Practical Sharpness-Aware Minimization Cannot Converge All the Way to Optima
This work addresses a gap in understanding SAM's practical limitations for optimization in machine learning, highlighting that widely used configurations may not achieve full convergence, which is incremental but crucial for algorithm design.
The paper investigates the convergence properties of Sharpness-Aware Minimization (SAM) under practical configurations with constant perturbation size and gradient normalization, finding that it often fails to converge to global minima or stationary points, with stochastic SAM suffering an unavoidable additive term O(ρ²) limiting convergence to neighborhoods of optima.
Sharpness-Aware Minimization (SAM) is an optimizer that takes a descent step based on the gradient at a perturbation $y_t = x_t + ρ\frac{\nabla f(x_t)}{\lVert \nabla f(x_t) \rVert}$ of the current point $x_t$. Existing studies prove convergence of SAM for smooth functions, but they do so by assuming decaying perturbation size $ρ$ 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 $ρ$ 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 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 $\tilde Θ(\frac{1}{T^2})$, the convergence bound of stochastic SAM suffers an inevitable additive term $O(ρ^2)$, indicating convergence only up to neighborhoods of optima. In fact, such $O(ρ^2)$ factors arise for stochastic SAM in all the settings we consider, and also for deterministic SAM in nonconvex cases; importantly, we prove by examples that such terms are 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.