改进高维采样偏差分析，揭示稀疏交互下偏差的局部化特性
A hierarchical entropy method for the delocalization of bias in high-dimensional Langevin Monte Carlo
- 基于分层熵分析，突破原有对强对数凸性的依赖
- 在相对熵下消除对数因子，证明低维边缘偏差仅与局部维度相关
- 拓展至弱相互作用分布，适用于更广泛的高维采样场景
无调整Langevin算法广泛用于复杂高维分布的采样，但存在偏差，通常在平方Wasserstein距离下随维度线性增长。然而Chen等（2024）发现：对于具有稀疏相互作用的一类分布，其低维边缘的偏差仅与局部维度相关，而非全维。本文在稀疏交互条件下强化该结果：移除了对数因子，改用相对熵（即KL散度）衡量距离，并放宽了强对数凸性假设。此外，我们进一步将该局域化现象推广至弱相互作用分布。证明方法基于对边缘相对熵的分层分析，受作者近期混沌传播研究启发。
原文摘要 · Abstract (English)
The unadjusted Langevin algorithm is widely used for sampling from complex high-dimensional distributions. It is well known to be biased, with the bias typically scaling linearly with the dimension when measured in squared Wasserstein distance. However, the recent paper of Chen et al. (2024) identifies an intriguing new delocalization effect: For a class of distributions with sparse interactions, the bias between low-dimensional marginals scales only with the lower dimension, not the full dimension. In this work, we strengthen the results of Chen et al. (2024) in the sparse interaction regime by removing a logarithmic factor, measuring distance in relative entropy (a.k.a. KL-divergence), and relaxing the strong log-concavity assumption. In addition, we expand the scope of the delocalization phenomenon by showing that it holds for a class of distributions with weak interactions. Our proofs are based on a hierarchical analysis of the marginal relative entropies, inspired by the authors' recent work on propagation of chaos.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。