On the Two Faces of Adam in Separable Linear Classification

Chen Fan, Csaba Szepesvári

We consider the behavior of deterministic, full-batch, bias-corrected Adam in separable linear classification with softmax parametrization under log-loss. In this setting, under a wide range of conditions Adam is known to approach max-norm-margin optimality when its stability constant $ε$ is zero, while with a positive $ε$, it is known to approach Euclidean-margin optimality. Our main contribution is the quantitative description of Adam's behavior for small fixed positive $ε$. We give sufficient conditions under which an Adam-trained classifier nearly maximizes the max-norm margin before the updates become gradient-like. We also show that the classifier reaches a fixed target Euclidean margin only much later. Specifically, we show that for polynomially decreasing stepsizes with exponent \(a\), where \(1/3<a<1\), the updates become approximately proportional to the negative gradient after $Θ(\log(1/ε)^{1/(1-a)})$ iterations. At that time, the classifier still nearly maximizes the max-norm margin. Reaching a fixed target Euclidean margin above that of every max-norm-optimal classifier, but below the optimum, is shown to require $ε^{-Θ(1)/(1-a)}$ iterations. Under inverse-linear…