Minor First, Major Last: A Depth-Induced Implicit Bias of Sharpness-Aware Minimization
For theorists studying implicit bias in deep learning, this work reveals that SAM's finite-time dynamics can differ from infinite-time analyses, providing a concrete example where the latter is insufficient.
This paper studies the implicit bias of Sharpness-Aware Minimization (SAM) in training deep linear diagonal networks for binary classification, revealing that SAM induces a 'sequential feature amplification' phenomenon where minor coordinates are learned first, then major ones, contrasting with gradient descent. For depth 2, ℓ∞-SAM's limit direction depends on initialization, while ℓ2-SAM matches ℓ1 max-margin but with distinct finite-time dynamics.
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.