Understanding SAM through Minimax Perspective

Abstract

Sharpness-Aware Minimization (SAM) empirically boosts generalization by seeking parameters that minimize the worst-case loss in a small neighborhood, yet existing theory explains its behavior under either Polyak-Lojasiewicz (PL) condition or upper bounded perturbation radius. We revisit SAM through the bilevel minimax problem $\min_{\theta}\max_{\|\Delta\|\le\rho}l(\theta+\Delta)$ and derive a $(\theta,\Delta)$ gradient flow ODE whose equilibria coincide with the problem’s optimality conditions. A Lyapunov argument-free of convexity assumptions, quantifies how the optimality gap depends on the radius~$\rho$ and local curvature. Discretizing the flow yields a Multi-step SAM algorithm that recovers classical SAM as $\rho\to 0$. Moreover, our analysis and the resulting algorithm remain valid even for large $\rho$, providing guidance for aggressive neighborhood exploration. Experiments on synthetic objectives and CIFAR-10 validate the predicted gains from multiple inner updates, bridging the gap between SAM’s minimax intuition and its practical implementation.