Semi-Supervised Ordinal Regression Based on Empirical Risk Minimization

Taira Tsuchiya, Nontawat Charoenphakdee, Issei Sato, Masashi Sugiyama

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, has a restriction on the model choice, and has no theoretical guarantee. To overcome these problems, we propose a novel generic framework for semi-supervised ordinal regression based on the empirical risk minimization (ERM) principle that is applicable to optimizing all of the metrics mentioned above. Also, our framework has flexible choices of models, surrogate losses, and optimization algorithms. Moreover, our framework does not require a geometric assumption on unlabeled data such as the cluster assumption or manifold assumption. We provide an estimation error bound to show that our risk estimator is consistent. Finally, we conduct experiments to show the usefulness of our framework.

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