An ordinary differential equation-based solution path algorithm

Abstract

Efron, Hastie, Johnstone, and Tibshirani [(2004), ‘Least Angle Regression (with discussions)’, The Annals of Statistics, 32, 409–499] proposed least angle regression (LAR), a solution path algorithm for the least squares regression. They pointed out that a slight modification of the LAR gives the LASSO [Tibshirani, R. (1996), ‘Regression Shrinkage and Selection Via the Lasso’, Journal of the Royal Statistical Society, Series B, 58, 267–288] solution path. However, it is largely unknown how to extend this solution path algorithm to models beyond the least squares regression. In this work, we propose an extension of the LAR for generalised linear models and the quasi-likelihood model by showing that the corresponding solution path is piecewise given by solutions of ordinary differential equation (ODE) systems. Our contribution is twofold. First, we provide a theoretical understanding on how the corresponding solution path propagates. Second, we propose an ODE-based algorithm to obtain the whole solution path.