Minimax rates for conditional density estimation via empirical entropy

We consider the task of estimating a conditional density using i.i.d. samples from joint distribution, which is fundamental problem with applications in both classification and uncertainty quantification for regression. For estimation, minimax rates have been characterized general classes terms uniform (metric) entropy, well-studied notion statistical capacity. When applying these results to use entropy -- infinite when covariate space unbounded suffers curse dimensionality can lead suboptimal rates. Consequently, estimation cannot be classical results. resolve this well-specified models, obtaining matching (within logarithmic factors) upper lower bounds on Kullback--Leibler risk empirical Hellinger class. The allows us appeal concentration arguments based local Rademacher complexity, contrast leads large, potentially nonparametric captures correct dependence complexity space. Our require only that densities are bounded above, do not they below or otherwise satisfy any tail conditions.