Convergence of Adaptive Stochastic Mirror Descent

In this article, we present a family of adaptive stochastic optimization methods, which are associated with mirror maps that are widely used to capture the geometry properties of optimization problems during iteration processes. The well-known adaptive moment estimation (Adam)-type algorithm falls into the family when the mirror maps take the form of temporal adaptation. In the context of convex objective functions, we show that with proper step sizes and hyperparameters, the average regret can achieve the convergence rate ${\mathcal { O}}(T^{-(1/2)})$ after T iterations under some standard assumptions. We further improve it to $O(T^{-1}\log T)$ when the objective functions are strongly convex. In the context of smooth objective functions (not necessarily convex), based on properties of the strongly convex differentiable mirror map, our algorithms achieve convergence rates of order ${\mathcal { O}}(T^{-(1/2)})$ up to a logarithmic term, requiring large or increasing hyperparameters that are coincident with practical usage of Adam-type algorithms. Thus, our work gives explanations for the selection of the hyperparameters in Adam-type algorithms’ implementation.