Privately Learning High-Dimensional Distributions

Gautam Kamath, Jerry Li, Vikrant Singhal, Jonathan Ullman
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:1853-1902, 2019.

Abstract

We present novel, computationally efficient, and differentially private algorithms for two fundamental high-dimensional learning problems: learning a multivariate Gaussian and learning a product distribution over the Boolean hypercube in total variation distance. The sample complexity of our algorithms nearly matches the sample complexity of the optimal non-private learners for these tasks in a wide range of parameters, showing that privacy comes essentially for free for these problems. In particular, in contrast to previous approaches, our algorithm for learning Gaussians does not require strong a priori bounds on the range of the parameters. Our algorithms introduce a novel technical approach to reducing the sensitivity of the estimation procedure that we call recursive private preconditioning.

Cite this Paper

BibTeX
@InProceedings{pmlr-v99-kamath19a, title = {Privately Learning High-Dimensional Distributions}, author = {Kamath, Gautam and Li, Jerry and Singhal, Vikrant and Ullman, Jonathan}, booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, pages = {1853--1902}, year = {2019}, editor = {Beygelzimer, Alina and Hsu, Daniel}, volume = {99}, series = {Proceedings of Machine Learning Research}, month = {25--28 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v99/kamath19a/kamath19a.pdf}, url = {https://proceedings.mlr.press/v99/kamath19a.html}, abstract = {We present novel, computationally efficient, and differentially private algorithms for two fundamental high-dimensional learning problems: learning a multivariate Gaussian and learning a product distribution over the Boolean hypercube in total variation distance. The sample complexity of our algorithms nearly matches the sample complexity of the optimal non-private learners for these tasks in a wide range of parameters, showing that privacy comes essentially for free for these problems. In particular, in contrast to previous approaches, our algorithm for learning Gaussians does not require strong a priori bounds on the range of the parameters. Our algorithms introduce a novel technical approach to reducing the sensitivity of the estimation procedure that we call recursive private preconditioning.} }
Endnote
%0 Conference Paper %T Privately Learning High-Dimensional Distributions %A Gautam Kamath %A Jerry Li %A Vikrant Singhal %A Jonathan Ullman %B Proceedings of the Thirty-Second Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Alina Beygelzimer %E Daniel Hsu %F pmlr-v99-kamath19a %I PMLR %P 1853--1902 %U https://proceedings.mlr.press/v99/kamath19a.html %V 99 %X We present novel, computationally efficient, and differentially private algorithms for two fundamental high-dimensional learning problems: learning a multivariate Gaussian and learning a product distribution over the Boolean hypercube in total variation distance. The sample complexity of our algorithms nearly matches the sample complexity of the optimal non-private learners for these tasks in a wide range of parameters, showing that privacy comes essentially for free for these problems. In particular, in contrast to previous approaches, our algorithm for learning Gaussians does not require strong a priori bounds on the range of the parameters. Our algorithms introduce a novel technical approach to reducing the sensitivity of the estimation procedure that we call recursive private preconditioning.
APA
Kamath, G., Li, J., Singhal, V. & Ullman, J.. (2019). Privately Learning High-Dimensional Distributions. Proceedings of the Thirty-Second Conference on Learning Theory, in Proceedings of Machine Learning Research 99:1853-1902 Available from https://proceedings.mlr.press/v99/kamath19a.html.

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