Firefly Monte Carlo: Exact MCMC with Subsets of Data (1403.5693v1)

Abstract: Markov chain Monte Carlo (MCMC) is a popular and successful general-purpose tool for Bayesian inference. However, MCMC cannot be practically applied to large data sets because of the prohibitive cost of evaluating every likelihood term at every iteration. Here we present Firefly Monte Carlo (FlyMC) an auxiliary variable MCMC algorithm that only queries the likelihoods of a potentially small subset of the data at each iteration yet simulates from the exact posterior distribution, in contrast to recent proposals that are approximate even in the asymptotic limit. FlyMC is compatible with a wide variety of modern MCMC algorithms, and only requires a lower bound on the per-datum likelihood factors. In experiments, we find that FlyMC generates samples from the posterior more than an order of magnitude faster than regular MCMC, opening up MCMC methods to larger datasets than were previously considered feasible.

• The paper introduces the FlyMC algorithm that uses auxiliary variables to reduce full-data likelihood evaluations while preserving exact posterior distributions.
• It details a method utilizing tight lower bounds and Bernoulli variables to bypass unnecessary computations, yielding over an order of magnitude speed improvement.
• The study demonstrates practical applications in areas like logistic regression and softmax classification, highlighting its potential across diverse data-intensive domains.

Overview of Firefly Monte Carlo: Exact MCMC with Subsets of Data

The paper "Firefly Monte Carlo: Exact MCMC with Subsets of Data" by Dougal Maclaurin and Ryan P. Adams elaborates on a novel approach to Markov chain Monte Carlo (MCMC) methodology designed to accommodate large datasets in Bayesian inference without incurring prohibitive computational costs associated with traditional methods. The Firefly Monte Carlo (FlyMC) algorithm innovates by leveraging subsets of data through an auxiliary variable representation, yet still maintains consistency with the exact posterior distribution.

In the field of Bayesian inference, MCMC methods offer significant robustness and flexibility, enabling practitioners to deal with models where closed-form solutions to posterior distributions are elusive. However, the expansive datasets typical of current applications introduce computational challenges, largely tied to the requirement of evaluating each likelihood term for every iteration of MCMC. FlyMC addresses this challenge by introducing an auxiliary variable mechanism that effectively sidesteps the need to consume the entirety of the dataset at each iteration. This mechanism employs Bernoulli variables associated with each datum, selectively activating data points necessary to evolve the posterior distribution without approximation errors prevalent in other subsets-based approaches.

FlyMC's operational foundations rest on the introduction of lower bounds on likelihood terms, $B_n(\theta)$, minimizing the computational burden. The paper describes a strategy for choosing tight lower bounds to efficiently evaluate the posterior given subsets of data, and the authors notably achieve sampling speeds "more than an order of magnitude faster" than conventional MCMC. The introduction of these auxiliary variables does not perturb the marginal distributions but rather optimizes the sampling process by reducing computationally expensive likelihood evaluations in large datasets.

The implications of this research are profound for applications across various scientific domains where Bayesian inference is utilized. These include, but are not limited to, fields such as genomics, image recognition, and natural language processing, where data-intensive computations are inherently involved. By enabling faster and scalable MCMC sampling, FlyMC opens Bayesian methods to datasets previously considered impractical due to computational constraints. Although the paper portrays a selection of experimental evaluations demonstrating FlyMC's efficacy, particularly in logistic regression, softmax classification, and robust regression scenarios, it acknowledges the necessity for careful selection of the auxiliary variable bounds tailored to each specific application.

Looking forward, FlyMC establishes a baseline for further exploration into auxiliary variable techniques in MCMC, inviting exploration of broader bounds and pseudo-marginal approaches to enhance efficiency without sacrificing accuracy in posterior evaluations. This integration of theoretical developments with practical improvements presents a promising avenue for AI and machine learning models requiring substantial data-intensive analyses.

In conclusion, Firefly Monte Carlo exemplifies an advancement in the MCMC domain, expanding its applicability to larger datasets, while maintaining rigorous adherence to Bayesian principles. The results showcased in this paper are pivotal for encouraging widespread adoption of FlyMC in practice, stimulating further research to harness the benefits of auxiliary variables within various MCMC frameworks possible in AI solutions moving forward.