The log-concave maximum likelihood estimator (MLE) problem answers: for a set of points @math , which log-concave density maximizes their likelihood? We present a characterization of the log-concave MLE that leads to an algorithm with runtime @math to compute a log-concave distribution whose log-likelihood is at most @math less than that of the MLE, and @math is parameter of the problem that is bounded by the @math norm of the vector of log-likelihoods the MLE evaluated at @math .
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 @math , for all @math . Prior to this work, no finite sample upper bound was known for this estimator in more than @math dimensions. In more detail, we prove that for any @math and @math , given @math samples drawn from an unknown log-concave density @math on @math , the MLE outputs a hypothesis @math that with high probability is @math -close to @math , in squared Hellinger loss. A sample complexity lower bound of @math 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 @math factor. The research of Richard J. Samworth was supported by an EPSRC Early Career Fellowship and a grant from the Leverhulme Trust. We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on @math , for all @math . Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of @math . In more detail, we give an estimator that, for any @math and @math , draws @math samples from an unknown target log-concave density on @math , and outputs a hypothesis that (with high probability) is @math -close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of @math for this problem.