Successive Ray Refinement and Its Application to Coordinate Descent for LASSO

Coordinate descent is one of the most popular approaches for solving Lasso and its extensions due to its simplicity and efficiency. When applying coordinate descent to solving Lasso, we update one coordinate at a time while fixing the remaining coordinates. Such an update, which is usually easy to compute, greedily decreases the objective function value. In this paper, we aim to improve its computational efficiency by reducing the number of coordinate descent iterations. To this end, we propose a novel technique called Successive Ray Refinement (SRR). SRR makes use of the following ray continuation property on the successive iterations: for a particular coordinate, the value obtained in the next iteration almost always lies on a ray that starts at its previous iteration and passes through the current iteration. Motivated by this ray-continuation property, we propose that coordinate descent be performed not directly on the previous iteration but on a refined search point that has the following properties: on one hand, it lies on a ray that starts at a history solution and passes through the previous iteration, and on the other hand, it achieves the minimum objective function value among all the points on the ray. We propose two schemes for defining the search point and show that the refined search point can be efficiently obtained. Empirical results for real and synthetic data sets show that the proposed SRR can significantly reduce the number of coordinate descent iterations, especially for small Lasso regularization parameters.