LR-GLM: High-Dimensional Bayesian Inference Using Low-Rank Data Approximations

TL;DR

LR-GLM introduces a low-rank data approximation method for high-dimensional Bayesian inference in generalized linear models, significantly reducing computational costs while maintaining approximation quality, enabling analysis of large-scale datasets.

Contribution

The paper presents LR-GLM, a novel low-rank approximation approach that accelerates Bayesian inference in high-dimensional GLMs, with theoretical guarantees and practical effectiveness.

Findings

Reduces inference time by a factor proportional to the parameter dimension.

Provides rigorous bounds on approximation quality.

Demonstrates effectiveness on large real-world datasets.

Abstract

Due to the ease of modern data collection, applied statisticians often have access to a large set of covariates that they wish to relate to some observed outcome. Generalized linear models (GLMs) offer a particularly interpretable framework for such an analysis. In these high-dimensional problems, the number of covariates is often large relative to the number of observations, so we face non-trivial inferential uncertainty; a Bayesian approach allows coherent quantification of this uncertainty. Unfortunately, existing methods for Bayesian inference in GLMs require running times roughly cubic in parameter dimension, and so are limited to settings with at most tens of thousand parameters. We propose to reduce time and memory costs with a low-rank approximation of the data in an approach we call LR-GLM. When used with the Laplace approximation or Markov chain Monte Carlo, LR-GLM provides a full Bayesian posterior approximation and admits running times reduced by a full factor of the parameter dimension. We rigorously establish the quality of our approximation and show how the choice of rank allows a tunable computational-statistical trade-off. Experiments support our theory and demonstrate the efficacy of LR-GLM on real large-scale datasets.