Two jobs in the loop

Both halves are now in place, so here is the whole loop in one piece.

A training step on one bottle has four parts:

1. Send the bottle forwards through the weights and make a price guess.
2. Compare the guess with the real price. That gap is the error.
3. Walk backwards from the error, multiplying at each crossing, until every weight has its effect.
4. Move every weight a small step in the direction that lowers the error.

Part three is backpropagation. Part four is gradient descent. One measures, the other moves.

Then the next bottle arrives and the four parts run again.

It is worth seeing how little of this is new. The Chapter 1 machine ran the same loop. Guess, measure the error, work out each weight's effect, nudge. The difference is that in Chapter 1 part three was trivial, so it never looked like a separate job. Every weight touched the price directly, so a weight's effect was just its input value, sitting right there in the tasting.

A hidden weight has steps in the way. Its effect on the price has to travel through a unit and an output weight before it arrives. Backpropagation is what fills that gap, and it fills it by multiplying by whatever stands in the way at each crossing.

That is the honest summary of the whole chapter. The learning rule did not change. The route from a weight to the error got longer, and something had to walk it.