The Dynamics of Sharpness-Aware Minimization: Bouncing Across Ravines and Drifting Towards Wide Minima

We consider Sharpness-Aware Minimization (SAM), a gradient-based optimization method for deep networks that has exhibited performance improvements on image and language prediction problems. We show that when SAM is applied with a convex quadratic objective, for most random initializations it converges to a cycle that oscillates between either side of the minimum in the direction with the largest curvature, and we provide bounds on the rate of convergence. In the non-quadratic case, we show that such oscillations effectively perform gradient descent, with a smaller step-size, on the spectral norm of the Hessian. In such cases, SAM's update may be regarded as a third derivative -- the derivative of the Hessian in the leading eigenvector direction -- that encourages drift toward wider minima.