|
Abstract:
|
We consider the task of estimating a conditional density using i.i.d. samples from the joint distribution. Existing minimax rates for general classes rely on the (metric) entropy of the class of joint densities, which is a fundamental and well-studied notion of statistical capacity. However, applying these results to estimating conditional densities can be arbitrarily suboptimal due to their dependence on uniform entropy, which is infinite when the covariate space is unbounded and suffers from the curse of dimensionality. In this work, we resolve this problem for well-specified models, obtaining matching upper and lower bounds on the minimax KL risk in terms of the empirical Hellinger entropy of the conditional density class. In contrast to uniform entropy, empirical entropy provides the correct dependence on the size of the covariate space. We only require that the conditional densities are bounded above, but not that they are bounded below or otherwise satisfy any tail conditions.
|
