gradient descent algorithm (AsySGD-qM) for the general non-convex optimization problem, and prove that AsySGD-qM can achieve a sublinear convergence rate to stationary points. The experimental results on various large-scale datasets confirm the fast convergence of our AsySGHT-qM, AsySPG-qM and AsySGD-qM through concrete realizations of SVRG and SAGA. [ABSTRACT FROM AUTHOR]