The Statistical Complexity of Early Stopped Mirror Descent

Recently there has been a surge of interest in understanding implicit regularization properties of iterative gradient-based optimization algorithms. In this paper, we study the statistical guarantees on the excess risk achieved by early stopped unconstrained mirror descent algorithms applied to the unregularized empirical risk with squared loss for linear models and kernel methods. We identify a link between offset Rademacher complexities and potential-based analysis of mirror descent that allows disentangling statistics from optimization in the analysis of such algorithms. Our main result characterizes the statistical performance of the path traced by the iterates of mirror descent in terms of offset complexities of certain function classes depending only on the choice of the mirror map, initialization point, step-size, and number of iterations. We apply our theory to recover, in a rather clean and elegant manner, some of the recent results in the implicit regularization literature, while also showing how to improve upon them in some settings.