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

Tengyuan Liang

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arXiv · 1911.00730

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

Tengyuan Liang

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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.

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Tengyuan Liang. 2019-11-02. Estimating Certain Integral Probability Metric (IPM) is as Hard as Estimating under the IPM. https://arxiv.org/abs/1911.00730

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