统一给出对数凹分布采样的近乎最优收敛速度
A unified complexity bound for logconcave sampling
- 基于指数提升与进出算法，统一分析采样复杂度
- 新推导的Poincaré常数界使收敛率近乎最优
- 适用于约束与良好条件情形，适合理论研究者
我们为从热启动开始，使用进出算法结合指数提升，对任意对数凹分布进行采样，给出了一个简洁、统一且近乎紧致的复杂度上界。分析中的关键创新是提升了被提升分布的Poincaré常数的上界。由此得到的收敛速率在受限情形（如高斯分布在凸体上）和良好条件情形（如强对数凹且光滑密度）下均近乎最优。
原文摘要 · Abstract (English)
We give a simple, unified, and nearly tight bound for sampling arbitrary logconcave distributions from a warm start using the In-and-Out algorithm along with exponential lifting. The main new ingredient in the analysis is an improved bound on the Poincaré constant of a lifted distribution. As a consequence, the resulting convergence rate is nearly tight for both constrained settings (e.g., Gaussian restricted to a convex body) and well-conditioned settings (e.g., strongly logconcave and smooth densities).
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