Generalized Isotonic Regression

TL;DR

This paper introduces a unified recursive algorithm for isotonic regression applicable to any convex differentiable loss function, extending previous methods and demonstrating effectiveness on various data types.

Contribution

It develops a general recursive partitioning algorithm for isotonic regression under convex differentiable loss functions, unifying and extending prior approaches.

Findings

Efficiently solves isotonic regression for any convex differentiable loss.

Demonstrates improved fitting on count data with Poisson loss.

Shows robustness in isotonic regression with Huber's loss.

Abstract

We present a computational and statistical approach for fitting isotonic models under convex differentiable loss functions. We offer a recursive partitioning algorithm which provably and efficiently solves isotonic regression under any such loss function. Models along the partitioning path are also isotonic and can be viewed as regularized solutions to the problem. Our approach generalizes and subsumes two previous results: the well-known work of Barlow and Brunk (1972) on fitting isotonic regressions subject to specially structured loss functions, and a recursive partitioning algorithm (Spouge et al 2003) for the case of standard (l2-loss) isotonic regression. We demonstrate the advantages of our generalized algorithm on both real and simulated data in two settings: fitting count data using negative Poisson log-likelihood loss, and fitting robust isotonic regression using Huber's loss.