Learning Multivariate Log-concave Distributions

Abstract: 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 $\mathbb{R}<sup>d$, for all $d \geq 1$. Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of $d&gt;3$. In more detail, we give an estimator that, for any $d \ge 1$ and $\epsilon&gt;0$, draws $$\tilde{O}_d \left( (1/\epsilon)^{(d+5)/2} \right)$$ samples from an unknown target log-concave density on $\mathbb{R}<sup>d$, and outputs a hypothesis that (with high probability) is $\epsilon$-close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of $$\Omega_d \left( (1/\epsilon)^{(d+1)/2} \right)$$ for this problem.