Learning Multivariate Log-concave Distributions

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 $\mathbb{R}^d$, 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 $d>3$. In more detail, we give an estimator that, for any $d \ge 1$ and $\epsilon>0$, draws $\tilde{O}d \left( (1/\epsilon)^{(d+5)/2} \right)$ samples from an unknown target log-concave density on $\mathbb{R}^d$, 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 $\Omegad \left( (1/\epsilon)^{(d+1)/2} \right)$ for this problem.