The Dynamics of Sharpness-Aware Minimization: Bouncing Across Ravines and Drifting Towards Wide Minima
This provides theoretical insights into SAM's optimization dynamics, which is incremental for researchers in deep learning optimization.
The paper analyzes Sharpness-Aware Minimization (SAM), showing that for convex quadratic objectives, it typically converges to an oscillatory cycle near minima with bounds on convergence rates, and in non-quadratic cases, it acts like gradient descent on the Hessian's spectral norm, promoting drift toward wider 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.