Convergence of online mirror descent

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

6 Scopus Citations
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Original languageEnglish
Pages (from-to)343-373
Journal / PublicationApplied and Computational Harmonic Analysis
Issue number1
Online published22 May 2018
Publication statusPublished - Jan 2020


In this paper we consider online mirror descent (OMD), a class of scalable online learning algorithms exploiting data geometric structures through mirror maps. Necessary and sufficient conditions are presented in terms of the step size sequence {ηt}t for the convergence of OMD with respect to the expected Bregman distance induced by the mirror map. The condition is limt→∞ ⁡η= 0,∑t=1 η= ∞ in the case of positive variances. It is reduced to ∑t=1 η= ∞ in the case of zero variance for which linear convergence may be achieved by taking a constant step size sequence. A sufficient condition on the almost sure convergence is also given. We establish tight error bounds under mild conditions on the mirror map, the loss function, and the regularizer. Our results are achieved by some novel analysis on the one-step progress of OMD using smoothness and strong convexity of the mirror map and the loss function.

Research Area(s)

  • Bregman distance, Convergence analysis, Learning theory, Mirror descent, Online learning