LR-GLM: High-Dimensional Bayesian Inference Using Low-Rank Data Approximations
This work addresses a computational problem for statisticians and data scientists dealing with large-scale datasets, offering a tunable trade-off between computational efficiency and statistical accuracy, though it is incremental as it builds on existing Bayesian GLM methods.
The paper tackles the computational bottleneck in Bayesian inference for high-dimensional generalized linear models (GLMs), where existing methods have cubic time complexity limiting scalability to tens of thousands of parameters, and proposes LR-GLM, a low-rank data approximation method that reduces running times by a factor of the parameter dimension while providing full Bayesian posterior approximations.
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.