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.
翻译:我们考虑的是使用联合分布的 i.d. 样本来估计一个条件密度的任务,这是在分类和不确定性的回归量化方面的应用的一个根本问题。对于联合密度估计,对于一般密度类别来说,最小值率的特征是统一的(计量) 英特罗比,这是一个经过仔细研究的统计能力概念。在将这些结果应用于有条件密度估计时,使用统一的英特罗比 -- -- 当共变空间不受限制并受到维度的诅咒时,它将是无限的 -- -- 可能导致不优化率。因此,使用这些古典结果不能说明有条件密度估计的迷你速率。我们解决了这个问题,在定义明确的模型中,在最小密度类别(对数)的(对数)英特罗比值(对数)上下对一般密度类别进行匹配(对数),这是对统计能力进行充分研究的一个概念概念概念概念概念。在将这些结果应用于有条件密度类别时,使用统一的英特罗比可以让我们根据当地Rademacher 复杂程度来进行集中论论,这与统一的英特罗比- 导致大型、潜在对大型、潜在非偏差等级分类的精确度的精确度的精确度的比要求。