Sharpness-Aware Minimization (SAM) is a recent optimization framework aiming to improve the deep neural network generalization, through obtaining flatter (i.e. less sharp) solutions. As SAM has been numerically successful, recent papers have studied the theoretical aspects of the framework and have shown SAM solutions are indeed flat. However, there has been limited theoretical exploration regarding statistical properties of SAM. In this work, we directly study the statistical performance of SAM, and present a new theoretical explanation of why SAM generalizes well. To this end, we study two statistical problems, neural networks with a hidden layer and kernel regression, and prove under certain conditions, SAM has smaller prediction error over Gradient Descent (GD). Our results concern both convex and non-convex settings, and show that SAM is particularly well-suited for non-convex problems. Additionally, we prove that in our setup, SAM solutions are less sharp as well, showing our results are in agreement with the previous work. Our theoretical findings are validated using numerical experiments on numerous scenarios, including deep neural networks.
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