Asymptotic optimality in stochastic optimization

John C. Duchi, Feng Ruan

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We study local complexity measures for stochastic convex optimization problems, providing a local minimax theory analogous to that of Hájek and Le Cam for classical statistical problems. We give complementary optimality results, developing fully online methods that adaptively achieve optimal convergence guarantees. Our results provide function-specific lower bounds and convergence results that make precise a correspondence between statistical difficulty and the geometric notion of tilt-stability from optimization. As part of this development, we show how variants of Nesterov's dual averaging-a stochastic gradient-based procedure-guarantee finite time identification of constraints in optimization problems, while stochastic gradient procedures fail. Additionally, we highlight a gap between problems with linear and nonlinear constraints: standard stochastic-gradient-based procedures are suboptimal even for the simplest nonlinear constraints, necessitating the development of asymptotically optimal Riemannian stochastic gradient methods.

Original languageEnglish (US)
Pages (from-to)21-48
Number of pages28
JournalAnnals of Statistics
Volume49
Issue number1
DOIs
StatePublished - Feb 2021

Keywords

  • Convex analysis
  • Local asymptotic minimax theory
  • Manifold identification
  • Stochastic gradients

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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