Partial identification of the maximum mean discrepancy with mismeasured data

Ron Nafshi, Maggie Makar
Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence, PMLR 244:2623-2645, 2024.

Abstract

Nonparametric estimates of the distance between two distributions such as the Maximum Mean Discrepancy (MMD) are often used in machine learning applications. However, the majority of existing literature assumes that error-free samples from the two distributions of interest are available.We relax this assumption and study the estimation of the MMD under $\epsilon$-contamination, where a possibly non-random $\epsilon$ proportion of one distribution is erroneously grouped with the other. We show that under $\epsilon$-contamination, the typical estimate of the MMD is unreliable. Instead, we study partial identification of the MMD, and characterize sharp upper and lower bounds that contain the true, unknown MMD. We propose a method to estimate these bounds, and show that it gives estimates that converge to the sharpest possible bounds on the MMD as sample size increases, with a convergence rate that is faster than alternative approaches. Using three datasets, we empirically validate that our approach is superior to the alternatives: it gives tight bounds with a low false coverage rate.

Cite this Paper

BibTeX
@InProceedings{pmlr-v244-nafshi24a, title = {Partial identification of the maximum mean discrepancy with mismeasured data}, author = {Nafshi, Ron and Makar, Maggie}, booktitle = {Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence}, pages = {2623--2645}, year = {2024}, editor = {Kiyavash, Negar and Mooij, Joris M.}, volume = {244}, series = {Proceedings of Machine Learning Research}, month = {15--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v244/main/assets/nafshi24a/nafshi24a.pdf}, url = {https://proceedings.mlr.press/v244/nafshi24a.html}, abstract = {Nonparametric estimates of the distance between two distributions such as the Maximum Mean Discrepancy (MMD) are often used in machine learning applications. However, the majority of existing literature assumes that error-free samples from the two distributions of interest are available.We relax this assumption and study the estimation of the MMD under $\epsilon$-contamination, where a possibly non-random $\epsilon$ proportion of one distribution is erroneously grouped with the other. We show that under $\epsilon$-contamination, the typical estimate of the MMD is unreliable. Instead, we study partial identification of the MMD, and characterize sharp upper and lower bounds that contain the true, unknown MMD. We propose a method to estimate these bounds, and show that it gives estimates that converge to the sharpest possible bounds on the MMD as sample size increases, with a convergence rate that is faster than alternative approaches. Using three datasets, we empirically validate that our approach is superior to the alternatives: it gives tight bounds with a low false coverage rate.} }
Endnote
%0 Conference Paper %T Partial identification of the maximum mean discrepancy with mismeasured data %A Ron Nafshi %A Maggie Makar %B Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2024 %E Negar Kiyavash %E Joris M. Mooij %F pmlr-v244-nafshi24a %I PMLR %P 2623--2645 %U https://proceedings.mlr.press/v244/nafshi24a.html %V 244 %X Nonparametric estimates of the distance between two distributions such as the Maximum Mean Discrepancy (MMD) are often used in machine learning applications. However, the majority of existing literature assumes that error-free samples from the two distributions of interest are available.We relax this assumption and study the estimation of the MMD under $\epsilon$-contamination, where a possibly non-random $\epsilon$ proportion of one distribution is erroneously grouped with the other. We show that under $\epsilon$-contamination, the typical estimate of the MMD is unreliable. Instead, we study partial identification of the MMD, and characterize sharp upper and lower bounds that contain the true, unknown MMD. We propose a method to estimate these bounds, and show that it gives estimates that converge to the sharpest possible bounds on the MMD as sample size increases, with a convergence rate that is faster than alternative approaches. Using three datasets, we empirically validate that our approach is superior to the alternatives: it gives tight bounds with a low false coverage rate.
APA
Nafshi, R. & Makar, M.. (2024). Partial identification of the maximum mean discrepancy with mismeasured data. Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 244:2623-2645 Available from https://proceedings.mlr.press/v244/nafshi24a.html.

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