  The paper investigates the fundamental convergence properties of
Sharpness-Aware Minimization (SAM), a recently proposed gradient-based
optimization method (Foret et al., 2021) that significantly improves the
generalization of deep neural networks. The convergence properties including
the stationarity of accumulation points, the convergence of the sequence of
gradients to the origin, the sequence of function values to the optimal value,
and the sequence of iterates to the optimal solution are established for the
method. The universality of the provided convergence analysis based on inexact
gradient descent frameworks (Khanh et al., 2023b) allows its extensions to the
normalized versions of SAM such as VaSSO (Li & Giannakis, 2023), RSAM (Liu et
al., 2022), and to the unnormalized versions of SAM such as USAM
(Andriushchenko & Flammarion, 2022). Numerical experiments are conducted on
classification tasks using deep learning models to confirm the practical
aspects of our analysis.
