Necessary and Sufficient Conditions for Adaptive, Mirror, and Standard Gradient Methods

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 $\ell_p$-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.