A Class of Dimension-free Metrics for the Convergence of Empirical Measures

Abstract: This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of probability metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is critical for high-dimensional analysis and stands in contrast to classical metrics ({\it e.g.}, the Wasserstein metric). The proposed metrics fall into the category of integral probability metrics, for which we specify criteria of test function spaces to guarantee the property of being free of CoD. Examples of the selected test function spaces include the reproducing kernel Hilbert spaces, Barron space, and flow-induced function spaces. Three applications of the proposed metrics are presented: 1. The convergence of empirical measure in the case of random variables; 2. The convergence of $n$-particle system to the solution to McKean-Vlasov stochastic differential equation; 3. The construction of an $\varepsilon$-Nash equilibrium for a homogeneous $n$-player game by its mean-field limit. As a byproduct, we prove that, given a distribution close to the target distribution measured by our metric and a certain representation of the target distribution, we can generate a distribution close to the target one in terms of the Wasserstein metric and relative entropy. Overall, we show that the proposed class of metrics is a powerful tool to analyze the convergence of empirical measures in high dimensions without CoD.