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