The Mathematics of Gradient Descent
The Variance That Keeps a Deep Network Alive
Last timeA Step Size Per Coordinate
Initial weights have one job: keep the scale of the signal steady through every layer. Being a few percent off compounds into a factor of hundreds by layer fifty.
Everything in this course has assumed a starting point and then asked how to
move from it. This last lesson asks what the starting point has to be, and the
answer is not a matter of taste. A deep network initialised at the wrong scale
produces either no gradient at all or an infinite one, and neither can be
recovered from by tuning anything covered so far.
What one layer does to the scale
Take a layer with no activation function yet: , where has
components and the weights are drawn independently with mean zero.
Each output is a sum of terms, each a weight times an input. Independent
zero-mean terms add their variances, so:
- the fan-in, the number of inputs to the layer
- the variance of a single weight, the quantity being chosen
- the variance of an input component, which is the scale arriving from the previous layer
For the output to have the same scale as the input, the multiplier must be one:
The lesson stops here
9 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