  In big data analysis, a simple task such as linear regression can become very
challenging as the variable dimension $p$ grows. As a result, variable
screening is inevitable in many scientific studies. In recent years, randomized
algorithms have become a new trend and are playing an increasingly important
role for large scale data analysis. In this article, we combine the ideas of
variable screening and random partitioning to propose a new iterative variable
screening method. For moderate sized $p$ of order $O(n^{2-\delta})$, we propose
a basic algorithm that adopts a distribution-based thresholding rule. For very
large $p$, we further propose a two-stage procedure. This two-stage procedure
first performs a random partitioning to divide predictors into subsets of
manageable size of order $O(n^{2-\delta})$ for variable screening, where
$\delta >0$ can be an arbitrarily small positive number. Random partitioning is
repeated a few times. Next, the final estimate of variable subset is obtained
by integrating results obtained from multiple random partitions. Simulation
studies show that our method works well and outperforms some renowned
competitors. Real data applications are presented. Our algorithms are able to
handle predictors in the size of millions.
