Beyond the regret minimization barrier: an optimal algorithm for stochastic strongly-convex optimization

Elad Hazan, Satyen Kale
Proceedings of the 24th Annual Conference on Learning Theory, PMLR 19:421-436, 2011.

Abstract

We give a novel algorithm for stochastic strongly-convex optimization in the gradient oracle model which returns an $O(\frac1T)$-approximate solution after $T$ gradient updates. This rate of convergence is optimal in the gradientoracle model. This improves upon the previously known best rate of $O(\frac{\log(T)}{T})$, 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 $\Omega(\log(T))$ 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v19-hazan11a, title = {Beyond the regret minimization barrier: an optimal algorithm for stochastic strongly-convex optimization}, author = {Hazan, Elad and Kale, Satyen}, booktitle = {Proceedings of the 24th Annual Conference on Learning Theory}, pages = {421--436}, year = {2011}, editor = {Kakade, Sham M. and von Luxburg, Ulrike}, volume = {19}, series = {Proceedings of Machine Learning Research}, address = {Budapest, Hungary}, month = {09--11 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v19/hazan11a/hazan11a.pdf}, url = {https://proceedings.mlr.press/v19/hazan11a.html}, abstract = {We give a novel algorithm for stochastic strongly-convex optimization in the gradient oracle model which returns an $O(\frac1T)$-approximate solution after $T$ gradient updates. This rate of convergence is optimal in the gradientoracle model. This improves upon the previously known best rate of $O(\frac{\log(T)}{T})$, 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 $\Omega(\log(T))$ 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.} }
Endnote
%0 Conference Paper %T Beyond the regret minimization barrier: an optimal algorithm for stochastic strongly-convex optimization %A Elad Hazan %A Satyen Kale %B Proceedings of the 24th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2011 %E Sham M. Kakade %E Ulrike von Luxburg %F pmlr-v19-hazan11a %I PMLR %P 421--436 %U https://proceedings.mlr.press/v19/hazan11a.html %V 19 %X We give a novel algorithm for stochastic strongly-convex optimization in the gradient oracle model which returns an $O(\frac1T)$-approximate solution after $T$ gradient updates. This rate of convergence is optimal in the gradientoracle model. This improves upon the previously known best rate of $O(\frac{\log(T)}{T})$, 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 $\Omega(\log(T))$ 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.
RIS
TY - CPAPER TI - Beyond the regret minimization barrier: an optimal algorithm for stochastic strongly-convex optimization AU - Elad Hazan AU - Satyen Kale BT - Proceedings of the 24th Annual Conference on Learning Theory DA - 2011/12/21 ED - Sham M. Kakade ED - Ulrike von Luxburg ID - pmlr-v19-hazan11a PB - PMLR DP - Proceedings of Machine Learning Research VL - 19 SP - 421 EP - 436 L1 - http://proceedings.mlr.press/v19/hazan11a/hazan11a.pdf UR - https://proceedings.mlr.press/v19/hazan11a.html AB - We give a novel algorithm for stochastic strongly-convex optimization in the gradient oracle model which returns an $O(\frac1T)$-approximate solution after $T$ gradient updates. This rate of convergence is optimal in the gradientoracle model. This improves upon the previously known best rate of $O(\frac{\log(T)}{T})$, 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 $\Omega(\log(T))$ 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. ER -
APA
Hazan, E. & Kale, S.. (2011). Beyond the regret minimization barrier: an optimal algorithm for stochastic strongly-convex optimization. Proceedings of the 24th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 19:421-436 Available from https://proceedings.mlr.press/v19/hazan11a.html.

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