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.

翻译：Langevin算法常被用于贝叶斯推断中的后验分布采样。然而在许多实际问题中，后验分布往往包含不可微分量，这给标准Langevin算法带来挑战，因为其在每次迭代中需要评估能量函数的梯度。为此，一种常见的解决方法是利用邻近算子，但需要每次迭代求解一个邻近子问题。传统做法是精确求解该子问题，由于迭代过程中需要反复求解，计算开销极大。我们提出一种近似原对偶不动点算法来求解该子问题，该方法仅需要子问题的近似解，从而大幅降低计算成本。我们提供了所提方法的理论分析，并通过数值算例验证了其性能。