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

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.