[simiones]

It's incredible to me how widely this is misunderstood.

The universal function approximator theorem only applies for continuous functions. Non-continuous functions can only be approximated to the extent that they are of the same "class" as the activation function.

Additionally, the theorem only proves that for any given continuous function, there exists a particular NN with particular weight that can approximate that function to a given precision. Training is not necessarily possible, and the same NN isn't guaranteed to approximate any other function to some desired precision.

It seems pretty obvious to me that most interesting behaviors in the real world can't be modelled by a mathematical function at all (that is, for each input having a single output); if we further restrict to continuous functions, or step functions, or whatever restriction we get from our chosen activation function.

[ben_w]

> The universal function approximator theorem only applies for continuous functions. Non-continuous functions can only be approximated to the extent that they are of the same "class" as the activation function.

Yes, and?

> Training is not necessarily possible

That would be surprising, do you have any examples?

> and the same NN isn't guaranteed to approximate any other function to some desired precision.

Well duh. Me speaking English doesn't mean I can tell 你好[0] from 泥壕[1] when spoken.

> It seems pretty obvious to me that most interesting behaviours in the real world can't be modelled by a mathematical function at all (that is, for each input having a single output)

I think all of physics would disagree with you there, what with it being built up from functions where each input has a single output. Even Heisenberg uncertainty and quantised results from the Stern-Gerlach setup can be modelled that way in silico to high correspondence with reality, despite the result of testing the Bell inequality meaning there can't be a hidden variable.

[0] Nǐ hǎo, meaning "hello"

[1] Ní háo, which google says is "mud trench", but I wouldn't know

[simiones]

> Yes, and?

It means that there is no guarantee that, given a non-continuous function function f(x), there exists an NN that approximates it over its entire domain withing some precision p.

> That would be surprising, do you have any examples?

Do you know of a universal algorithm that can take a continuous function and a target precision, and return an NN architecture (number of layers, number of neurons per layer) and a starting set of weights for an NN, and a training set, such that training the NN will reach the final state?

All I'm claiming is that there is no known algorithm of this kind, and also that the existence of such an algorithm is not guaranteed by any known theorem.

> Well duh. Me speaking English doesn't mean I can tell 你好[0] from 泥壕[1] when spoken.

My point was relevant because we are discussing whether an NN might be equivalent to the human brain, and using the Universal Approximation Theorem to try to decide this. So what I'm saying is that even if "knowning English" were a continuous function and "knowing French" were a continuous function, so by the theorem we know there are NNs that can approximate either one, there is no guarantee that there exists a single NN which can approximate both. There might or might not be one, but the theorem doesn't promise one must exist.

> I think all of physics would disagree with you there, what with it being built up from functions where each input has a single output.

It is built up of them, but there doesn't exist a single function that represents all of physics. You have different functions for different parts of physics. I'm not saying it's not possible a single function could be defined, but I also don't think it's proven that all of physics could be represented by a single function.

[ben_w]

> It means that there is no guarantee that, given a non-continuous function function f(x), there exists an NN that approximates it over its entire domain withing some precision p.

And why is this important?

> Do you know of a universal algorithm that can take a continuous function and a target precision, and return an NN architecture (number of layers, number of neurons per layer) and a starting set of weights for an NN, and a training set, such that training the NN will reach the final state?

> All I'm claiming is that there is no known algorithm of this kind, and also that the existence of such an algorithm is not guaranteed by any known theorem.

I think so: the construction proof of the claim that they are universal function approximators seems to meet those requirements.

Even better: it just goes direct to giving you the weights and biases.

> My point was relevant because we are discussing whether an NN might be equivalent to the human brain, and using the Universal Approximation Theorem to try to decide this. So what I'm saying is that even if "knowning English" were a continuous function and "knowing French" were a continuous function, so by the theorem we know there are NNs that can approximate either one, there is no guarantee that there exists a single NN which can approximate both. There might or might not be one, but the theorem doesn't promise one must exist.

I still don't understand your point. It still doesn't seem to matter?

If any organic brain can't do $thing, surely it makes no difference either way whether or not that $thing can or can't be done by whatever function is used by an ANN?

> It is built up of them, but there doesn't exist a single function that represents all of physics. You have different functions for different parts of physics. I'm not saying it's not possible a single function could be defined, but I also don't think it's proven that all of physics could be represented by a single function.

I could point you to this: https://www.youtube.com/watch?v=PHiyQID7SBs

But that would be unfair, given the QM/GR incompatibility.

That said, ultimately I think the onus is on you to demonstrate that it can't be done when all the (known) parts not only already exist separately in such a form, but also, AFAICT, we don't even have a way to describe any possible alternative that wouldn't be made of functions.

[simiones]

> And why is this important?

Since we know non-continuous functions are used in describing various physical phenomena, it opens the gate to the possibility that there are physical phenomena that NNs might not be able to learn.

And while piece-wise continuous functions may still be ok, fully discontinuous functions are much harder.

> I think so: the construction proof of the claim that they are universal function approximators seems to meet those requirements.

Oops, you're right, I was too generous. If we know the function, we can easily create the NN, no learning step needed.

The actual challenge I had in mind was to construct an NN for a function which we do not know, but can only sample, such as the "understand English" function. Since we don't know the exact function, we can't use the method from the proof to even construct the network architecture (since we don't know ahead of time how many bumps there are are, we don't know how many hidden neurons to add).

And note that this is an extremely important limitation. After all, if the UAF was good enough, we wouldn't need DL or different network architectures for different domains at all: a single hidden layer is all you need to approximate any continuous function, right?

> If any organic brain can't do $thing, surely it makes no difference either way whether or not that $thing can or can't be done by whatever function is used by an ANN?

Organic brains can obviously learn both English and French. Arguably GPT-4 can too, so maybe this is not the best example.

But the general doubt remains: we know humans express knowledge in a way that doesn't seem contingent upon that knowledge being a single continuous mathematical function. Since the universal function approximator theorem only proves that for each continuous function there exists an NN which approximates it, this theorem doesn't prove that NNs are equivalent to human brains, even in principle.

> That said, ultimately I think the onus is on you to demonstrate that it can't be done when all the (known) parts not only already exist separately in such a form, but also, AFAICT, we don't even have a way to describe any possible alternative that wouldn't be made of functions.

The way physical theories are normally defined is as a set of equations that model a particular process. QM has the Schrodinger equation or its more advanced forms. Classical mechanics has Newton's laws of motion. GR has the Einstein equations. Fluid dynamics has the Navier-Stokes equations. Each of these is defined in terms of mathematical functions: but they are different functions. And yet many humans know all of them.

As we established earlier, the UFA theorem proves that some NN can approximate one function. For 5 functions you can use 5 NNs. But you can't necessarily always combine these into a single NN that can approximate all 5 functions at once. It's trivial if they are simply 5 easily distinguishable inputs which you can combine into a single 5-input function, but not as easy if they are harder to distinguish, or if you don't know that you should model them as different inputs ahead of time.

By the way, there is also an example of a pretty well known mathematical object used in physics that is not actually a proper function - the so-called Dirac delta function. It's not hard to approximate this with an NN at all, but it does show that physics is not strictly speaking limited to functions.

Edit to add: I agree with you that the GP is wrong to claim that the behavior exhibited by some organisms is impossible to explain if we assumed that the brain was equivalent to an (artificial) neural network.

I'm only trying to argue that the reverse is also not proven: that we don't have any proof that an ANN must be equivalent to a human/animal brain in computational power.

Overall, my position is that we just don't know to what extent brains and ANNs correspond to each other.