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Any Deep ReLU Network Is Shallow

arxiv.org

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Re: Any Deep ReLU Network Is Shallow

#61
post #60

Earlier quoted context omitted.

Ah, get your point and misunderstood. I thought you were talking about comparing the neural network to a lookup table, not modeling the network itself. In that case the proof is only true if you consider the digital implementation of the neural network. Since it's a continuous function this proof would be impossible mathematically, as the domain is not enumerable. But if you consider only every possible float32 for e…

You're right, but the idea of looking things up instead of computing them can be useful when we are constrained by the available compute power. I'm not talking about simple lookup tables, of course, but if you look at recent trends in large foundational models, there's a lot of interest in efficient access to external information, or ways to pay attention to the inputs selectively, rather than in all-to-all fashion (…

Oh yeah I didn't see the current discussion as related to that but I find the topic of fact databases for LLMs pretty interesting, thanks for drawing the analogy.

Re: Any Deep ReLU Network Is Shallow

#62
post #49

Earlier quoted context omitted.

They (suspiciously but not necessarily intentionally) avoid calling attension to it but: > Let ({fω}ω∈Ω,Ω) be the decomposition that corresponds to N. Label the parts 1,...,p. p is the number of regions the deep network divides R^n into, and is in general exponential in the network depth. (Eg consider R^1 = [-1,+1], L[k](x) = 2*abs(x)-1 ∀k (where abs(x) = x - 2*ReLU(-x), IIRC), which divides R^1 into 2^D equal segmen…

I would assume the distinction between trained and untrained networks is a constant factor at best.

I meant 'trained' networks in contrast to totally-synthetic intentionally-pathological networks like my 2abs-1 example. (It could be that only pathological cases give bad results - similar to how (Hindley-Milner) type inference is EXPTIME-complete but works fine for ~all real world programs - but that seems unlikely.)
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