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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?

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    $\begingroup$ "Ranking" per Everitt (at least, an earlier edition) just means " the process of sorting a set of variables into ascending or descending order". So, what do you mean by ranking? $\endgroup$ – Peter Flom Oct 28 '12 at 17:24
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    $\begingroup$ I mean ranking in the context of machine learning: en.wikipedia.org/wiki/Learning_to_rank $\endgroup$ – Fabian Pedregosa Oct 28 '12 at 17:36
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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

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    $\begingroup$ Can you recommend some ranking algorithms ? I'm facing kinda ranking problem but still don't know which algorithms can deal with it. Thanks. $\endgroup$ – Catbuilts Apr 8 at 8:38
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It's a great question! In general the difference between statistics and machine-learning or the approaches of other fields to "our" questions can be hard to understand, because there's a zoo of terms associated to each field.

So, for example, when people found out that backprop neural nets were "just" doing a nonlinear regression of some kind, that was like a major finding among researchers.

I think this is the same kind of thing: there are just a lot of techniques people have come up with to attack the same problem. Ordinal logistic is one.

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