A Unified Stability Analysis of SAM vs SGD: Role of Data Coherence and Emergence of Simplicity Bias
This work addresses a foundational theoretical gap in machine learning by connecting data structure to optimization dynamics, though it is incremental in building on prior flatness theories.
The paper tackled the problem of understanding why optimization algorithms like SGD and SAM prefer flatter minima in deep learning by developing a linear stability framework for two-layer ReLU networks, revealing that a coherence measure explains stability and preference for certain minima.
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.