Asymptotic normality and optimality in nonsmooth stochastic approximation

Polyak and Juditsky [30] famously showed that the stochastic gradient method for minimizing smooth and strongly convex functions enjoys a central limit theorem: the error between the running average of the iterates and the minimizer, normalized by the square root of the iteration counter, converges to a normal random vector. Moreover, the covariance matrix of the limiting distribution is in a precise sense "optimal" among any estimation procedure. A long standing open question is whether similar guarantees - asymptotic normality and optimality - exist for nonsmooth optimization and, more generally, for equilibrium problems. In this work, we obtain such guarantees under mild conditions that hold both in concrete circumstances (e.g.