For nonsmooth stochastic approximation with a local smooth-manifold structure, the online batch-means estimator attains covariance estimation rate O(sqrt(d) n^{-1/8+eps}), matching the smooth strongly convex case up to logarithmic factors.
Online covariance matrix estimation in stochastic gradient descent
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Online Covariance Estimation in Nonsmooth Stochastic Approximation
For nonsmooth stochastic approximation with a local smooth-manifold structure, the online batch-means estimator attains covariance estimation rate O(sqrt(d) n^{-1/8+eps}), matching the smooth strongly convex case up to logarithmic factors.