Abstract: Given two probability measures, P and Q defined on a measurable space, S, the integral probability metric (IPM) is defined as where F is a class of real-valued bounded measurable functions on S. By appropriately choosing F, various popular distances between P and Q, including the Kantorovich metric, Fortet-Mourier metric, dual-bounded Lipschitz distance (also called the Dudley metric), total variation distance, and kernel distance, can be obtained. In this paper, we consider the problem of estimating γ F from finite random samples drawn i.i.d. from P and Q. Although the above mentioned distances cannot be computed in closed form for every P and Q, we show their empirical estimators to be easily computable, and strongly consistent (except for the total-variation distance). We further analyze their rates of convergence. Based on these results, we discuss the advantages of certain choices of F (and therefore the corresponding IPMs) over others-in particular, the kernel distance is shown to have three favorable properties compared with the other mentioned distances: it is computationally cheaper, On the empirical estimation of integral probability metrics 1551 the empirical estimate converges at a faster rate to the population value, and the rate of convergence is independent of the dimension d of the space (for S = R d ). We also provide a novel interpretation of IPMs and their empirical estimators by relating them to the problem of binary classification: while the IPM between class-conditional distributions is the negative of the optimal risk associated with a binary classifier, the smoothness of an appropriate binary classifier (e.g., support vector machine, Lipschitz classifier, etc.) is inversely related to the empirical estimator of the IPM between these class-conditional distributions.
REF analyze the sample complexity of evaluating integral probability metrics.
On the empirical estimation of integral probability metrics
Electronic Journal of Statistics
Mathematics