Remarks on Loss Function of Threshold Method for Ordinal Regression Problem

Ordinal regression (OR, or called ordinal classification) is the classification of ordinal data in which the underlying target variable is labeled from a categorical ordinal scale that is considered to be equipped with a natural ordinal relation for the underlying explanatory variable, as formalized in Section 2.1. The ordinal scale is typically formed as a graded summary of objective indicators like age groups {'0-9', '10-19',..., '90-99', '100-'} or graded evaluation of subjectivity like human rating {'excellent', 'good', 'average', 'bad', 'terrible'}. OR techniques are employed in a variety of practical applications, for example, age estimation (Niu et al., 2016; Cao et al., 2020), information retrieval (Liu, 2011), movie rating (Yu et al., 2006), and questionnaire survey (Bürkner and Vuorre, 2019). Threshold methods are popular for OR problems as a simple way to capture the ordinal relation of ordinal data, and have been studied vigorously in machine learning research (Shashua and Levin, 2003; Lin and Li, 2006; Chu and Keerthi, 2007; Lin and Li, 2012; Li and Lin, 2007; Pedregosa et al., 2017; Yamasaki, 2023). Those methods learn a one-dimensional transformation (1DT) of the observation of the explanatory variable so that an observation with a larger class label tends to have a larger 1DT value; they then assign a label prediction to the learned 1DT according to the rank of an interval to which the 1DT belongs among intervals on the real line separated by threshold parameters.