Polynomial time guarantees for sampling based posterior inference in high-dimensional generalised linear models

TL;DR

This paper establishes polynomial-time guarantees for a gradient-based MCMC method in high-dimensional generalized linear models, enabling efficient posterior inference even with non-log-concave likelihoods.

Contribution

It introduces a novel proof approach that relies solely on local likelihood conditions, providing nonasymptotic guarantees without M-estimation theory.

Findings

Guarantees scale polynomially with key algorithmic quantities.

Applicable to density estimation, nonparametric regression, and inverse problems from PDEs.

Demonstrates practical efficiency in high-dimensional Bayesian inference.

Abstract

The problem of computing posterior functionals in general high-dimensional statistical models with possibly non-log-concave likelihood functions is considered. Based on the proof strategy of Nickl and Wang (2022), but using only local likelihood conditions and without relying on M-estimation theory, nonasymptotic statistical and computational guarantees are provided for a gradient based MCMC algorithm. Given a suitable initialiser, these guarantees scale polynomially in key algorithmic quantities. The abstract results are applied to several concrete statistical models, including density estimation, nonparametric regression with generalised linear models and a canonical statistical non-linear inverse problem from PDEs.