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

We study the problem of estimating mixtures of Gaussians under the constraint of differential privacy (DP). Our main result is that $\tilde{O}(k^2 d^4 \log(1/\delta) / \alpha^2 \varepsilon)$ samples are sufficient to estimate a mixture of $k$ Gaussians 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 [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].