The Statistical Complexity of Early-Stopped Mirror Descent
This provides theoretical insights into implicit regularization for researchers in optimization and machine learning, though it is incremental as it builds on existing literature.
The paper tackles the problem of understanding the statistical guarantees of early-stopped mirror descent algorithms for linear models and kernel methods, showing that it yields excess risk bounds in terms of offset Rademacher complexities and recovers or improves upon recent results in implicit regularization.
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 the squared loss for linear models and kernel methods. By completing an inequality that characterizes convexity for the squared loss, we identify an intrinsic link between offset Rademacher complexities and potential-based convergence analysis of mirror descent methods. Our observation immediately yields excess risk guarantees for 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 the number of iterations. We apply our theory to recover, in a clean and elegant manner via rather short proofs, some of the recent results in the implicit regularization literature, while also showing how to improve upon them in some settings.