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

2019/11/02 by Tengyuan Liang, Liang, Tengyuan · 5 citations
Computer Science · Mathematics · #Bayesian Modeling and Causal Inference #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Machine Learning (stat.ML) #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1911.00730

openalex publication_date 2019/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

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