Fast and Flexible ADMM Algorithms for Trend Filtering

TL;DR

This paper introduces a fast, robust, and flexible ADMM algorithm for trend filtering, improving scalability and numerical stability, and extending to related problems with available software implementations.

Contribution

It develops a highly efficient ADMM routine for trend filtering that surpasses existing methods in robustness and flexibility, enabling broader practical use.

Findings

Competitive with interior point methods in speed

More numerically stable than existing algorithms

Extensible to sparse and isotonic trend filtering

Abstract

This paper presents a fast and robust algorithm for trend filtering, a recently developed nonparametric regression tool. It has been shown that, for estimating functions whose derivatives are of bounded variation, trend filtering achieves the minimax optimal error rate, while other popular methods like smoothing splines and kernels do not. Standing in the way of a more widespread practical adoption, however, is a lack of scalable and numerically stable algorithms for fitting trend filtering estimates. This paper presents a highly efficient, specialized ADMM routine for trend filtering. Our algorithm is competitive with the specialized interior point methods that are currently in use, and yet is far more numerically robust. Furthermore, the proposed ADMM implementation is very simple, and importantly, it is flexible enough to extend to many interesting related problems, such as sparse trend filtering and isotonic trend filtering. Software for our method is freely available, in both the C and R languages.