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

Kimon Antonakopoulos (1) , Panayotis Mertikopoulos (1, 2)

1 POLARIS - Performance analysis and optimization of LARge Infrastructures and Systems
2 Criteo AI Lab

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 objective, 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, 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.

Mots clés: Convex optimization; Adaptive methods; Mirror descent; Relative Lipschitz continuity; Relative Lipschitz smoothness

Domaines: Optimisation et contrôle [math.OC]

Kimon Antonakopoulos, Panayotis Mertikopoulos. Adaptive first-order methods revisited: Convex optimization without Lipschitz requirements. NeurIPS 2021 - 35th International Conference on Neural Information Processing Systems, Dec 2021, Virtual, Unknown Region. ⟨hal-03342998⟩