Md. Asif Uddin

Proposition 33 of 39 in the corpus

Learning changes parameters, not rules.

Training does not write new logic. It moves numbers inside a structure that was fixed before the first example arrived, which is why the architecture decides what is learnable at all.

Depends on

Training as a path through parameter spaceTwo parameter axes with a sequence of points joined into a path, starting at a randomly initialised point and ending at a settled one. Every point on the path is the same model with different numbers in it.parameter spaceθ₁θ₂θ₀θ*Same architectureat every point.Only the numbersmove.
Fig. 3 — Training as a path through parameter space. Every point on the path is the same architecture holding different numbers.

Demonstration

The phrase the model learned to do X invites a picture of a system acquiring a rule, in something like the way a person acquires one. The mechanism is narrower than that, and the narrowness is the useful part.

At initialisation the model already has its full structure: every layer, every connection, every operation in its final arrangement. Training changes none of it. What training does is move a point through the space of possible parameter settings — the same architecture, at every step, holding different numbers.

So when a network appears to acquire a rule, the rule was always expressible; the search merely found the region where it is expressed. And when a network cannot acquire a rule, there are two quite different explanations, and telling them apart is the whole job:

  1. No setting of the parameters expresses it. A single linear layer cannot express xor at any θ. More data will not help. More training will not help. The family is wrong.
  2. Some setting expresses it, but the search did not arrive. The architecture is adequate and the optimisation, the data or the objective is not.

The first is a statement about the architecture, and it is usually provable. The second is a statement about a training run, and it is usually only testable. Practitioners spend a great deal of time on the second while the answer lies in the first.

Corollary

Inductive bias is what this proposition makes precise. A convolution shares weights across positions and so cannot learn a position-specific rule the way a dense layer can — that is not a limitation discovered during training but a decision made before it. Choosing an architecture is choosing what will be easy, what will be hard, and what will be impossible.

Sources

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