A model is a function with a second argument.

Demonstration

Write it down and most of the mystery leaves:

ŷ = f(x ; θ)

Two arguments, not one. The first, x, comes from the world and the model has no say in it. The second, θ, is a list of numbers the model carries around, and it is the only thing training is permitted to touch.

Fix θ and you have an ordinary function — deterministic, inspectable, no more mysterious than a polynomial. Vary θ and you have a family of functions, one per setting of the numbers. The architecture is the shape of that family; training is a search inside it.

This is worth being pedantic about, because almost every confused question about deep learning is a confusion about which of the two arguments is which. “Why did the model answer that?” is a question about x and about which member of the family got selected. “Why can’t the model do this at all?” is usually a question about the family — about what no setting of θ could express.

A useful consequence: nothing in the definition requires f to be complicated. Linear regression fits it exactly. So does a lookup table with a billion rows. The interesting choices are which family you pick and how you search it, and neither is settled by calling something a model.

Corollary

Two claims that sound similar are entirely different in this frame. This model cannot do X is a claim about the family, and is falsified by exhibiting a θ that works. This training run did not learn X is a claim about the search, and is falsified by a better run. Keep them apart and most arguments about capability become tractable.

The two-argument form is equation (I.1.1) of this chapter, and it is the single sentence the rest of Book I elaborates. Every later architecture changes what f is; none of them changes that θ is the only argument training may touch.