Beyond the regret minimization barrier: Optimal algorithms for stochastic strongly-convex optimization

Elad Hazan, Satyen Kale

Research output: Contribution to journalArticlepeer-review

157 Scopus citations


We give novel algorithms for stochastic strongly-convex optimization in the gradient oracle model which return a O( 1/T )-approximate solution after T iterations. The first algorithm is deterministic, and achieves this rate via gradient updates and historical averaging. The second algorithm is randomized, and is based on pure gradient steps with a random step size. This rate of convergence is optimal in the gradient oracle model. This improves upon the previously known best rate of O( log(T)/T ), which was obtained by applying an online strongly-convex optimization algorithm with regret O(log(T)) to the batch setting. We complement this result by proving that any algorithm has expected regret of Ω(log(T)) in the online stochastic strongly-convex optimization setting. This shows that any online-to-batch conversion is inherently suboptimal for stochastic strongly-convex optimization. This is the first formal evidence that online convex optimization is strictly more difficult than batch stochastic convex optimization.

Original languageEnglish (US)
Pages (from-to)2489-2512
Number of pages24
JournalJournal of Machine Learning Research
StatePublished - Jul 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Systems Engineering
  • Statistics and Probability
  • Artificial Intelligence


  • Convex optimization
  • Online Learning
  • Regret minimization
  • Stochastic gradient descent


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