Agnostic Distribution Learning via Compression

We prove that Θ̃(kd2/ε2) samples are necessary and sufficient for learning a mixture of k Gaussians in Rd, up to error ε in total variation distance. This improves both the known upper bound and lower bound for this problem. For mixtures of axis-aligned Gaussians, we show that Õ(kd/ε2) samples suffice, matching a known lower bound. Moreover, these results hold in an agnostic learning setting as well. The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in Rd has an efficient sample compression.