Abstract

We study the impact of the constraint set and gradient geometry on the convergence of online and stochastic methods for convex optimization, providing a characterization of the geometries for which stochastic gradient and adaptive gradient methods are (minimax) optimal. In particular, we show that when the constraint set is quadratically convex, diagonally pre-conditioned stochastic gradient methods are minimax optimal. We further provide a converse that shows that when the constraints are not quadratically convex—for example, any -ball for p < 2—the methods are far from optimal. Based on this, we can provide concrete recommendations for when one should use adaptive, mirror or stochastic gradient methods.