(The) convergence analysis of sharpness-aware minimization under practical settings

Abstract

importantly, we prove by examples that such terms are \emph{unavoidable}. Our results highlight vastly different characteristics of SAM with vs.\ without decaying perturbation size or gradient normalization, and suggest that the intuitions gained from one version may not apply to the other.

Sharpness-Aware Minimization (SAM) is an optimizer that takes a descent step based on the gradient at a perturbation $y_t = x_t + \rho \frac{\nabla f(x_t)}{\| \nabla f(x_t) \|}$ of the current point~$x_t$. Existing studies prove convergence of SAM for smooth functions, but they do so by assuming decaying perturbation size $\rho$ and/or no gradient normalization in $y_t$, which is detached from practice. To address this gap, we study deterministic/stochastic versions of SAM with practical configurations (i.e., constant $\rho$ and gradient normalization in $y_t$) and explore their convergence properties on smooth functions with (non)convexity assumptions. Perhaps surprisingly, in many scenarios, we find out that SAM has \emph{limited} capability to converge to global minima or stationary points. For smooth strongly convex functions, we show that while deterministic SAM enjoys tight global convergence rates of $\tilde \Theta(\frac{1}{T^2})$, the convergence bound of stochastic SAM suffers an \emph{inevitable} additive term $\mathcal O(\rho^2)$, indicating convergence only up to \emph{neighborhoods} of optima. In fact, such $\mathcal O(\rho^2)$ factors arise for stochastic SAM in all the settings we consider, and also for deterministic SAM in nonconvex cases