Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures

We describe a general technique that yields the first em Statistical Query lower bounds for a range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning of a single unknown Gaussian distribution. For each of these problems, we show a em super-polynomial gap between the (information-theoretic) sample complexity and the computational complexity of em any Statistical Query algorithm for the problem. Our SQ lower bound for Problem (1) is qualitatively matched by known learning algorithms for GMMs. Our lower bound for Problem (2) implies that the accuracy of the robust learning algorithm in~citeDiakonikolasKKLMS16 is essentially best possible among all polynomial-time SQ algorithms. Our SQ lower bounds are attained via a unified moment-matching technique that is useful in other contexts and may be of broader interest. Our technique yields nearly-tight lower bounds for a number of related unsupervised estimation problems. Specifically, for the problems of (3) robust covariance estimation in spectral norm, and (4) robust sparse mean estimation, we establish a quadratic em statistical--computational tradeoff for SQ algorithms, matching known upper bounds. Finally, our technique can be used to obtain tight sample complexity lower bounds for high-dimensional em testing problems. Specifically, for the classical problem of robustly em testing an unknown mean (known covariance) Gaussian, our technique implies an information-theoretic sample lower bound that scales em linearly in the dimension. Our sample lower bound matches the sample complexity of the corresponding robust em learning problem and separates the sample complexity of robust testing from standard (non-robust) testing.