CPDHG: A gradient-based stochastic method for efficiently solving deblurring convex inverse problems

Abstract

We present a novel Composite PDHG method for addressing a class of deterministic and stochastic saddle-point problems (SPP), in particular designed for efficiently fixing large-scale, non-smooth convex inverse problems. The basic idea of this algorithm lies in its integration of gradient updates with proximal operators, permitting it to handle the smooth component of the objective function effectively, a capability that distinguishes it from the Chambolle–Pock algorithm, which is tailored exclusively for non-smooth functions.By leveraging a strategy that updates a randomly selected subset of dual variables in each iteration, the Stochastic Composite PDHG (SCPDHG) considerably reduces computational overhead while maintaining convergence stability. We establish its almost-sure convergence through foundational lemmas inspired by the framework of the Three-Operator Splitting with Stochastic Primal-Dual Hybrid Gradient (TOS-SPDHG). Additionally, we exhibit the practical utility of SCPDHG inside the context of image deblurring, illustrating its robustness and adaptability across complex data environments. This work highlights the capacity of SCPDHG to enhance performance and scalability in challenging real-world applications.