[this answer](https://datascience.stackexchange.com/a/23468/25583) to another question references a paper [(Ordinal Regression with Multiple Output CNN for Age Estimation)](https://www.cv-foundation.org/openaccess/content_cvpr_2016/papers/Niu_Ordinal_Regression_With_CVPR_2016_paper.pdf) which utilizes a clever representation of the labels to measure error with cross entropy.

The paper presents the total error as the sum of errors in predicting whether or not the "rank" of a sample $x_i$ is greater than rank $k_i$. 

In other words, we would generate predictions of vectors with elements $r(x_i) > k_i$, representing the prediction of the classifier for whether or not the rank of the sample is greater than each rank. This becomes a multiclass classification problem and error functions for that problem can be utilized. 

E.g.: error = sum of the individual binary classifier loss functions