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The strategy of early stopping is a regularization technique based onchoosing a stopping time for an iterative algorithm. Focusing on non-parametricregression in a reproducing kernel Hilbert space, we analyze the early stoppingstrategy for a form of gradient-descent applied to the least-squares lossfunction. We propose a data-dependent stopping rule that does not involvehold-out or cross-validation data, and we prove upper bounds on the squarederror of the resulting function estimate, measured in either the $L^2(P)$ and$L^2(P_n)$ norm. These upper bounds lead to minimax-optimal rates for variouskernel classes, including Sobolev smoothness classes and other forms ofreproducing kernel Hilbert spaces. We show through simulation that our stoppingrule compares favorably to two other stopping rules, one based on hold-out dataand the other based on Stein's unbiased risk estimate. We also establish atight connection between our early stopping strategy and the solution path of akernel ridge regression estimator.

Early stopping and non-parametric regression: An optimal data-dependent stopping rule