Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities

We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on R, for all d ≥ 4. Prior to this work, no finite sample upper bound was known for this estimator in more than 3 dimensions. In more detail, we prove that for any d ≥ 1 and > 0, given Õd((1/ )) samples drawn from an unknown log-concave density f0 on R, the MLE outputs a hypothesis h that with high probability is -close to f0, in squared Hellinger loss. A sample complexity lower bound of Ωd((1/ )) was previously known for any learning algorithm that achieves this guarantee. We thus establish that the sample complexity of the log-concave MLE is near-optimal, up to an Õ(1/ ) factor. ∗Supported by NSF Award CCF-1652862 (CAREER) and a Sloan Research Fellowship. †Supported by NSF Award CCF-1453472 (CAREER) and NSF grant CCF-1423230. ar X iv :1 80 2. 10 57 5v 1 [ m at h. ST ] 2 8 Fe b 20 18