Coordinate descent for variance-component models

Variance-component models are an indispensable tool for statisticians wanting to capture both random and fixed model effects. They have applications in a wide range of scientific disciplines. While maximum likelihood estimation (MLE) is the most popular method for estimating the variance-component model parameters, it is numerically challenging for large data sets. In this article, we consider the class of coordinate descent (CD) algorithms for computing the MLE. We show that a basic implementation of coordinate descent is numerically costly to implement and does not easily satisfy the standard theoretical conditions for convergence. We instead propose two parameter-expanded versions of CD, called PX-CD and PXI-CD. These novel algorithms not only converge faster than existing competitors (MM and EM algorithms) but are also more amenable to convergence analysis. PX-CD and PXI-CD are particularly well-suited for large data sets—namely, as the scale of the model increases, the performance gap between the parameter-expanded CD algorithms and the current competitor methods increases.