Question regarding how preferences are modeled with BPR loss

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Hi all!

First of all, thank you so much for the effort in this lib. Works really well :)

However, while using it with some implicit data (in fact, count data) I had a question on how the training is done with BPR loss. Let me explain:

As you already know: for modelling preferences with BPR loss, a pairwise loss is used; so each training instance is a triplet in the form of `(user, item1, item2)`; where the model learns the preference between the two items for that specific user.

Imagine that I have an implicit dataset with play counts for movies (how many times each user has watched a specific movie). To make things easier, let's imagine that I only have a single user, and the following interaction data:

- Movie A, watched 8 times
- Movie B, watched 2 times
- Movie C, watched 2 times
- Movie D, watched 0 times (this is, no interaction)

I was wondering how the training triplets would look like with such dataset. We know the preference for all possible triplet combinations, except for the one between B and C (as their play count is exactly the same; so we cannot infer any preference between them).

However by skimming through the source code, I would say that the way the *negative sampling* works in LightFM with BPR loss would yield only the following training triplets:

- `(user, Movie A, Movie D)`
- `(user, Movie B, Movie D)`
- `(user, Movie C, Movie D)`

Which is correct (as BPR assumes preference on all watched movies over the unwatched one). However, the model would me missing some valuable information: our user prefers Movie A over B and C also (as he/she has watched it way more times).

But for what I have seen in the source code, non-zero items are only compared against zero (unwatched) ones, therefore missing the *prefer A over B/D* training instances.

Is that right? If that is the case, would you be interested in a contribution to include those additional preferences as part of the training procedure?

Also, does it work the same way for WARP loss?

Thanks a ton!

Start by tracing the source code's negative-sampling paths for BPR and WARP and compare their behavior with the count-data example in the issue. Confirm how equal and differing nonzero interactions are handled; done requires an agreed training design for additional preferences and tests covering the resulting triplets.

Written by the indexing model from the issue text.