Estimating Certain Integral Probability Metrics (IPMs) Is as Hard as Estimating under the IPMs

Tengyuan Liang ()
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Tengyuan Liang: University of Chicago - Booth School of Business

No 2020-153, Working Papers from Becker Friedman Institute for Research In Economics

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.

Pages: 15 pages
Date: 2020
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