On the Consistency of Ordinal Regression Methods

TL;DR

This paper characterizes Fisher consistency for various ordinal regression surrogate loss functions, derives excess risk bounds, and introduces a novel surrogate that outperforms existing methods on multiple datasets.

Contribution

It provides a comprehensive Fisher consistency analysis for ordinal regression surrogates, introduces a new surrogate for squared error, and empirically demonstrates its effectiveness.

Findings

Fisher consistency characterized by the derivative at zero for a broad family of surrogates.

Derived excess risk bounds for surrogate absolute error.

New surrogate for squared error outperforms least squares on most datasets.

Abstract

Many of the ordinal regression models that have been proposed in the literature can be seen as methods that minimize a convex surrogate of the zero-one, absolute, or squared loss functions. A key property that allows to study the statistical implications of such approximations is that of Fisher consistency. Fisher consistency is a desirable property for surrogate loss functions and implies that in the population setting, i.e., if the probability distribution that generates the data were available, then optimization of the surrogate would yield the best possible model. In this paper we will characterize the Fisher consistency of a rich family of surrogate loss functions used in the context of ordinal regression, including support vector ordinal regression, ORBoosting and least absolute deviation. We will see that, for a family of surrogate loss functions that subsumes support vector ordinal regression and ORBoosting, consistency can be fully characterized by the derivative of a real-valued function at zero, as happens for convex margin-based surrogates in binary classification. We also derive excess risk bounds for a surrogate of the absolute error that generalize existing risk bounds for binary classification. Finally, our analysis suggests a novel surrogate of the squared error loss. We compare this novel surrogate with competing approaches on 9 different datasets. Our method shows to be highly competitive in practice, outperforming the least squares loss on 7 out of 9 datasets.