Asymptotically Exact, Embarrassingly Parallel MCMC

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

Global consensus Monte Carlo

To conduct Bayesian inference with large data sets, it is often convenient or necessary to distribute the data across multiple machines. We consider a likelihood function expressed as a product of terms, each associated with a subset of the data. Inspired by global variable consensus optimisation, we introduce an instrumental hierarchical model associating auxiliary statistical parameters with each term, which are conditionally independent given the top-level parameters. One of these top-level parameters controls the unconditional strength of association between the auxiliary parameters. This model leads to a distributed MCMC algorithm on an extended state space yielding approximations of posterior expectations. A trade-off between computational tractability and fidelity to the original model can be controlled by changing the association strength in the instrumental model. We further propose the use of a SMC sampler with a sequence of association strengths, allowing both the automatic determination of appropriate strengths and for a bias correction technique to be applied. In contrast to similar distributed Monte Carlo algorithms, this approach requires few distributional assumptions. The performance of the algorithms is illustrated with a number of simulated examples

Distributed Bayesian Matrix Factorization with Limited Communication

Bayesian matrix factorization (BMF) is a powerful tool for producing low-rank representations of matrices and for predicting missing values and providing confidence intervals. Scaling up the posterior inference for massive-scale matrices is challenging and requires distributing both data and computation over many workers, making communication the main computational bottleneck. Embarrassingly parallel inference would remove the communication needed, by using completely independent computations on different data subsets, but it suffers from the inherent unidentifiability of BMF solutions. We introduce a hierarchical decomposition of the joint posterior distribution, which couples the subset inferences, allowing for embarrassingly parallel computations in a sequence of at most three stages. Using an efficient approximate implementation, we show improvements empirically on both real and simulated data. Our distributed approach is able to achieve a speed-up of almost an order of magnitude over the full posterior, with a negligible effect on predictive accuracy. Our method outperforms state-of-the-art embarrassingly parallel MCMC methods in accuracy, and achieves results competitive to other available distributed and parallel implementations of BMF.Comment: 28 pages, 8 figures. The paper is published in Machine Learning journal. An implementation of the method is is available in SMURFF software on github (bmfpp branch): https://github.com/ExaScience/smurf

Patterns of Scalable Bayesian Inference

Datasets are growing not just in size but in complexity, creating a demand for rich models and quantification of uncertainty. Bayesian methods are an excellent fit for this demand, but scaling Bayesian inference is a challenge. In response to this challenge, there has been considerable recent work based on varying assumptions about model structure, underlying computational resources, and the importance of asymptotic correctness. As a result, there is a zoo of ideas with few clear overarching principles. In this paper, we seek to identify unifying principles, patterns, and intuitions for scaling Bayesian inference. We review existing work on utilizing modern computing resources with both MCMC and variational approximation techniques. From this taxonomy of ideas, we characterize the general principles that have proven successful for designing scalable inference procedures and comment on the path forward

Non-Reversible Parallel Tempering: a Scalable Highly Parallel MCMC Scheme

Parallel tempering (PT) methods are a popular class of Markov chain Monte Carlo schemes used to sample complex high-dimensional probability distributions. They rely on a collection of $N$ interacting auxiliary chains targeting tempered versions of the target distribution to improve the exploration of the state-space. We provide here a new perspective on these highly parallel algorithms and their tuning by identifying and formalizing a sharp divide in the behaviour and performance of reversible versus non-reversible PT schemes. We show theoretically and empirically that a class of non-reversible PT methods dominates its reversible counterparts and identify distinct scaling limits for the non-reversible and reversible schemes, the former being a piecewise-deterministic Markov process and the latter a diffusion. These results are exploited to identify the optimal annealing schedule for non-reversible PT and to develop an iterative scheme approximating this schedule. We provide a wide range of numerical examples supporting our theoretical and methodological contributions. The proposed methodology is applicable to sample from a distribution $\pi$ with a density $L$ with respect to a reference distribution $\pi_0$ and compute the normalizing constant. A typical use case is when $\pi_0$ is a prior distribution, $L$ a likelihood function and $\pi$ the corresponding posterior.Comment: 74 pages, 30 figures. The method is implemented in an open source probabilistic programming available at https://github.com/UBC-Stat-ML/blangSD