Regularization aims to improve prediction performance of a given statistical modeling approach by moving to a second approach which achieves worse training error but is expected to have fewer degrees of freedom, i.e., better agreement between trainin
g and prediction error. We show here, however, that this expected behavior does not hold in general. In fact, counter examples are given that show regularization can increase the degrees of freedom in simple situations, including lasso and ridge regression, which are the most common regularization approaches in use. In such situations, the regularization increases both training error and degrees of freedom, and is thus inherently without merit. On the other hand, two important regularization scenarios are described where the expected reduction in degrees of freedom is indeed guaranteed: (a) all symmetric linear smoothers, and (b) linear regression versus convex constrained linear regression (as in the constrained variant of ridge regression and lasso).
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 func
tion. 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 Hubers loss.
Isotonic regression is a nonparametric approach for fitting monotonic models to data that has been widely studied from both theoretical and practical perspectives. However, this approach encounters computational and statistical overfitting issues in
higher dimensions. To address both concerns, we present an algorithm, which we term Isotonic Recursive Partitioning (IRP), for isotonic regression based on recursively partitioning the covariate space through solution of progressively smaller best cut subproblems. This creates a regularized sequence of isotonic models of increasing model complexity that converges to the global isotonic regression solution. The models along the sequence are often more accurate than the unregularized isotonic regression model because of the complexity control they offer. We quantify this complexity control through estimation of degrees of freedom along the path. Success of the regularized models in prediction and IRPs favorable computational properties are demonstrated through a series of simulated and real data experiments. We discuss application of IRP to the problem of searching for gene--gene interactions and epistasis, and demonstrate it on data from genome-wide association studies of three common diseases.
