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Asymptotically Exact, Embarrassingly Parallel MCMC
Willie Neiswanger, Chong Wang, Eric Xing
TL;DR
MCMC can be costly because large datasets require extensive computation, burn-in, and distributed coordination. The paper partitions data across machines, samples subposteriors independently with classical MCMC, and combines the outputs; it provides theoretical guarantees and empirical evidence for faster burn-in and asymptotically correct sampling.
Problem
MCMC sampling can require O(N) operations per sample, lengthy burn-in, and substantial communication when data are distributed across machines.
Method
The algorithm runs MCMC independently on data subsets and combines the resulting subposterior samples to estimate their product, which is proportional to the full-data posterior.
Results
The paper provides theoretical guarantees for asymptotically correct full-posterior samples and reports empirical speed-ups for burn-in and sampling.
Takeaways & Limitations
The procedure supports distributed data settings with little intercommunication and is particularly suitable for MapReduce.
Takeaways & Limitations
The method currently applies primarily to real, unconstrained posterior samples and does not yet extend to infinite-dimensional models or distributions over the simplex.
Abstract
from arXiv · showhide
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.
1. Introduction
MCMC offers asymptotically exact posterior sampling but can be slow because computation scales with dataset size, burn-in can be lengthy, and distributed data creates communication challenges. The paper proposes independently sampling subset posteriors and combining them to accelerate burn-in and sampling while preserving asymptotic exactness.
- Motivation: MCMC guarantees asymptotically exact posterior recovery, but drawing samples commonly requires O(N) operations and potentially many burn-in steps.These costs become difficult when data must be stored and processed across multiple machines.
- Existing approaches: Parallel chains still process the entire dataset and do not accelerate burn-in because every chain must complete full burn-in.This approach speeds sampling through multiple chains but leaves the burn-in workload unchanged.
- Existing approaches: Distributed MCMC methods can process data subsets but require substantial inter-machine communication at each iteration.Waiting for external information can greatly increase computation time.
- Proposed approach: The proposed procedure runs MCMC independently on each data subset and then combines the resulting samples to construct full-posterior samples.It targets quicker burn-in and sampling when data are partitioned across machines.
- Design criteria: The method requires each machine to access only part of the data and communicate only during combination, while supporting any MCMC method and provably asymptotically exact samples.The combination stage is presented as a post-processing step compatible with existing MCMC software and MapReduce.
- Contributions: The paper introduces subposteriors, develops combination methods, proves theoretical guarantees, and reports empirical speed-ups for burn-in and sampling.These contributions are organized around applying MCMC independently to subsets and combining the outputs.
2. Embarrassingly Parallel MCMC
The algorithm partitions data into subsets, samples subposteriors independently in parallel, and combines those samples to approximate the product of subposteriors. Because that product is proportional to the full-data posterior, communication is confined to the combination stage.
- Algorithm: The method partitions N i.i.d. data points into M subsets and samples each subset’s subposterior in parallel.The subposterior uses the data subset with an underweighted prior.
- Algorithm: Each machine samples from its subposterior independently, without communication during local MCMC.The procedure permits different classical MCMC methods to be used on the separate machines.
- Combination: The combination stage estimates the product p1···pM of subposterior densities to produce full-posterior samples.The product is proportional to p(θ|xN).
- Communication: Only the combination stage requires communication, and the local sampling and combination steps need not be iterated.Sampling each subposterior can typically use the same procedure as sampling from the full-data posterior.
3. Combining Subposterior Samples
The paper combines samples from independently estimated subposterior densities to approximate or asymptotically recover the full posterior. It presents parametric, nonparametric, and semiparametric estimators with different accuracy and convergence properties.
- The combination procedure estimates the product of subposterior densities and samples from that estimate to approach the full posterior.Consistency of the density-product estimator yields asymptotically exact samples, while estimator error determines convergence toward the true posterior.
- Parametric estimator: The parametric estimator uses Gaussian approximations of subposteriors, offering quick convergence but potentially producing asymptotically biased samples.The bias is especially relevant when the posterior is strongly non-Gaussian.
- Nonparametric estimator: The nonparametric estimator uses Gaussian-kernel density estimates whose product defines a mixture of T^M Gaussian components with unnormalized weights.The estimator is asymptotically exact but may converge slowly in high dimensions.
- Nonparametric estimator: An independent Metropolis within Gibbs sampler efficiently selects mixture components by proposing one subposterior index at a time and accepting according to mixture weights.This avoids enumerating all T^M components when generating samples.
- Nonparametric estimator: The nonparametric combination can mix slowly when its acceptance rate is low, motivating repeated combination over groups of subposteriors.Pairwise application reduces the computational burden while addressing slow mixing.
- Semiparametric estimator: The semiparametric estimator combines a Gaussian approximation with a nonparametric correction, retaining near-Gaussian behavior for few samples while converging to the true density.Its resulting procedure is asymptotically exact and balances the earlier estimators’ convergence and accuracy properties.
4. Method Complexity
The combination stage has quadratic dependence on the number of machines in its direct form, while pairwise repetition reduces this dependence. Communication can be performed online with unidirectional sample transfers.
- O(dTM^2) operations are required to produce T samples in d dimensions from M data subsets using the direct nonparametric or semiparametric procedures.The pairwise variation instead requires O(dTM) operations.
- O(dTM) scalars are communicated, consisting of T samples from each of M machines.The transfer can occur online as MCMC runs.
- Online combination lets parallel MCMC and combination proceed concurrently while transferring only one sample at a time.Communication is unidirectional, and machines do not pause waiting for information during parallel sampling.
5. Theoretical Results
The paper establishes that nonparametric and semiparametric density-product estimators become asymptotically correct as subposterior sample size increases. Consistency follows from bias and variance bounds, with the mean-squared error controlled under stated smoothness and boundedness assumptions.
- The nonparametric and semiparametric procedures generate asymptotically exact full-posterior samples as T, the number of samples per subposterior, grows.The paper proves estimator correctness and bounds its convergence rate.
- The analysis assumes samples lie in a bounded subset of R^d, uses bandwidth h, and requires bounded subposterior densities.The bandwidth is annealed to zero as T approaches infinity.
- The estimator bias tends to zero as the bandwidth shrinks.The bias bound uses a term proportional to h^β under the stated smoothness conditions.
- The estimator variance tends to zero as the sample count grows and the bandwidth shrinks.The variance analysis uses bounds involving T and h^d.
- The mean-squared error equals variance plus squared bias, so the estimator is consistent.Theorem 5.3 specifies h ≍ T^-1/(2β+d) for the stated MSE bound.
6. Method Scope
The theoretical results and algorithms apply to finite-dimensional real-valued posterior distributions, including several common model classes and both unimodal and multimodal posteriors. They do not yet extend to infinite-dimensional models or distributions over the simplex.
- The method covers finite-dimensional posterior distributions over real spaces, including generalized linear, mixture, hierarchical, and graphical models.The scope includes unconstrained variables.
- The method applies to both unimodal and multimodal posterior densities.A multimodal example is identified in Section 8.2.
- The theory does not yet extend to infinite-dimensional models or distributions over the simplex.Examples include nonparametric Bayesian models and topics in latent Dirichlet allocation.
7. Related Work
Prior parallel MCMC methods often target specific models or retain synchronization requirements, whereas this approach is intended for more general settings with communication only during final combination. Consensus Monte Carlo is the closest comparison but differs in prior reweighting, sample combination, and theoretical guarantees.
- Earlier parallel MCMC methods target specific models, including topic models and nonparametric mixture models.These approaches still require synchronization to be correct or approximately correct.
- The proposed approach targets more general model settings and requires no synchronization until the final combination stage.This contrasts with parallel methods that synchronize during computation.
- Consensus Monte Carlo also partitions data and runs MCMC independently across machines, giving roughly the same time complexity.The comparison identifies it as the most relevant prior work.
- Unlike Consensus Monte Carlo, the proposed method explicitly reweights the prior during sampling and combines subposterior samples without simply averaging them.The paper also states that Consensus Monte Carlo has few theoretical guarantees.
- Consensus Monte Carlo serves as one of the paper's experimental baselines.
8. Empirical Study
Experiments across logistic regression, real-world classification, and Gaussian mixtures show that density-product combinations accelerate convergence while preserving posterior fidelity better than averaging or pooling.
- Experimental setup: The experiments compare single full-data chains, duplicate full-data chains, three density-product estimators, subposterior averaging, and pooling across Bayesian models.Subposterior samples were generated independently on worker machines and transferred for final combination.
- Logistic regression: The parametric density product overlaps the true logistic-regression posterior more closely than subposterior averaging, whose error increases as M grows.The comparison uses M=10 and M=20 data subsets.
- Logistic regression: The method achieves faster convergence than duplicate full-data chains because partitioning lets each sampler take cheaper steps and parallelizes burn-in.Each machine uses only a fraction of the data, unlike duplicate full-data chains.
- Computational cost: All batch combination procedures complete in under twenty seconds on a 2.5GHz Intel Core i5 with 16GB memory.The timing refers to a Matlab implementation of the combination algorithms.
- Logistic regression: Density-product combinations reduce posterior error faster than a single full-data chain, while subposterior averaging and pooling produce biased solutions.This pattern is reported for logistic regression and the timing comparisons.
- Gaussian mixture models: In Gaussian mixtures, asymptotically exact combinations converge quickly to low posterior error, whereas subpostAvg and subpostPool perform worse and some procedures miss multimodality.The posterior samples represent a two-dimensional marginal for one mean parameter.
9. Discussion and Future Work
The paper presents embarrassingly parallel MCMC with theoretical guarantees and empirical speedups, while identifying a current scope boundary for posterior samples.
- Discussion: The algorithm provides theoretical guarantees for samples combined from independently run subposterior MCMC chains.The combination stage follows parallel sampling with little intermachine communication.
- Discussion: Experiments demonstrate faster burn-in and asymptotically correct sampling in settings where data are partitioned across machines with little intercommunication.The authors identify MapReduce as an especially suitable setting.
- Future work: The current algorithm works primarily when posterior samples are real, unconstrained values, with extensions to more general settings left for future work.This is the paper's stated scope boundary.
- Empirical evidence: Figure 5 reports posterior error versus time separately for Gaussian mixture and Poisson-gamma hierarchical models.The Gaussian mixture results are shown on the left and the Poisson-gamma results on the right.