Unified algorithms for distributed regularized linear regression model

In recent years, distributed statistical models have received increasing attention for large-scale data analysis. On the one hand, data sets come from multiple data sources, and are stored in different locations due to limited bandwidth and storage, or privacy protocols, directly centralizing all data together is impossible. On the other hand, the size of data is so large that it is difficult or inefficient to analyze data together. There are two main research aspects to using distributed statistical models to analyze large-scale data. The first one is to study the statistical convergence rate under some mild assumptions. The second one is to establish fast and efficient optimization algorithms considering the property of the loss function. There is a lot of research on the first aspect, but relatively little research on the second one. Motivated by this, we consider the construction of unified algorithms for distributed linear regression with different losses and regularizers. As a result, we designed two type methods, proximal alternating direction method of multipliers (pADMM) and distributed accelerated proximal gradient method with line-search (DAPGL). In order to demonstrate the efficiency of the proposed algorithms, we perform numerical experiments on the distributed Huber-Lasso model and Huber-Group-Lasso model. In view of the numerical results, we can observe that these two algorithms are more competitive than some of state-of-art algorithms. In particular, DAPGL algorithm performs better than pADMM in most cases.