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.
Accelerated Primal-Dual Proximal Gradient Splitting Methods for Convex-Concave Saddle-Point Problems
National Center for Applied Mathematics in Chongqing, Chongqing Normal University（重庆师范大学应用数学中心）；Chongqing Research Institute of Big Data, Peking University（北京大学重庆大数据研究院）
A primal-dual algorithm for image reconstruction with input-convex neural network regularizers
Department of Mathematical Sciences, University of Bath, UK（英国巴斯大学数学科学系）；Department of Electronics & Electrical Communication Engineering, Indian Institute of Technology (IIT), Kharagpur, India（印度理工学院Kharagpur分校电子与电气通信工程系）
Stochastic Primal-Dual Hybrid Gradient Algorithm with Arbitrary Sampling and Imaging Applications
CMAP, Ecole Polytechnique, CNRS（应用数学中心，巴黎综合理工学院，法国国家科学研究中心）；Department for Applied Mathematics and Theoretical Physics, University of Cambridge（应用数学与理论物理系，剑桥大学）；Visual Computing Center & Extreme Computing Research Center, KAUST（视觉计算中心与极端计算研究中心，阿卜杜拉国王科技大学）；School of Mathematics, University of Edinburgh（数学学院，爱丁堡大学）；The Alan Turing Institute（阿兰·图灵研究所）
