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.