Minor First, Major Last: A Depth-Induced Implicit Bias of Sharpness-Aware Minimization

TL;DR

This paper investigates the implicit bias of Sharpness-Aware Minimization (SAM) in training linear diagonal networks, revealing depth-dependent behaviors and a phenomenon called sequential feature amplification.

Contribution

It provides a theoretical analysis of SAM's implicit bias in linear models, highlighting differences from gradient descent and introducing the concept of sequential feature amplification.

Findings

For linear models, SAM recovers the max-margin classifier.

For depth 2, SAM's limit depends on initialization and can differ from GD.

In finite time, $ ext{l}_2$-SAM exhibits sequential feature amplification.

Abstract

We study the implicit bias of Sharpness-Aware Minimization (SAM) when training $L$-layer linear diagonal networks on linearly separable binary classification. For linear models ($L=1$), both $\ell_\infty$- and $\ell_2$-SAM recover the $\ell_2$ max-margin classifier, matching gradient descent (GD). However, for depth $L = 2$, the behavior changes drastically -- even on a single-example dataset. For $\ell_\infty$-SAM, the limit direction depends critically on initialization and can converge to $\mathbf{0}$ or to any standard basis vector, in stark contrast to GD, whose limit aligns with the basis vector of the dominant data coordinate. For $\ell_2$-SAM, we show that although its limit direction matches the $\ell_1$ max-margin solution as in the case of GD, its finite-time dynamics exhibit a phenomenon we call "sequential feature amplification", in which the predictor initially relies on minor coordinates and gradually shifts to larger ones as training proceeds or initialization increases. Our theoretical analysis attributes this phenomenon to $\ell_2$-SAM's gradient normalization factor applied in its perturbation, which amplifies minor coordinates early and allows major ones to dominate later, giving a concrete example where infinite-time implicit-bias analyses are insufficient. Synthetic and real-data experiments corroborate our findings.