Adaptive first-order methods revisited: Convex optimization without Lipschitz requirements
This addresses a gap in optimization for problems with singular objectives like Fisher markets and Poisson tomography, enabling adaptive methods where existing ones fail.
The authors tackled convex optimization problems lacking Lipschitz continuity or smoothness by proposing adaptive mirror descent (AdaMir), achieving min-max optimal rates for relatively continuous or smooth problems, including stochastic cases.
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.