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

TL;DR

This paper develops a unified stability analysis framework for SGD and SAM in deep learning, highlighting how data coherence influences the preference for flat minima and simplicity bias.

Contribution

It introduces a linear stability framework incorporating data coherence to explain the dynamics and solution preferences of SGD and SAM in neural networks.

Findings

Data coherence explains stability and minima preference.

SAM promotes flatter, simpler solutions compared to SGD.

The framework applies to two-layer ReLU networks.

Abstract

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.