Hallucination are not generalization beyond the training data but interpolations gone wrong.
LLMs are in fact good at generalizing beyond their training set, if they wouldn’t generalize at all we would call that over-fitting, and that is not good either. What we are talking about here is simply a bias and I suspect biases like these are simply a limitation of the technology. Some of them we can get rid of, but—like almost all statistical modelling—some biases will always remain.
What, may I ask, is the difference between "generalization" and "interpolation"? As far as I can tell, the two are exactly the same thing.
In which case the only way I can read your point is that hallucinations are specifically incorrect generalizations. In which case, sure if that's how you want to define it. I don't think it's a very useful definition though, nor one that is universally agreed upon.
I would say a hallucination is any inference that goes beyond the compressed training data represented in the model weights + context. Sometimes these inferences are correct, and yes we don't usually call that hallucination. But from a technical perspective they are the same -- the only difference is the external validity of the inference, which may or may not be knowable.
Biases in the training data are a very important, but unrelated issue.
Interpolation and generalization are two completely different constructs. Interpolation is when you have two data points and make a best guess where a hypothetical third point should fit between them. Generalization is when you have a distribution which describes a particular sample, and you apply it with some transformation (e.g. a margin of error, a confidence interval, p-value, etc.) to a population the sample is representative of.
Interpolation is a much narrower construct then generalization. LLMs are fundamentally much closer to curve fitting (where interpolation is king) then they are to hypothesis testing (where samples are used to describe populations), though they certainly do something akin to the latter to.
The bias I am talking about is not a bias in the training data, but bias in the curve fitting, probably because of mal-adjusted weights, parameters, etc. And since there are billions of them, I am very skeptical they can all be adjusted correctly.
I assumed you were speaking by analogy, as LLMs do not work by interpolation, or anything resembling that. Diffusion models, maybe you can make that argument. But GPT-derived inference is fundamentally different. It works via model building and next token prediction, which is not interpolative.
As for bias, I don’t see the distinction you are making. Biases in the training data produce biases in the weights. That’s where the biases come from: over-fitting (or sometimes, correct fitting) of the training data. You don’t end up with biases at random.
What I meant was that what LLMs are doing is very similar to curve fitting, so I think it is not wrong to call it interpolation (curve fitting is a type of interpolation, but not all interpolation is curve fitting).
As for bias, sampling bias is only one many types of biases. I mean the UNIX program YES(1) has a bias towards outputting the string y despite not sampling any data. You can very easily and deliberately program a bias into everything you like. I am writing a kanji learning program using SSR and I deliberately bias new cards towards the end of the review queue to help users with long review queues empty it quicker. There is no data which causes that bias, just program it in there.
I don‘t know enough about diffusion models to know how biases can arise, but with unsupervised learning (even though sampling bias is indeed very common) you can get a bias because you are using wrong, mal-adjusted, to many parameters, etc. even the way your data interacts during training can cause a bias, heck even by random one of your parameters hits an unfortunate local maxima yielding a mal-adjusted weight, which may cause bias in your output.
LLMs are in fact good at generalizing beyond their training set, if they wouldn’t generalize at all we would call that over-fitting, and that is not good either. What we are talking about here is simply a bias and I suspect biases like these are simply a limitation of the technology. Some of them we can get rid of, but—like almost all statistical modelling—some biases will always remain.