One-Step Sampling via MCMC Distillation

Markov chain Monte Carlo (MCMC) enables sampling from unnormalized distributions, but obtaining well-mixed samples often requires many sequential transitions. We study MCMC distillation, which amortizes these iterative refinements into an implicit neural sampler that generates samples in a single forward pass. Starting from the current generator, we apply short MCMC refinements and train the generator to match the resulting distribution using sample-based discrepancies, without requiring target samples or evaluating the generator density. We establish a unified weak convergence framework based on a finite-window error decomposition that separates reference mixing, accumulated distillation error, and transition bias. We investigate sliced Wasserstein distance, MMD with Riesz and Matérn kernel, and Sinkhorn divergence as potential sample-based divergences in this theoretical framework. For unadjusted Langevin algorithm (ULA) teachers, we derive explicit sufficient conditions for weak convergence to the target and conditional rates for expectation errors. Experiments on synthetic and high-dimensional Bayesian inference tasks achieves comparable results to ULA teachers, demonstrating the effectiveness of repeated distillation across different sample-based discrepancies.