A Unified Stability Analysis of SAM vs SGD: Role of Data Coherence and Emergence of Simplicity Bias

Understanding the dynamics of optimization in deep learning is increasingly important as models scale. While stochastic gradient descent (SGD) and its variants reliably find solutions that generalize well, the mechanisms driving this generalization remain unclear. Notably, these algorithms often prefer flatter or simpler minima-particularly in overparameterized settings. Prior work has linked flatness to generalization, and methods like Sharpness-Aware Minimization (SAM) explicitly encourage flatness, but a unified theory connecting data structure, optimization dynamics, and the nature of learned solutions is still lacking. In this work, we develop a linear stability framework that analyzes the behavior of SGD, random perturbations, and SAM-particularly in two-layer ReLU networks. Central to our analysis is a coherence measure that quantifies how gradient curvature aligns across data points, revealing why certain minima are stable and favored during training. (Code are available in: https://github.com/changwk1001/ Stability_Analysis_and_Simplicity-Bias.git)

Assuming w 0 ∼ N (0, I), we reduce to analyzing the quantity

, which captures the contraction or expansion behavior of the iterates under the sequence of update matrices. See more details discussion of assumption in appendix A.

The system is said to be linearly stable at w ⋆ under a given optimization method if the expected squared norm E[∥w k ∥ 2 ] remains bounded as k → ∞. A sufficient condition for this is that the spectral norm of the average update matrix E[ Ĵ⊤ t Ĵt ] is strictly less than 1. For full-batch gradient descent, this reduces to requiring η < 2/λ max (H).

More generally, in the presence of stochasticity and structure in the data, one can derive stability conditions involving both the Hessian spectrum and how curvature is distributed across examples. This motivates the use of a data-dependent coherence measure, which we introduce next.