# Semi-Supervised Ordinal Regression Based on Empirical Risk Minimization

## TL;DR

This paper introduces a flexible semi-supervised ordinal regression framework based on empirical risk minimization, capable of optimizing various metrics with theoretical guarantees and no restrictive assumptions on unlabeled data.

## Contribution

It proposes a novel, general framework for semi-supervised ordinal regression that supports multiple metrics, model choices, and provides theoretical consistency guarantees.

## Key findings

- Framework effectively optimizes multiple evaluation metrics.
- Theoretical analysis confirms estimator consistency.
- Experimental results demonstrate practical usefulness.

## Abstract

Ordinal regression is aimed at predicting an ordinal class label. In this paper, we consider its semi-supervised formulation, in which we have unlabeled data along with ordinal-labeled data to train an ordinal regressor. There are several metrics to evaluate the performance of ordinal regression, such as the mean absolute error, mean zero-one error, and mean squared error. However, the existing studies do not take the evaluation metric into account, have a restriction on the model choice, and have no theoretical guarantee. To overcome these problems, we propose a novel generic framework for semi-supervised ordinal regression based on the empirical risk minimization principle that is applicable to optimizing all of the metrics mentioned above. Besides, our framework has flexible choices of models, surrogate losses, and optimization algorithms without the common geometric assumption on unlabeled data such as the cluster assumption or manifold assumption. We further provide an estimation error bound to show that our risk estimator is consistent. Finally, we conduct experiments to show the usefulness of our framework.