In this paper, we investigate the non-asymptotic stationary convergence
behavior of Stochastic Mirror Descent (SMD) for nonconvex optimization. We
focus on a general class of nonconvex nonsmooth stochastic optimization
problems, in which the objective can be decomposed into a relatively weakly
convex function (possibly non-Lipschitz) and a simple non-smooth convex
regularizer. We prove that SMD, without the use of mini-batch, is guaranteed to
converge to a stationary point in a convergence rate of $
\mathcal{O}(1/\sqrt{t}) $. The efficiency estimate matches with existing
results for stochastic subgradient method, but is evaluated under a stronger
stationarity measure. Our convergence analysis applies to both the original SMD
and its proximal version, as well as the deterministic variants, for solving
relatively weakly convex problems.

该论文研究了随机镜像下降法在非凸优化中的非渐近稳态收敛性，特别关注了一类非凸非光滑的随机优化问题，其中目标函数可以分解为一个相对弱凸函数（可能是非 Lipschitz）和简单的非光滑凸规则化函数。论文证明，SMD 算法在收敛速率为 $O（1/√t）$ 的同时，无需使用小批量就能保证收敛到一个稳定点。