Estimating Certain Integral Probability Metric (IPM) is as Hard as Estimating under the IPM

In this note, we study the minimax optimal rates for estimati ng the Integral Probability Metrics (IPMs) between probability measures based on samples. IPMs are widely used in both statistics and machine learning, with applications in nonparametric t wo-sample tests [ 23; 10 ], inferring the transportation cost (the Wasserstein-1 metric) from one se t of samples to another [ 22; 20 ], and with more recent appearances in rigorous investigations on the g enerative adversarial networks (GANs) [ 1; 17; 14; 21; 25; 3 ].