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

Abstract

We study the minimax optimal rates for estimating a range of Integral Probability Metrics (IPMs) between two unknown probability measures, based on $n$ independent samples from them. Curiously, we show that estimating the IPM itself between probability measures, is not significantly easier than estimating the probability measures under the IPM. We prove that the minimax optimal rates for these two problems are multiplicatively equivalent, up to a $\log \log (n)/\log (n)$ factor.

arXiv comment15 pages. arXiv admin note: substantial text overlap with arXiv:1908.10324

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• Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional
• Nonparametric Density Estimation under Adversarial Losses
• How Well Generative Adversarial Networks Learn Distributions
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