Elad Hazan and Satyen Kale
In Journal of Machine Learning Research (JMLR), 2014. Also in proceedings of 24th Conference on Learning Theory (COLT), 2011

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 \(\Omega(\log(T))\) 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.