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

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 stronglyconvex 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.