  We study stochastic convex optimization under infinite noise variance.
Specifically, when the stochastic gradient is unbiased and has uniformly
bounded $(1+\kappa)$-th moment, for some $\kappa \in (0,1]$, we quantify the
convergence rate of the Stochastic Mirror Descent algorithm with a particular
class 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 or
normalization, which have been extensively used in several recent empirical and
theoretical works. We complement our convergence results with
information-theoretic lower bounds showing that no other algorithm using only
stochastic first-order oracles can achieve improved rates. Our results have
several interesting consequences for devising online/streaming stochastic
approximation algorithms for problems arising in robust statistics and machine
learning.
