  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.
