Early stopping of iterative algorithms is an algorithmic regularization method to avoid over-fitting in estimation and classification. In this paper, we show that early stopping can also be applied to obtain the minimax optimal testing in a general non-parametric setup. Specifically, a Wald-type test statistic is obtained based on an iterated estimate produced by functional gradient descent algorithms in a reproducing kernel Hilbert space. A notable contribution is to establish a ``sharp'' stopping rule: when the number of iterations achieves an optimal order, testing optimality is achievable; otherwise, testing optimality becomes impossible. As a by-product, a similar sharpness result is also derived for minimax optimal estimation under early stopping. All obtained results hold for various kernel classes, including Sobolev smoothness classes and Gaussian kernel classes.

Summary: This is a very interesting paper that provides an alternative approach to nonparametric testing. Instead of imposing a penalty to encourage bias-variance tradeoff, the paper proposed to do an "early stopping" to achieve that. The reason why this approach would work is that when applying a gradient ascent/descent approach, even the optimal solution will overfit the data, on the way to the optimal, there is a sweet spot where the variability has been removed a lot while the bias is still not too large (have not yet overfit the data). Overall, this is a nice and well-written paper and its idea is worth spreading in the community of machine learning and statistics. In Theorem 3.1 and soon after it, there is a rule for testing H0 using the asymptotic distribution.

Early stopping of iterative algorithms is an algorithmic regularization method to avoid over-fitting in estimation and classification. In this paper, we show that early stopping can also be applied to obtain the minimax optimal testing in a general non-parametric setup. Specifically, a Wald-type test statistic is obtained based on an iterated estimate produced by functional gradient descent algorithms in a reproducing kernel Hilbert space. A notable contribution is to establish a sharp'' stopping rule: when the number of iterations achieves an optimal order, testing optimality is achievable; otherwise, testing optimality becomes impossible. As a by-product, a similar sharpness result is also derived for minimax optimal estimation under early stopping. All obtained results hold for various kernel classes, including Sobolev smoothness classes and Gaussian kernel classes.

In the presence of weak overall correlation, it may be useful to investigate if the correlation is significantly and substantially more pronounced over a subpopulation. Two different testing procedures are compared. Both are based on the rankings of the values of two variables from a data set with a large number n of observations. The first maintains its level against Gaussian copulas; the second adapts to general alternatives in the sense that that the number of parameters used in the test grows with n . An analysis of wine quality illustrates how the methods detect heterogeneity of association between chemical properties of the wine, which are attributable to a mix of different cultivars.