What the extra layer buys

Look again at what the second layer did on the last slide, because it is the reason depth is worth having.

The pairing unit needed to know whether a bottle is a matched pairing. It did not go back to the tastes to work that out. It took the four wine-type signals already sitting in the layer below and grouped them: these two are matched, these two are not.

That is reuse. The wine types were built once, and then two different ideas were built on top of them.

Now picture the same job with no middle layer. A matched unit wired straight to the tastes has to work out the whole combination itself, from cabernet and plummy and viognier and apricotty. The mismatched unit sitting beside it has to do the same work over again, separately, because there is nothing in between for it to borrow.

The illustration puts the two side by side. Both reach the same two ideas. On the left, each idea is derived from raw tastes on its own. On the right, one shared row is built first and both ideas are assembled from it.

With four wines that is a fair fight, and the right-hand machine is not smaller. It has more wires, not fewer.

The saving appears when the ideas multiply. Add a third pairing idea, then a fourth, then fifty. On the left, each new idea starts from nothing and needs its own full set of wires down to the raw tastes. On the right, each new idea is a short list of wine types to combine, because the hard part is already built and sitting there.

That is the shape of it. A layer that has to rebuild everything from raw facts grows with the difficulty of each idea. A layer that can reuse what is beneath it grows with how many pieces each idea needs.

One more honest note, because this is often overstated. A single hidden layer, made wide enough, can represent a remarkable range of patterns, this wine problem included. Depth is not a power that shallow networks can never have. It is a cheaper route to the same place, and for hard problems it is cheaper by enough to be the difference between a network you can build and one you cannot.