Convergence of online mirror descent

Yunwen Lei, Ding Xuan Zhou*

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

16 Citations (Scopus)


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→∞⁡ηt=0,∑t=1 ηt=∞ in the case of positive variances. It is reduced to ∑t=1 ηt=∞ 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.

Original languageEnglish
Pages (from-to)343-373
Number of pages31
JournalApplied and Computational Harmonic Analysis
Issue number1
Publication statusPublished - Jan 2020

Scopus Subject Areas

  • Applied Mathematics

User-Defined Keywords

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


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