Online and stochastic Douglas-Rachford splitting method for large scale machine learning

TL;DR

This paper extends the Douglas-Rachford splitting method to online and stochastic settings for large-scale machine learning, providing theoretical guarantees and demonstrating effectiveness through experiments.

Contribution

First to generalize Douglas-Rachford splitting to online and stochastic frameworks with proven regret bounds and convergence rates.

Findings

Online DRs has an $O(1)$ regret bound.

Stochastic DRs converges at an $O(1/\sqrt{T})$ rate.

Numerical experiments confirm theoretical results.

Abstract

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.