Pathwise coordinate optimization for sparse learning: Algorithm and theory
The Annals of Statistics · 2018 · Volume 46 · Issue 1
Abstract
The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: warm start initialization, active set updating and strong rule for coordinate preselection. Such a complex algorithmic structure grants superior empirical performance, but also poses significant challenge to theoretical analysis. To tackle this long lasting problem, we develop a new theory showing that these three features play pivotal roles in guaranteeing the outstanding statistical and computational performance of the pathwise coordinate optimization framework. Particularly, we analyze the existing pathwise coordinate optimization algorithms and provide new theoretical insights into them. The obtained insights further motivate the development of several modifications to improve the pathwise coordinate optimization framework, which guarantees linear convergence to a unique sparse local optimum with optimal statistical properties in parameter estimation and support recovery. This is the first result on the computational and statistical guarantees of the pathwise coordinate optimization framework in high dimensions. Thorough numerical experiments are provided to support our theory.
Keywords
62F30, 62J12, 90C26, 90C52, active set, global linear convergence, Nonconvex sparse learning, optimal statistical rates of convergence, oracle property, pathwise coordinate optimization, strong rule
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