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

• URL: http://arxiv.org/abs/2202.11632v1
• Date: Wed, 23 Feb 2022 17:08:40 GMT
• Title: Mirror Descent Strikes Again: Optimal Stochastic Convex Optimization under Infinite Noise Variance
• Authors: Nuri Mert Vural, Lu Yu, Krishnakumar Balasubramanian, Stanislav Volgushev, Murat A. Erdogdu
• Abstract summary: We quantify the convergence rate of the Mirror Descent algorithm with a class of uniformly convex mirror maps. This algorithm does not require any explicit gradient clipping or normalization. We complement our results with information-theoretic lower bounds showing that no other algorithm using only first-order oracles can achieve improved rates.
• Score: 17.199063087458907
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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