LRGLM: HighDimensional Bayesian Inference Using LowRank Data Approximations
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:63156324, 2019.
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 highdimensional problems, the number of covariates is often large relative to the number of observations, so we face nontrivial 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 lowrank approximation of the data in an approach we call LRGLM. When used with the Laplace approximation or Markov chain Monte Carlo, LRGLM 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 computationalâ€“statistical tradeoff. Experiments support our theory and demonstrate the efficacy of LRGLM on real largescale datasets.
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