凸集的比率覆盖与最优混合密度估计
Ratio Covers of Convex Sets and Optimal Mixture Density Estimation
AI总结：
本文研究了基于Kullback-Leibler散度的密度估计问题，提出在有限密度字典下，通过凸集覆盖理论和混合密度估计方法，获得最优的高概率保证，改进了传统密度比有界时的估计率。
摘要：
我们研究了基于Kullback-Leibler散度的密度估计问题：给定一个来自未知密度p*的i.i.d.样本，目标是构造一个估计器p^，使得KL(p*,p^)在高概率下较小。我们考虑两种基本设置，涉及有限密度字典：(i) 模型聚合，其中p*属于字典；(ii) 凸集聚合（混合密度估计），其中p*是字典中密度的混合。关键的是，我们不假设基础密度：它们的比值可能无界，其支持集可能不同。对于两种问题，我们确定了在字典大小、样本大小和置信水平方面的最佳高概率保证。这些最优速率高于当密度比有界时可达到的速率；对于混合密度估计，它们在离散分布的特殊情况下与现有下界一致。我们的混合分析依赖于两个新的覆盖结果。首先，我们提供了一个分布无关的混合密度类的局部Hellinger熵的精确上界。其次，我们证明了一个凸集的最优比率覆盖定理：对于每个凸紧集K⊆R_+^d，存在一个子集A⊆K，其元素数不超过2^{O(d)}，使得K中的每个元素都被A中的元素在坐标上无界地支配。这个几何结果具有独立价值；值得注意的是，它为具有凸可行集的多目标优化中的ε-近似帕累托集提供了新的基数估计。
英文摘要：
We study density estimation in Kullback-Leibler divergence: given an i.i.d. sample from an unknown density $p^\star$, the goal is to construct an estimator $\widehat{p}$ such that $\mathrm{KL}(p^\star,\widehat{p})$ is small with high probability. We consider two fundamental settings involving a finite dictionary of densities: (i) model aggregation, where $p^\star$ belongs to the dictionary, and (ii) convex aggregation (mixture density estimation), where $p^\star$ is a mixture of densities from the dictionary. Crucially, we make no assumption on the base densities: their ratios may be unbounded and their supports may differ. For both problems, we identify the best possible high-probability guarantees in terms of the dictionary size, sample size, and confidence level. These optimal rates are higher than those achievable when density ratios are bounded by absolute constants; for mixture density estimation, they match existing lower bounds in the special case of discrete distributions. Our analysis of the mixture case hinges on two new covering results. First, we provide a sharp, distribution-free upper bound on the local Hellinger entropy of the class of mixtures of $M$ distributions. Second, we prove an optimal ratio covering theorem for convex sets: for every convex compact set $K \subset \mathbb{R}_+^d$, there exists a subset $A \subset K$ with at most $2^{O(d)}$ elements such that each element of $K$ is coordinate-wise dominated by an element of $A$ up to a universal constant factor. This geometric result is of independent interest; notably, it yields new cardinality estimates for $\varepsilon$-approximate Pareto sets in multi-objective optimization with convex feasible set.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
