Adaptive first-order methods revisited: Convex optimization without Lipschitz requirements

Abstract: We propose a new family of adaptive first-order methods for a class of convex minimization problems that may fail to be Lipschitz continuous or smooth in the standard sense. Specifically, motivated by a recent flurry of activity on non-Lipschitz (NoLips) optimization, we consider problems that are continuous or smooth relative to a reference Bregman function - as opposed to a global, ambient norm (Euclidean or otherwise). These conditions encompass a wide range of problems with singular objectives, such as Fisher markets, Poisson tomography, D-design, and the like. In this setting, the application of existing order-optimal adaptive methods - like UnixGrad or AcceleGrad - is not possible, especially in the presence of randomness and uncertainty. The proposed method - which we call adaptive mirror descent (AdaMir) - aims to close this gap by concurrently achieving min-max optimal rates in problems that are relatively continuous or smooth, including stochastic ones.