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We study stochastic convex optimization under infinite noise variance.Specifically, when the stochastic gradient is unbiased and has uniformlybounded $(1+\kappa)$-th moment, for some $\kappa \in (0,1]$, we quantify theconvergence rate of the Stochastic Mirror Descent algorithm with a particularclass of uniformly convex mirror maps, in terms of the number of iterations,dimensionality and related geometric parameters of the optimization problem.Interestingly this algorithm does not require any explicit gradient clipping ornormalization, which have been extensively used in several recent empirical andtheoretical works. We complement our convergence results withinformation-theoretic lower bounds showing that no other algorithm using onlystochastic first-order oracles can achieve improved rates. Our results haveseveral interesting consequences for devising online/streaming stochasticapproximation algorithms for problems arising in robust statistics and machinelearning.

Mirror Descent Strikes Again: Optimal Stochastic Convex Optimization under Infinite Noise Variance