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(k²d log³/²(1/δ)/α²ε) samples are sufficient to learn a mixture of k axis-aligned Gaussians in R^d 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 …