Threshold methods are popular for ordinal regression problems, which are
classification problems for data with a natural ordinal relation. They learn a
one-dimensional transformation (1DT) of observations of the explanatory
variable, and then assign label predictions to the observations by thresholding
their 1DT values. In this paper, we study the influence of the underlying data
distribution and of the learning procedure of the 1DT on the classification
performance of the threshold method via theoretical considerations and
numerical experiments. Consequently, for example, we found that threshold
methods based on typical learning procedures may perform poorly when the
probability distribution of the target variable conditioned on an observation
of the explanatory variable tends to be non-unimodal. Another instance of our
findings is that learned 1DT values are concentrated at a few points under the
learning procedure based on a piecewise-linear loss function, which can make
difficult to classify data well.

通过理论考虑和数值实验，本文研究了阈值法在阶序回归问题中的分类性能与数据分布以及一维转换学习程序之间的影响。研究发现，当目标变量在解释变量条件下的概率分布趋向于非单峰分布时，基于典型学习程序的阈值法可能表现不佳。此外，基于分段线性损失函数的学习程序得到的一维转换值集中于少数几个点，这可能导致难以对数据进行良好的分类。

关于序回归问题阈值方法的损失函数评注

Remarks on Loss Function of Threshold Method for Ordinal Regression  Problem

Ordinal regression (OR) is classification of ordinal data in which the
underlying categorical target variable has a natural ordinal relation for the
underlying explanatory variable. For $K$-class OR tasks, threshold methods
learn a one-dimensional transformation (1DT) of the explanatory variable so
that 1DT values for observations of the explanatory variable preserve the order
of label values $1,\ldots,K$ for corresponding observations of the target
variable well, and then assign a label prediction to the learned 1DT through
threshold labeling, namely, according to the rank of an interval to which the
1DT belongs among intervals on the real line separated by $(K-1)$ threshold
parameters. In this study, we propose a parallelizable algorithm to find the
optimal threshold labeling, which was developed in previous research, and
derive sufficient conditions for that algorithm to successfully output the
optimal threshold labeling. In a numerical experiment we performed, the
computation time taken for the whole learning process of a threshold method
with the optimal threshold labeling could be reduced to approximately 60\,\% by
using the proposed algorithm with parallel processing compared to using an
existing algorithm based on dynamic programming.

该研究提出了一种并行化算法，用于找到最优的阈值标签，通过使用该算法和并行处理相比使用基于动态规划的现有算法，可以将阈值方法的整个学习过程的计算时间缩短到约 60％。