Private Statistical Estimation of Many Quantiles

Clément Lalanne, Aurélien Garivier, Rémi Gribonval
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:18399-18418, 2023.

Abstract

This work studies the estimation of many statistical quantiles under differential privacy. More precisely, given a distribution and access to i.i.d. samples from it, we study the estimation of the inverse of its cumulative distribution function (the quantile function) at specific points. For instance, this task is of key importance in private data generation. We present two different approaches. The first one consists in privately estimating the empirical quantiles of the samples and using this result as an estimator of the quantiles of the distribution. In particular, we study the statistical properties of the recently published algorithm introduced by (Kaplan et al., 2022) that privately estimates the quantiles recursively. The second approach is to use techniques of density estimation in order to uniformly estimate the quantile function on an interval. In particular, we show that there is a tradeoff between the two methods. When we want to estimate many quantiles, it is better to estimate the density rather than estimating the quantile function at specific points.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-lalanne23a, title = {Private Statistical Estimation of Many Quantiles}, author = {Lalanne, Cl\'{e}ment and Garivier, Aur\'{e}lien and Gribonval, R\'{e}mi}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {18399--18418}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/lalanne23a/lalanne23a.pdf}, url = {https://proceedings.mlr.press/v202/lalanne23a.html}, abstract = {This work studies the estimation of many statistical quantiles under differential privacy. More precisely, given a distribution and access to i.i.d. samples from it, we study the estimation of the inverse of its cumulative distribution function (the quantile function) at specific points. For instance, this task is of key importance in private data generation. We present two different approaches. The first one consists in privately estimating the empirical quantiles of the samples and using this result as an estimator of the quantiles of the distribution. In particular, we study the statistical properties of the recently published algorithm introduced by (Kaplan et al., 2022) that privately estimates the quantiles recursively. The second approach is to use techniques of density estimation in order to uniformly estimate the quantile function on an interval. In particular, we show that there is a tradeoff between the two methods. When we want to estimate many quantiles, it is better to estimate the density rather than estimating the quantile function at specific points.} }
Endnote
%0 Conference Paper %T Private Statistical Estimation of Many Quantiles %A Clément Lalanne %A Aurélien Garivier %A Rémi Gribonval %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-lalanne23a %I PMLR %P 18399--18418 %U https://proceedings.mlr.press/v202/lalanne23a.html %V 202 %X This work studies the estimation of many statistical quantiles under differential privacy. More precisely, given a distribution and access to i.i.d. samples from it, we study the estimation of the inverse of its cumulative distribution function (the quantile function) at specific points. For instance, this task is of key importance in private data generation. We present two different approaches. The first one consists in privately estimating the empirical quantiles of the samples and using this result as an estimator of the quantiles of the distribution. In particular, we study the statistical properties of the recently published algorithm introduced by (Kaplan et al., 2022) that privately estimates the quantiles recursively. The second approach is to use techniques of density estimation in order to uniformly estimate the quantile function on an interval. In particular, we show that there is a tradeoff between the two methods. When we want to estimate many quantiles, it is better to estimate the density rather than estimating the quantile function at specific points.
APA
Lalanne, C., Garivier, A. & Gribonval, R.. (2023). Private Statistical Estimation of Many Quantiles. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:18399-18418 Available from https://proceedings.mlr.press/v202/lalanne23a.html.

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