PASS-GLM: polynomial approximate sufficient statistics for scalable Bayesian GLM inference

TL;DR

PASS-GLM introduces polynomial approximate sufficient statistics enabling scalable Bayesian inference for GLMs, with theoretical guarantees and competitive empirical performance on large datasets.

Contribution

The paper presents PASS-GLM, a novel polynomial approximation method for scalable Bayesian inference in GLMs with theoretical error bounds and simple distributed algorithms.

Findings

Achieves scalable Bayesian inference on large datasets with 40 million points.

Provides theoretical guarantees on the quality of estimates and uncertainty.

Performs competitively with existing methods like MCMC and SGD.

Abstract

Generalized linear models (GLMs) -- such as logistic regression, Poisson regression, and robust regression -- provide interpretable models for diverse data types. Probabilistic approaches, particularly Bayesian ones, allow coherent estimates of uncertainty, incorporation of prior information, and sharing of power across experiments via hierarchical models. In practice, however, the approximate Bayesian methods necessary for inference have either failed to scale to large data sets or failed to provide theoretical guarantees on the quality of inference. We propose a new approach based on constructing polynomial approximate sufficient statistics for GLMs (PASS-GLM). We demonstrate that our method admits a simple algorithm as well as trivial streaming and distributed extensions that do not compound error across computations. We provide theoretical guarantees on the quality of point (MAP) estimates, the approximate posterior, and posterior mean and uncertainty estimates. We validate our approach empirically in the case of logistic regression using a quadratic approximation and show competitive performance with stochastic gradient descent, MCMC, and the Laplace approximation in terms of speed and multiple measures of accuracy -- including on an advertising data set with 40 million data points and 20,000 covariates.