Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples
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We study the problem of estimating mixtures of Gaussians under the constraint of differential privacy (DP). Our main result is that Õ(k^2 d^4 log(1/δ) / α^2 ε) samples are sufficient to estimate a mixture of k Gaussians up to total variation distance α while satisfying (ε, δ)-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 [BKSW19] 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 [AAL21].
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