在有限 f-散度下的分区函数估计
Partition Function Estimation under Bounded f-Divergence
AI总结：
在有限 f-散度下，研究了分区函数估计的样本复杂性，通过整合覆盖轮廓提出新方法，统一了重要性采样等方法的分析，并扩展到厚尾区域。
摘要：
我们研究了在给定提案分布和目标分布的未归一化密度比的样本访问下，估计分区函数的统计复杂性。尽管分区函数估计是一个经典问题，但现有保证通常依赖于关于领域或模型几何结构的结构性假设。我们提供了一种通用的信息论特征，其仅依赖于提案分布和目标分布之间的关系。我们的分析引入了整合覆盖轮廓，这是一个功能，量化了目标质量在密度比大的区域中的分布情况。我们展示了整合覆盖紧密刻画了乘法分区函数估计的样本复杂性，并提供了匹配的下界。我们进一步将这些界限用 f-散度表达，从而得到依赖于 f 增长率的精确相变，恢复经典结果作为特殊情况，同时扩展到厚尾区域。匹配的下界在所有区域中建立了紧致性。作为应用，我们推导了重要性采样和自归一化重要性采样的改进有限样本保证，并展示了在相同散度约束下近似采样和计数的复杂性之间的严格分离。我们的结果统一并扩展了重要性采样、拒绝采样和厚尾均值估计的先前分析，提供了一个最小假设的分区函数估计理论。在过程中我们引入了新的技术工具，包括覆盖和 f-散度之间的新联系以及经典 Paley-Zygmund 不等式的推广。
英文摘要：
We study the statistical complexity of estimating partition functions given sample access to a proposal distribution and an unnormalized density ratio for a target distribution. While partition function estimation is a classical problem, existing guarantees typically rely on structural assumptions about the domain or model geometry. We instead provide a general, information-theoretic characterization that depends only on the relationship between the proposal and target distributions. Our analysis introduces the integrated coverage profile, a functional that quantifies how much target mass lies in regions where the density ratio is large. We show that integrated coverage tightly characterizes the sample complexity of multiplicative partition function estimation and provide matching lower bounds. We further express these bounds in terms of $f$-divergences, yielding sharp phase transitions depending on the growth rate of f and recovering classical results as a special case while extending to heavy-tailed regimes. Matching lower bounds establish tightness in all regimes. As applications, we derive improved finite-sample guarantees for importance sampling and self-normalized importance sampling, and we show a strict separation between the complexity of approximate sampling and counting under the same divergence constraints. Our results unify and generalize prior analyses of importance sampling, rejection sampling, and heavy-tailed mean estimation, providing a minimal-assumption theory of partition function estimation. Along the way we introduce new technical tools including new connections between coverage and $f$-divergences as well as a generalization of the classical Paley-Zygmund inequality.
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