Multiply Along, Add Across
Last timeEvery Computation Is a Graph
Two rules cover everything. Along a chain the derivatives multiply, and where a value was used more than once the contributions add. The rest is bookkeeping about the order.
The two rules in this lesson are the entire mathematical content of
backpropagation. Everything after this is about applying them efficiently,
which turns out to be where all the interesting difficulties are.
Along a chain, multiply
Suppose a value passes through three operations in turn. A small change at the
start is scaled by the first operation, then by the second, then by the third.
- the value at the start of the chain
- what the first operation produced from it
- what the second produced from that
- the single number at the end of the whole computation
The consequence to notice now, because a later lesson is entirely about it: if
each factor happens to be around a half, ten links take the derivative down by
a factor of a thousand. Nothing has gone wrong. That is just what multiplying
ten numbers does.
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