TY - JOUR
T1 - Approximate primal-dual fixed-point based langevin algorithms for non-smooth convex potentials
AU - Cai, Ziruo
AU - Li, Jinglai
AU - Zhang, Xiaoqun
PY - 2024/3/4
Y1 - 2024/3/4
N2 - 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.
AB - 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.
KW - Bayesian inference
KW - Langevin alorithms
KW - non-smooth convex potentials
KW - proximity operators
UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-85189244733&partnerID=MN8TOARS
U2 - 10.4310/CMS.2024.v22.n3.a3
DO - 10.4310/CMS.2024.v22.n3.a3
M3 - Article
SN - 1539-6746
VL - 22
SP - 655
EP - 684
JO - Communications in Mathematical Sciences
JF - Communications in Mathematical Sciences
IS - 3
ER -