  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.
