Parallelizing MCMC Across the Sequence Length

2025-08-25

AI Summary

To address the low sampling efficiency of Markov Chain Monte Carlo (MCMC) methods caused by inherent sequential dependencies in long chains, this paper proposes the first general parallel-in-sequence MCMC framework. The core idea is to reformulate the state sequence as a fixed-point problem governed by a nonlinear recurrence, then solve it via parallel fixed-point iteration combined with quasi-Newton methods—thereby decoupling temporal ordering constraints across time steps. To this end, we design two memory-efficient parallel quasi-Newton algorithms, enabling parallelization of diverse samplers including Gibbs, Metropolis-Adjusted Langevin Algorithm (MALA), and Hamiltonian Monte Carlo (HMC). Experiments demonstrate speedups of tens to over one hundred times across multiple tasks: tens of parallel iterations suffice to generate hundreds of thousands of samples, achieving more than an order-of-magnitude improvement in overall performance and significantly overcoming the scalability bottleneck of conventional MCMC.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

Markov chain Monte Carlo (MCMC) methods are foundational algorithms for Bayesian inference and probabilistic modeling. However, most MCMC algorithms are inherently sequential and their time complexity scales linearly with the sequence length. Previous work on adapting MCMC to modern hardware has therefore focused on running many independent chains in parallel. Here, we take an alternative approach: we propose algorithms to evaluate MCMC samplers in parallel across the chain length. To do this, we build on recent methods for parallel evaluation of nonlinear recursions that formulate the state sequence as a solution to a fixed-point problem and solve for the fixed-point using a parallel form of Newton's method. We show how this approach can be used to parallelize Gibbs, Metropolis-adjusted Langevin, and Hamiltonian Monte Carlo sampling across the sequence length. In several examples, we demonstrate the simulation of up to hundreds of thousands of MCMC samples with only tens of parallel Newton iterations. Additionally, we develop two new parallel quasi-Newton methods to evaluate nonlinear recursions with lower memory costs and reduced runtime. We find that the proposed parallel algorithms accelerate MCMC sampling across multiple examples, in some cases by more than an order of magnitude compared to sequential evaluation.

Problem

• Parallelizing MCMC sampling across sequence length
• Reducing linear time complexity of sequential MCMC
• Developing parallel Newton methods for nonlinear recursions

Innovation

• Parallel Newton's method for MCMC sampling
• Fixed-point formulation for sequence parallelism
• Quasi-Newton methods reducing memory and runtime