Reach the incoming weights

Two crossings are done. Every output weight carries its effect, and hidden 1 itself is worth $25 for every extra 1 it sends. The weights coming into the unit are all that remain.

Press reach the incoming weights. The four weights feeding hidden 1 now show their values, and they have held those values since the bottle went in.

From here the problem is a Chapter 1 problem again. Hidden 1 is worth $25 for every 1 it sends, and the cabernet weight feeds it directly.

So imagine adding 1 to the cabernet weight, the disc reading .20. It would become 1.20. Again, this is a measurement, not a change we keep.

Be clear about what is not changing. Cabernet's input stays 1. That is a fact about the bottle, this wine is a cabernet, and no amount of training alters it. Only weights ever move.

Follow it through. Hidden 1 builds its total by multiplying each input by its weight and adding:

Before: 1 x .20 + 0 x .10 + 1 x .30 + 0 x .20 = .50

After: 1 x 1.20 + 0 x .10 + 1 x .30 + 0 x .20 = 1.50

Look at what makes up that .50. Cabernet contributes .20 and plummy contributes .30. Plummy is doing the larger share, and this bottle is a plummy cabernet, so both tastes are present and both have a say.

Now look at what happens in the second line. Plummy's .30 does not move. Neither do the two zeros. Only the term we nudged changes, from .20 to 1.20, and the total rises by exactly that 1.

That is the general shape. A hidden unit's signal is built from every input at once, but when we ask about one weight, everything else holds still and drops out of the difference.

One rule still stands between that total and what the unit sends: the response rule from slide 18, the floor at zero. The walk crosses it here without leaving a trace, because the floor leaves positive totals alone. 1.50 passes through exactly as .50 did, so hidden 1 would send 1 more.

It is worth seeing what the floor does when it is not so gentle. Slide 18 showed a unit pushed the other way: an apricotty viognier drove its total to -.40, and the floor turned that into 0. Nudge one of its weights a little on that bottle and the total becomes something like -.39, which the floor also turns into 0. The signal does not move, so the price does not move, and every incoming weight of that unit gets an effect of 0. A unit that stayed silent for a bottle is handed no share of the blame for it.

Our unit is wide awake, so its +1 goes through, and the last slide already priced that: every extra 1 that hidden 1 sends is worth $25.

So here is the whole chain in one line:

weight +1 makes hidden 1 +1, and 1 x 25 = $25.

That $25 is the cabernet weight's effect, and it now sits on the weight itself in the same place and style as the +$.50 on the output weight. Two captions sit under the picture, and each one starts by naming what gets moved, so read the one whose ring you are looking at. The left caption is the working behind the $25.

That is the point of the whole walk. These two weights sit in different layers, one beside the price and one three steps away from it, and both end up with the same kind of number: what the price would do if this weight moved by 1. The output weight is worth $.50. The cabernet weight is worth $25.

Notice why the two +1s line up. Cabernet's input is 1, so adding 1 to the weight passes through at full strength: the weight rises by 1, and hidden 1's total rises by 1 x 1 = 1.

Had cabernet sent 0.5 instead, the same +1 on the weight would have raised hidden 1's total by only 1 x 0.5 = 0.5, and the price by $12.50. That is the Chapter 1 rule exactly: a weight whose input spoke softly has less effect.

Check it against slide 23. Raising the cabernet weight by .01 raised the price by $.25. Follow the chain: .01 into the unit, then across the output weight, .01 x 25 = .25. The same number, calculated instead of tested.

Three things are worth keeping.

The calculation starts at the last weights, the ones beside the price, and moves left one layer at a time. It cannot start at the front, because until the error has crossed the output weights, nothing earlier has a share of the blame yet.

Every weight is reached in one pass. The machine does not run a separate experiment per weight. One walk backwards hands every weight in the network its own effect, which is why this is affordable on a network with billions of them.

And nothing is carried backwards unchanged. At each crossing the running number is multiplied by whatever stands in the way. Crossing the output weight multiplied by 25. Reaching the cabernet weight multiplied by its input of 1.

That is the entire mechanism. Arrive with a rate, multiply by the local thing, leave with a new rate for whatever sits one step further back. Repeat until every weight has one.

That walk has a name. Sending the error back through the machine, crossing one set of weights at a time, is called .

The word is built from its job. To propagate something is to send it spreading through, the way a crack propagates through glass. Here the thing being sent through the machine is the error, and it is sent back, against the direction the bottle travelled.

It is worth being precise about what backpropagation does and does not do. It has not corrected anything. Not one weight has moved. All it has produced is a number for each weight, saying how the price, and with it the error, answers to that weight.

Something still has to take those numbers and change the machine.