  In this paper, we design and analyze a new family of adaptive subgradient
methods for solving an important class of weakly convex (possibly nonsmooth)
stochastic optimization problems. Adaptive methods that use exponential moving
averages of past gradients to update search directions and learning rates have
recently attracted a lot of attention for solving optimization problems that
arise in machine learning. Nevertheless, their convergence analysis almost
exclusively requires smoothness and/or convexity of the objective function. In
contrast, we establish non-asymptotic rates of convergence of first and
zeroth-order adaptive methods and their proximal variants for a reasonably
broad class of nonsmooth \& nonconvex optimization problems. Experimental
results indicate how the proposed algorithms empirically outperform stochastic
gradient descent and its zeroth-order variant for solving such optimization
problems.
