Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples

Mohammad Afzali, Hassan Ashtiani, Christopher Liaw
Proceedings of The 35th International Conference on Algorithmic Learning Theory, PMLR 237:47-73, 2024.

Abstract

We study the problem of estimating mixtures of Gaussians under the constraint of differential privacy (DP). Our main result is that $\text{poly}(k,d,1/\alpha,1/\varepsilon,\log(1/\delta))$ samples are sufficient to estimate a mixture of $k$ Gaussians in $\mathbb{R}^d$ up to total variation distance $\alpha$ while satisfying $(\varepsilon, \delta)$-DP. This is the first finite sample complexity upper bound for the problem that does not make any structural assumptions on the GMMs. To solve the problem, we devise a new framework which may be useful for other tasks. On a high level, we show that if a class of distributions (such as Gaussians) is (1) list decodable and (2) admits a “locally small” cover (Bun et al., 2021) with respect to total variation distance, then the class of its mixtures is privately learnable. The proof circumvents a known barrier indicating that, unlike Gaussians, GMMs do not admit a locally small cover (Aden-Ali et al., 2021b).

Cite this Paper

BibTeX
@InProceedings{pmlr-v237-afzali24a, title = {Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples}, author = {Afzali, Mohammad and Ashtiani, Hassan and Liaw, Christopher}, booktitle = {Proceedings of The 35th International Conference on Algorithmic Learning Theory}, pages = {47--73}, year = {2024}, editor = {Vernade, Claire and Hsu, Daniel}, volume = {237}, series = {Proceedings of Machine Learning Research}, month = {25--28 Feb}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v237/afzali24a/afzali24a.pdf}, url = {https://proceedings.mlr.press/v237/afzali24a.html}, abstract = {We study the problem of estimating mixtures of Gaussians under the constraint of differential privacy (DP). Our main result is that $\text{poly}(k,d,1/\alpha,1/\varepsilon,\log(1/\delta))$ samples are sufficient to estimate a mixture of $k$ Gaussians in $\mathbb{R}^d$ up to total variation distance $\alpha$ while satisfying $(\varepsilon, \delta)$-DP. This is the first finite sample complexity upper bound for the problem that does not make any structural assumptions on the GMMs. To solve the problem, we devise a new framework which may be useful for other tasks. On a high level, we show that if a class of distributions (such as Gaussians) is (1) list decodable and (2) admits a “locally small” cover (Bun et al., 2021) with respect to total variation distance, then the class of its mixtures is privately learnable. The proof circumvents a known barrier indicating that, unlike Gaussians, GMMs do not admit a locally small cover (Aden-Ali et al., 2021b). } }
Endnote
%0 Conference Paper %T Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples %A Mohammad Afzali %A Hassan Ashtiani %A Christopher Liaw %B Proceedings of The 35th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2024 %E Claire Vernade %E Daniel Hsu %F pmlr-v237-afzali24a %I PMLR %P 47--73 %U https://proceedings.mlr.press/v237/afzali24a.html %V 237 %X We study the problem of estimating mixtures of Gaussians under the constraint of differential privacy (DP). Our main result is that $\text{poly}(k,d,1/\alpha,1/\varepsilon,\log(1/\delta))$ samples are sufficient to estimate a mixture of $k$ Gaussians in $\mathbb{R}^d$ up to total variation distance $\alpha$ while satisfying $(\varepsilon, \delta)$-DP. This is the first finite sample complexity upper bound for the problem that does not make any structural assumptions on the GMMs. To solve the problem, we devise a new framework which may be useful for other tasks. On a high level, we show that if a class of distributions (such as Gaussians) is (1) list decodable and (2) admits a “locally small” cover (Bun et al., 2021) with respect to total variation distance, then the class of its mixtures is privately learnable. The proof circumvents a known barrier indicating that, unlike Gaussians, GMMs do not admit a locally small cover (Aden-Ali et al., 2021b).
APA
Afzali, M., Ashtiani, H. & Liaw, C.. (2024). Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples. Proceedings of The 35th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 237:47-73 Available from https://proceedings.mlr.press/v237/afzali24a.html.

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