The reason for deep learning is that shallow networks are very hard (or impossible) to train. In that sense, long time of training is evidence for shallow networks.
No it's because shallow networks can't express complex functions. If you think about it the shallowest network is pretty much a lookup table. They can theoretically model any function, but the number of parameters needed means in practice they can't. Deep networks can learn much more complex functions for the same number of parameters.
> They can theoretically model any function, but the number of parameters needed means in practice they can't.
Even theoretically, no they can't. They can theoretically model any continuos function.
Plus, even for continuous functions, the theorem only proves that, for any function, there exists some NN that approximates it to arbitrary precision. It is not known whether there is some base NN + finite training set that could be used to arrive at that target NN using some algorithm in a finite number of steps.
I'm not sure it is all that interesting of a distinction seeing as non-continuous functions can be approximated by continuous ones (basically the entire premise of a digital computer).
I don't think this is right at all. Digital computers express non-continuous functions, and they sometimes use those to approximate continuous functions.
For example, for a function f(x) defined on R with f(x) = -x if x < 0 and f(x) = 7+x if x >= 0, how would you approximate it by a continuous function g(x) with precision lower than, say, 1 (i.e. |f(x) - g(x)| < 1 for any x in R)?
And of course, there are functions with much worse discontinuities than this.
I mean… a 3 layer network is a Universal approximator… and you can very much do network distillation… it’s just that getting them wide enough to learn whatever we want them to isn’t computationally efficient. You end up with much larger matmuls which let’s say for simplicity exhibit cubic scaling in the dim. In contrast, you can stack layers and that comes much more computationally friendly because your matmuls are smaller.
Of course you then need to compensate with residuals, initialisation, normalisation, and all that, but it’s a small price to pay for scaling much much better with compute.
Yes that's exactly my point. A lookup table is a universal approximator. Good luck making AI with LUTs.
It's kind of like the halting problem or the no-free-lunch theorem. Interesting academic properties but they don't really have any practical significance and often confuse people into thinking that things like formal verification and lossless compression are impossible.
What the ratio for the number of parameters required to learn some complex function between a shallow network and a deep network (preferably as a function of the complexity)?
