Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities
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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 ≥ 1 and ϵ>0, given Õ_d((1/ϵ)^(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 ϵ-close to f_0, in squared Hellinger loss. A sample complexity lower bound of Ω_d((1/ϵ)^(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, up to an Õ(1/ϵ) factor.
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