Ordinal Regression – What Is the Difference Between Ordinal Regression and Ranking?

Question

In both ordinal regression and ranking you are learning from an ordered dependent variables, so my question is:

What is the difference in formulation (if any) between the ordinal regression problem and a learning to rank problem?

Answer 1

3 years after, I answer to my own question.

For me, the main difference is in what is the output of the models in the different problems. In ordinal regression, the task is to predict a label for a given sample, hence the output of a prediction is a label (as is the case for example in multiclass classification). On the other hand, in the problem of learning to rank, the output is an order of a sequence of samples. That is, a the output of a ranking model can be seen as a permutation that makes the samples to have labels as ordered as possible. Hence, unlike the ordinal regression model, the ranking algorithm is not able to predict a class label. Because of this, the input of a ranking model does not need to specify class labels, but only a partial order between the samples (see e.g. [0] for an application of this). In this sense, ranking is an easier problem than ordinal regression: from the numerical labels you can construct an order, but not necessarily the other way round.

This is better explained with an example. Suppose that we have the following pairs of (sample, label): $\{(x_1, 1), (x_2, 2), (x_3, 2)\}$. Given this input, a ranking model will predict an order of this sequence of samples. For example, for a ranking algorithms, the permutations $(1, 2, 3) \to (1, 2, 3)$ and $(1, 2, 3) \to (1, 3, 2)$ are predictions with perfect score since the labels of both sequences $\{(x_1, 1), (x_2, 2), (x_3, 2)\}$ and $\{(x_1, 1), (x_3, 2), (x_2, 2)\}$ are ordered. On the other hand, an ordinal regression would predict a label for each of the samples, and in this case the prediction (1, 2, 2) would give a perfect score, but not (1, 2, 3) or (1, 3, 2).

[0] Optimizing Search Engines using Clickthrough Data Thorsten Joachims