  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.
