### Abstract

We give a novel algorithm for stochastic strongly-convex optimization in the gradient oracle model which returns an O(1/T)-approximate solution after T gradient updates. 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 lower bound holds even in the full-information setting which reveals more information to the algorithm than just gradients. 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 language | English (US) |
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Pages (from-to) | 421-436 |

Number of pages | 16 |

Journal | Journal of Machine Learning Research |

Volume | 19 |

State | Published - Jan 1 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

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