Privately Learning Mixtures of Axis-Aligned Gaussians

We consider the problem of learning mixtures of Gaussians under the constraint of approximate differential privacy. We prove that Õ(kd log(1/δ)/αε) samples are sufficient to learn a mixture of k axis-aligned Gaussians in R to within total variation distance α while satisfying (ε, δ)-differential privacy. This is the first result for privately learning mixtures of unbounded axis-aligned (or even unbounded univariate) Gaussians. If the covariance matrices of each of the Gaussians is the identity matrix, we show that Õ(kd/α + kd log(1/δ)/αε) samples are sufficient. Recently, the “local covering” technique of Bun, Kamath, Steinke, and Wu [BKSW19] has been successfully used for privately learning high-dimensional Gaussians with a known covariance matrix and extended to privately learning general high-dimensional Gaussians by Aden-Ali, Ashtiani, and Kamath [AAK21]. Given these positive results, this approach has been proposed as a promising direction for privately learning mixtures of Gaussians. Unfortunately, we show that this is not possible. We design a new technique for privately learning mixture distributions. A class of distributions F is said to be list-decodable if there is an algorithm that, given “heavily corrupted” samples from f ∈ F , outputs a list of distributions, F̂ , such that one of the distributions in F̂ approximates f . We show that if F is privately list-decodable, then we can privately learn mixtures of distributions in F . Finally, we show axis-aligned Gaussian distributions are privately list-decodable, thereby proving mixtures of such distributions are privately learnable.