Stochastic Primal-Dual Hybrid Gradient (SPDHG) is an algorithm to efficiently solve a wide class of nonsmooth large-scale optimization problems. In this paper we contribute to its theoretical foundations and prove its almost sure convergence for convex but neither necessarily strongly convex nor smooth functionals. We also prove its convergence for any sampling. In addition, we study SPDHG for parallel Magnetic Resonance Imaging reconstruction, where data from different coils are randomly selected at each iteration. We apply SPDHG using a wide range of random sampling methods and compare its performance across a range of settings, including mini-batch size and step size parameters. We show that the sampling can significantly affect the convergence speed of SPDHG and for many cases an optimal sampling can be identified.

翻译：随机原始-对偶混合梯度算法（SPDHG）是一类高效求解广泛非光滑大规模优化问题的算法。本文对其理论基础作出贡献，证明该算法在凸但既非强凸也非光滑泛函条件下的几乎必然收敛性，同时证明其对任意采样方式的收敛性。此外，我们研究了SPDHG在并行磁共振成像重建中的应用——每次迭代随机选取不同线圈的数据。我们采用多种随机采样方法应用SPDHG，并在小批量尺寸、步长参数等多种设置下比较其性能。研究表明，采样方式可显著影响SPDHG的收敛速度，且在多数情况下可确定最优采样策略。