Stochastic Primal-Dual Hybrid Gradient (SPDHG) is an algorithm proposed by Chambolle et al. (2018) 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, as well as for any random 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：随机原对偶混合梯度（SPDHG）是由Chambolle等人（2018）提出的一种算法，用于高效求解一类广泛的非光滑大规模优化问题。本文对其理论基础做出贡献，证明了该算法在凸泛函（既不必然强凸也不光滑）以及任意随机采样条件下几乎必然收敛。此外，我们研究了SPDHG在并行磁共振成像重建中的应用，其中每次迭代时从不同线圈中随机选择数据。我们采用广泛的随机采样方法应用SPDHG，并在多种设置（包括小批量大小和步长参数）下比较其性能。结果表明，采样方式能显著影响SPDHG的收敛速度，并且在许多情况下可以识别出最优采样策略。