Asynchronous and Distributed Data Augmentation for Massive Data Settings

TL;DR

This paper introduces an asynchronous, distributed data augmentation framework that significantly speeds up Bayesian inference in massive data settings by updating only a fraction of data subsets per iteration, maintaining accuracy and ergodicity.

Contribution

It develops a novel ADDA framework that extends traditional DA algorithms to distributed, asynchronous environments with theoretical guarantees and practical speed improvements.

Findings

ADDA achieves significant speed-up over traditional DA in massive data scenarios.

The Markov chain of ADDA is Harris ergodic, ensuring convergence to the correct distribution.

ADDA is proven to be geometrically ergodic, allowing for valid asymptotic inference.

Abstract

Data augmentation (DA) algorithms are widely used for Bayesian inference due to their simplicity. In massive data settings, however, DA algorithms are prohibitively slow because they pass through the full data in any iteration, imposing serious restrictions on their usage despite the advantages. Addressing this problem, we develop a framework for extending any DA that exploits asynchronous and distributed computing. The extended DA algorithm is indexed by a parameter $r \in (0, 1)$ and is called Asynchronous and Distributed (AD) DA with the original DA as its parent. Any ADDA starts by dividing the full data into $k$ smaller disjoint subsets and storing them on $k$ processes, which could be machines or processors. Every iteration of ADDA augments only an $r$-fraction of the $k$ data subsets with some positive probability and leaves the remaining $(1-r)$-fraction of the augmented data unchanged. The parameter draws are obtained using the $r$-fraction of new and $(1-r)$-fraction of old augmented data. For many choices of $k$ and $r$, the fractional updates of ADDA lead to a significant speed-up over the parent DA in massive data settings, and it reduces to the distributed version of its parent DA when $r=1$. We show that the ADDA Markov chain is Harris ergodic with the desired stationary distribution under mild conditions on the parent DA algorithm. We demonstrate the numerical advantages of the ADDA in three representative examples corresponding to different kinds of massive data settings encountered in applications. In all these examples, our DA generalization is significantly faster than its parent DA algorithm for all the choices of $k$ and $r$. We also establish geometric ergodicity of the ADDA Markov chain for all three examples, which in turn yields asymptotically valid standard errors for estimates of desired posterior quantities.