ZO-AdaMM: Zeroth-Order Adaptive Momentum Method for Black-Box Optimization
ZO-AdaMM: Zeroth-Order Adaptive Momentum Method for Black-Box Optimization
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发表时间：
2019-10
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通讯作者：
Xiangyi Chen;Sijia Liu;Kaidi Xu;Xingguo Li;Xue Lin;Mingyi Hong;David Cox
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作者：
Xiangyi Chen;Sijia Liu;Kaidi Xu;Xingguo Li;Xue Lin;Mingyi Hong;David Cox
自适应动量法（AdaMM）利用过去的梯度同时更新下降方向和学习率，已成为解决机器学习问题最流行的一阶优化方法之一。然而，AdaMM不适合解决黑盒优化问题，因为在黑盒优化问题中，显式梯度形式很难或不可获得。本文提出了一种零阶AdaMM （ZO-AdaMM）算法，将AdaMM算法推广到无梯度区域。我们证明了ZO-AdaMM对于凸和非凸优化的收敛速度大约比一阶AdaMM算法的收敛速度差$O(\sqrt{d})$，其中$d$为问题大小。特别是，我们深入理解了为什么马氏距离对ZO-AdaMM和其他adam型方法收敛的影响。作为副产品，我们的分析为理解非凸约束优化的自适应学习率方法迈出了第一步。此外，我们展示了两种应用，分别设计了来自黑盒神经网络的单图像和通用对抗性攻击。我们在ImageNet上进行了大量的实验，经验表明，与6美元最先进的ZO优化方法相比，ZO- adamm收敛到高精度解决方案的速度要快得多。
The adaptive momentum method (AdaMM), which uses past gradients to update descent directions and learning rates simultaneously, has become one of the most popular first-order optimization methods for solving machine learning problems. However, AdaMM is not suited for solving black-box optimization problems, where explicit gradient forms are difficult or infeasible to obtain. In this paper, we propose a zeroth-order AdaMM (ZO-AdaMM) algorithm, that generalizes AdaMM to the gradient-free regime. We show that the convergence rate of ZO-AdaMM for both convex and nonconvex optimization is roughly a factor of $O(\sqrt{d})$ worse than that of the first-order AdaMM algorithm, where $d$ is problem size. In particular, we provide a deep understanding on why Mahalanobis distance matters in convergence of ZO-AdaMM and other AdaMM-type methods. As a byproduct, our analysis makes the first step toward understanding adaptive learning rate methods for nonconvex constrained optimization.Furthermore, we demonstrate two applications, designing per-image and universal adversarial attacks from black-box neural networks, respectively. We perform extensive experiments on ImageNet and empirically show that ZO-AdaMM converges much faster to a solution of high accuracy compared with $6$ state-of-the-art ZO optimization methods.