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.
A nearly linearly convergent first-order method for nonsmooth functions with quadratic growth
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abstract
Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic growth. This work designs such a method for a wide class of nonsmooth and nonconvex locally Lipschitz functions, including max-of-smooth, Shapiro's decomposable class, and generic semialgebraic functions. The algorithm is parameter-free and derives from Goldstein's conceptual subgradient method.
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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.