Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality
This work provides a high-accuracy iteration complexity guarantee for the Proximal Sampler, which is significant for researchers and practitioners working with strongly log-concave and log-smooth target distributions.
The authors tackled the problem of mixing time guarantee for the Proximal Sampler algorithm and achieved an exponential convergence rate for strongly log-concave target probability distributions. This matches the convergence rate of the continuous-time Langevin dynamics.
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.