  In their seminal work, Polyak and Juditsky showed that stochastic
approximation algorithms for solving smooth equations enjoy a central limit
theorem. Moreover, it has since been argued that the asymptotic covariance of
the method is best possible among any estimation procedure in a local minimax
sense of H\'{a}jek and Le Cam. A long-standing open question in this line of
work is whether similar guarantees hold for important non-smooth problems, such
as stochastic nonlinear programming or stochastic variational inequalities. In
this work, we show that this is indeed the case.
