Asymptotically Exact, Embarrassingly Parallel MCMC (1311.4780v2)

Abstract: Communication costs, resulting from synchronization requirements during learning, can greatly slow down many parallel machine learning algorithms. In this paper, we present a parallel Markov chain Monte Carlo (MCMC) algorithm in which subsets of data are processed independently, with very little communication. First, we arbitrarily partition data onto multiple machines. Then, on each machine, any classical MCMC method (e.g., Gibbs sampling) may be used to draw samples from a posterior distribution given the data subset. Finally, the samples from each machine are combined to form samples from the full posterior. This embarrassingly parallel algorithm allows each machine to act independently on a subset of the data (without communication) until the final combination stage. We prove that our algorithm generates asymptotically exact samples and empirically demonstrate its ability to parallelize burn-in and sampling in several models.

• The paper introduces an embarrassingly parallel MCMC method that uses subposterior sampling to combine independent chain outputs into asymptotically exact full-data inference.
• The approach allows any traditional MCMC technique to be employed on partitioned data, significantly reducing computational burden and communication overhead.
• Empirical results demonstrate notable speed-ups and robust performance across logistic regression and Gaussian mixture models, even for challenging multimodal distributions.

Asymptotically Exact, Embarrassingly Parallel MCMC

The paper by Neiswanger, Wang, and Xing presents a novel methodology for conducting Markov chain Monte Carlo (MCMC) simulations in a parallel and communication-efficient manner. The central aim of the research is to address the challenges posed by large-scale data and the prohibitive computational cost of traditional MCMC methods. The authors propose an embarrassingly parallel approach that permits simultaneous MCMC sampling across multiple machines without the need for substantial inter-machine communication, thus significantly mitigating synchronization delays.

This methodology exploits the concept of subposterior sampling, where the data is partitioned randomly across different machines, and each machine independently performs MCMC sampling on its data subset. The samples from each machine are subsequently combined to yield asymptotically exact samples from the full posterior distribution, as articulated through rigorous theoretical guarantees.

Core Contributions

1. Subposterior Sampling and Combination: The paper introduces the notion of subposterior density, which forms the cornerstone of their embarrassingly parallel MCMC method. Each machine draws samples independently from a conditioned posterior given its data partition, and these subposteriors are combined to approximate the full-data posterior.
2. Flexibility in MCMC Implementation: The approach allows each machine to employ any traditional MCMC technique, enhancing its portability and ease of integration with existing software solutions. This makes the methodology amenable to various modeling scenarios typically encountered in Bayesian inference.
3. Asymptotic Exactness: Theoretical results are provided that ensure the combined samples achieve asymptotic correctness relative to the full-data posterior, as the sample size increases. This is akin to the convergence properties of traditional MCMC, but with the advantage of reduced computational burdens.
4. Efficiency and Communication Minimization: The structure ensures that the only point of inter-machine communication is during the final combination phase, effectively eliminating the bottleneck typically associated with parallel Bayesian sampling methods.
5. Derivation of Multiple Estimators: The authors develop parametric, nonparametric, and semiparametric estimators for the combination of subposterior samples, providing a spectrum of options based on the underlying distributional assumptions and the dimensionality of the problem at hand.

Empirical Results and Implications

Empirically, the paper demonstrates the efficacy of their method across several models, including logistic regression and Gaussian mixtures. The method shows notable speed-ups in both burn-in and sampling phases compared to conventional methods, without sacrificing sampling accuracy. Particularly interesting is the paper's exploration of its method's performance on multimodal and hierarchical models, where traditional approaches might struggle with mixing.

Theoretical and Practical Implications

The method expands the theoretical understanding of MCMC in distributed systems, showing that communication limitations can be effectively managed without compromising the integrity and robustness of the sampling process. From a practical standpoint, this research has implications for the deployment of Bayesian methods in large-scale data contexts, particularly in distributed computing environments such as cloud infrastructures employing MapReduce frameworks.

Speculation on Future Developments

Future research may explore extensions of this framework to more complex distributions, such as infinite-dimensional models, or those with constrained variables like those found in latent Dirichlet allocation. Furthermore, advancements can be expected in optimizing the accuracy-speed trade-offs inherent in the choice between parametric, nonparametric, or semiparametric combinations.

Overall, this paper contributes an innovative angle on parallel MCMC that balances computational efficiency with theoretical rigor, offering substantial promise for applications in big data and distributed machine learning.