Communications in Mathematical Sciences

Volume 22 (2024)

Number 3

Approximate primal-dual fixed-point based langevin algorithms for non-smooth convex potentials

Pages: 655 – 684

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n3.a3

Authors

Ziruo Cai (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

Jinglai Li (School of Mathematics, University of Birmingham, United Kingdom)

Xiaoqun Zhang (School of Mathematical Sciences, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China)

Abstract

The Langevin algorithms are frequently used to sample the posterior distributions in Bayesian inference. In many practical problems, however, the posterior distributions often consist of non-differentiable components, posing challenges for the standard Langevin algorithms, as they require to evaluate the gradient of the energy function in each iteration. To this end, a popular remedy is to utilize the proximity operator, and as a result one needs to solve a proximity subproblem in each iteration. The conventional practice is to solve the subproblems accurately, which can be exceedingly expensive, as the subproblem needs to be solved in each iteration. We propose an approximate primal-dual fixed-point algorithm for solving the subproblem, which only seeks an approximate solution of the subproblem and therefore reduces the computational cost considerably. We provide theoretical analysis of the proposed method and also demonstrate its performance with numerical examples.

Keywords

Bayesian inference, Langevin alorithms, non-smooth convex potentials, proximity operators

2010 Mathematics Subject Classification

62F15, 65C05, 68U10

Full Text (PDF format)

Received 6 November 2022

Received revised 10 April 2023

Accepted 16 August 2023

Published 4 March 2024