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 d , 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 ≥ 4 and > 0, givenÕ d ((1/ ) (d+3)/2 ) samples drawn from an unknown log-concave density f 0 on R d , the MLE outputs a hypothesis h that with high probability is -close to f 0 , in squared Hellinger loss. For any d ≥ 2, a sample complexity lower bound of Ω d ((1/ ) (d+1)/2 ) 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 for d ≥ 4, up to anÕ(1/ ) factor.
In recent work, Carpenter et al. REF showed a sample complexity upper bound ofÕ d (1/ǫ) (d+3)/2 for the multivariate logconcave MLE with respect to squared Hellinger loss, thus obtaining the first finite sample complexity upper bound for this estimator in dimension d ≥ 4.
Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities
COLT 2018 proceedings version
Mathematics Computer Science