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