Sparse online variational Bayesian regression

TL;DR

This paper introduces a scalable variational Bayesian method for sparse linear regression that efficiently handles large datasets and promotes sparsity, offering comparable performance to Bayesian LASSO at a fraction of the computational cost.

Contribution

The paper presents a novel online variational Bayesian approach for sparse regression with scale mixture priors, enabling efficient handling of large-scale data and promoting sparsity more strongly.

Findings

Method achieves comparable variable selection and uncertainty quantification to Bayesian LASSO.

Handles datasets with over 100,000 samples and millions of features within reasonable time.

Provides an efficient online implementation with linear or near-linear complexity.

Abstract

This work considers variational Bayesian inference as an inexpensive and scalable alternative to a fully Bayesian approach in the context of sparsity-promoting priors. In particular, the priors considered arise from scale mixtures of Normal distributions with a generalized inverse Gaussian mixing distribution. This includes the variational Bayesian LASSO as an inexpensive and scalable alternative to the Bayesian LASSO introduced in [65]. It also includes a family of priors which more strongly promote sparsity. For linear models the method requires only the iterative solution of deterministic least squares problems. Furthermore, for p unknown covariates the method can be implemented exactly online with a cost of $O(p^3)$ in computation and $O(p^2)$ in memory per iteration -- in other words, the cost per iteration is independent of n, and in principle infinite data can be considered. For large $p$ an approximation is able to achieve promising results for a cost of $O(p)$ per iteration, in both computation and memory. Strategies for hyper-parameter tuning are also considered. The method is implemented for real and simulated data. It is shown that the performance in terms of variable selection and uncertainty quantification of the variational Bayesian LASSO can be comparable to the Bayesian LASSO for problems which are tractable with that method, and for a fraction of the cost. The present method comfortably handles $n = 65536$, $p = 131073$ on a laptop in less than 30 minutes, and $n = 10^5$, $p = 2.1 \times 10^6$ overnight.