this answer to another question references a paper (Ordinal Regression with Multiple Output CNN for Age Estimation) 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