  Minimizing sum of two functions under a linear constraint is what we called
splitting problem. This convex optimization has wide applications in machine
learning problems, such as Lasso, Group Lasso and Sparse logistic regression. A
recent paper by Gu et al (2015) developed a Semi-Proximal-Based Strictly
Contractive Peaceman-Rachford Splitting Method (SPB-SPRSM), which is an
extension of Strictly Contractive Peaceman-Rachford Splitting Method (SPRSM)
proposed by He et al (2014). By introducing semi-proximal terms and using two
different relaxation factors, SPB-SPRSM showed a more flexiable applicability
comparing to its origin SPRSM and widely-used Alternating Direction Method of
Multipliers (ADMM) algorithm, although all of them have $O(1/t)$ convergence
rate. In this paper, we develop a stochastic version of SPB-SPRSM algorithm,
where only a subset of samples (even one sample) are used at each iteration.
The resulting algorithm, Stochastic SPB-SPRSM, is more flexiable than
Stochastic ADMM and other ADMM-based algorithms on both simulations and real
datasets. Moreover, we prove $O(1/\sqrt{t})$ convergence rate in ergodic sense,
which is the same with Stochastic ADMM algorithm under the same assumption. But
as shown in He et al (2014) that SPRSM based algorithms will always converge
faster than ADMM in apllication, our proposed algorithm will also preserve this
advantage.
