  Online and stochastic learning has emerged as powerful tool in large scale
optimization. In this work, we generalize the Douglas-Rachford splitting (DRs)
method for minimizing composite functions to online and stochastic settings (to
our best knowledge this is the first time DRs been generalized to sequential
version). We first establish an $O(1/\sqrt{T})$ regret bound for batch DRs
method. Then we proved that the online DRs splitting method enjoy an $O(1)$
regret bound and stochastic DRs splitting has a convergence rate of
$O(1/\sqrt{T})$. The proof is simple and intuitive, and the results and
technique can be served as a initiate for the research on the large scale
machine learning employ the DRs method. Numerical experiments of the proposed
method demonstrate the effectiveness of the online and stochastic update rule,
and further confirm our regret and convergence analysis.
