A Product of Many Small Numbers
Last timeThe Memory the Backward Pass Costs
The derivative reaching an early layer is a product of one factor per layer in between. Products of many numbers go to zero or to infinity, and almost every architectural trick is a response to that.
Everything in this course has concerned one operation at a time. This last lesson
is about what happens when there are hundreds of them in a row, and it is a
statement about products rather than about networks.
One factor per layer
Apply the chain rule down the depth of a network and the derivative reaching an
early layer is a product with one term for every layer above it. That is all.
- the local factor contributed by layer k, roughly how much it scales a small change
- the number of layers between here and the loss
- the derivative that finally arrives at the early layer
Why a hair matters
Products of numbers near one are unforgiving.
The lesson stops here
5 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents