Title: Implicit Bias of Per-sample Adam on Separable Data: Departure from the Full-batch Regime
Primary area: optimization
Keywords: Adam, implicit bias, separable data, adaptive algorithms, mini-batch

Abstract: Adam [Kingma & Ba, 2015] is the de facto optimizer in deep learning, yet its theoretical understanding remains limited. Prior analyses show that Adam favors solutions aligned with $\ell_\infty$-geometry, but these results are restricted to the full-batch regime. In this work, we study the implicit bias of incremental Adam (using one sample per step) for logistic regression on linearly separable data, and show that its bias can deviate from the full-batch behavior. To illustrate this, we construct a class of structured datasets where incremental Adam provably converges to the $\ell_2$-max-margin classifier, in contrast to the $\ell_\infty$-max-margin bias of full-batch Adam. For general datasets, we develop a proxy algorithm that captures the limiting behavior of incremental Adam as $\beta_2\to 1$ and characterize its convergence direction via a data-dependent dual fixed-point formulation. Finally, we prove that, unlike Adam, Signum [Bernstein et al., 2018] converges to the $\ell_\infty$-max-margin classifier for any batch size. Overall, our results highlight that the implicit bias of Adam crucially depends on both the batching scheme and the dataset, while Signum remains invariant.

TL;DR: We characterize the implicit bias of mini-batch Adam on separable data, showing that it can deviate from the $\ell_\infty$-max-margin classifier, whereas Signum consistently preserves the $\ell_\infty$-bias for any batch size.