Efficient MCMC Sampling with Dimension-Free Convergence Rate using ADMM-type Splitting
This work addresses the problem of unreliable MCMC sampling in high-dimensional scenarios for researchers and practitioners in statistics and machine learning, providing theoretical insights into an empirically promising method, though it is incremental as it focuses on analysis rather than introducing a new algorithm.
The paper tackles the computational intractability of exact Bayesian inference in high-dimensional models by theoretically analyzing the split Gibbs sampler, an ADMM-type MCMC method, and establishes explicit convergence rates under regularity conditions, supported by numerical illustrations.
Performing exact Bayesian inference for complex models is computationally intractable. Markov chain Monte Carlo (MCMC) algorithms can provide reliable approximations of the posterior distribution but are expensive for large datasets and high-dimensional models. A standard approach to mitigate this complexity consists in using subsampling techniques or distributing the data across a cluster. However, these approaches are typically unreliable in high-dimensional scenarios. We focus here on a recent alternative class of MCMC schemes exploiting a splitting strategy akin to the one used by the celebrated alternating direction of multipliers (ADMM) optimization algorithm. These methods appear to provide empirically state-of-the-art performance but their theoretical behavior in high dimension is currently unknown. In this paper, we propose a detailed theoretical study of one of these algorithms known as the split Gibbs sampler. Under regularity conditions, we establish explicit convergence rates for this scheme using Ricci curvature and coupling ideas. We support our theory with numerical illustrations.