证明了近端采样器在强对数凹分布下指数收敛，媲美连续朗之万动力学。
Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality
- 基于强数据压缩不等式，构建正向与逆向高斯信道的组合机制。
- 在强对数凹且光滑分布下，相对Fisher信息以指数速度收敛。
- 适合关注采样算法理论保证的优化与概率学习研究者。
本文研究近端采样器在相对Fisher信息下的混合时间保证，该算法是朗之万动力学的近似近端离散化。当目标分布为强对数凹时，相对Fisher信息沿近端采样器呈指数快速收敛，其速率与连续时间朗之万动力学一致。结合标准拒绝采样实现方式，该指数收敛率在目标分布强对数凹且对数光滑时，给出了近端采样器在相对Fisher信息下的高精度迭代复杂度保证。证明过程通过建立强对数凹条件下相对Fisher信息在高斯信道上的强数据处理不等式，以及针对特定分布在逆高斯信道上的数据处理不等式完成。正向与逆向高斯信道共同构成近端采样器，二者不等式共同导出相对Fisher信息的指数收敛性。
原文摘要 · Abstract (English)
We study the mixing time guarantee for sampling in relative Fisher information via the Proximal Sampler algorithm, which is an approximate proximal discretization of the Langevin dynamics. We show that when the target probability distribution is strongly log-concave, the relative Fisher information converges exponentially fast along the Proximal Sampler; this matches the exponential convergence rate of the relative Fisher information along the continuous-time Langevin dynamics for strongly log-concave target. When combined with a standard implementation of the Proximal Sampler via rejection sampling, this exponential convergence rate provides a high-accuracy iteration complexity guarantee for the Proximal Sampler in relative Fisher information when the target distribution is strongly log-concave and log-smooth. Our proof proceeds by establishing a strong data processing inequality for relative Fisher information along the Gaussian channel under strong log-concavity, and a data processing inequality along the reverse Gaussian channel for a special distribution. The forward and reverse Gaussian channels compose to form the Proximal Sampler, and these data processing inequalities imply the exponential convergence rate of the relative Fisher information along the Proximal Sampler.
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