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Adaptive gradient methods are typically used for training over-parameterizedmodels. To better understand their behaviour, we study a simplistic setting --smooth, convex losses with models over-parameterized enough to interpolate thedata. In this setting, we prove that AMSGrad with constant step-size andmomentum converges to the minimizer at a faster $O(1/T)$ rate. Wheninterpolation is only approximately satisfied, constant step-size AMSGradconverges to a neighbourhood of the solution at the same rate, while AdaGrad isrobust to the violation of interpolation. However, even for simple convexproblems satisfying interpolation, the empirical performance of both methodsheavily depends on the step-size and requires tuning, questioning theiradaptivity. We alleviate this problem by automatically determining thestep-size using stochastic line-search or Polyak step-sizes. With thesetechniques, we prove that both AdaGrad and AMSGrad retain their convergenceguarantees, without needing to know problem-dependent constants. Empirically,we demonstrate that these techniques improve the convergence and generalizationof adaptive gradient methods across tasks, from binary classification withkernel mappings to multi-class classification with deep networks.

Adaptive Gradient Methods Converge Faster with Over-Parameterization (but you should do a line-search)