Any Deep ReLU Network Is Shallow

The result in the paper is not an approximation result. The shallow network is of bounded width and is exactly equal to the deep one.

> Based on this proof, we provide an algorithm that, given a deep ReLU network, finds the explicit weights of the corresponding shallow network.

I’m out of date on the research, but I suspect the real value here is the algorithm. Sounds like some version of this could eventually help reduce inference time

    Corollary 1. Given a ReLU network N : Rn → Rm, the shallow 
    network S : Rn → Rm of depth L = 3, as specified in the proof 
    of Theorem 1, has width bounded by 
    max{2n + k, 2n + p, 2 · p · m}

this is the "universal approximator theorem".

https://cognitivemedium.com/magic_paper/assets/Hornik.pdf

> infinite number of [activations]

I don't think you need an infinite number of them, there is a relationship between "how close you want to get" and "how many of them you need".

I would just add the caveat that it is a single hidden layer. If you were to write this as a sequential model you would have something like:

nn.Sequential(

    Dense(many neurons),

    Dense(1)

)

From this, it's pretty easy to see it is "two" layers but also from your equation c_i and a_i denote two separate matrix multiplications.

This is like how you can write any boolean function as a (large) sum of minterms.

Thanks for the explanation. I've heard this before but think I get it now!

Isn't that just like a Taylor series?

I think you're being a little too quick to jump to the answer. There are plenty of functions where this isn't true. Two easy examples are jump discontinuities (one that looks like this ____----- but more wiggly) (known as piecewise continuous) and functions with hole. These cannot be approximated to arbitrary accuracy, even with an infinite number of them.

You need to pay close attention to the wording in the approximation theorems.

Hornik[0] which does the proof you're discussing says

> This paper rigorously establishes that standard multilayer feedforward networks with as few as one _hidden layer_ using arbitrary squashing functions are capable of approximating any __Borel measurable function__ from one __finite__ dimensional space to another to any desired degree of accuracy

The confusion is probably in the Borel sigma-algebra. Borel means that it includes all finite open intervals in the real numbers. So (0,1) but not [0,1]. Open means boundary is not included! The interval also needs to be continuous, so our discontinuities violate the assumptions. Funahashi's[1] and Cybenko's[2] also require continuous functions.

This is actually a really important thing that gets ignored because it "seems obvious." Or maybe we just see it too often. But it is __critical__ to pay attention to assumptions and limitations. There is a lot of that going off the rails lately with ML and it's going to give us some roadblocks (we're already seeing some[side note]). EVERYTHING (and I mean literally everything) has assumptions, and therefor biases[3]. Every evaluation method you use has a bias. Every learning method you use has a bias. Every dataset. Every architecture. Everything. This is because everything has a certain number of assumptions baked in. We try to reduce these and make them as sane as possible, but we should be aware of them. And in the case of the universal approximation, well the limitation is fairly meaningful. Data is not guaranteed to lie upon a smooth continuous manifold.

[0] https://cognitivemedium.com/magic_paper/assets/Hornik.pdf

[1] https://dx.doi.org/10.1016/0893-6080%2889%2990003-8

[2] https://link.springer.com/article/10.1007/BF02551274

[3] https://en.wikipedia.org/wiki/Bias_(statistics)

[side note] We actually see this a lot in evaluation, and this is why benchmarkism is so problematic. Because it causes us to look at results as hard signals instead of guides. Evaluation is fucking hard. No empirical results will ever give you the full answer: the map is not the territory. We saw a hugging face blog today[4] about the LLM results differing and the reason is because the different evaluation methods biased towards different models. This means the metrics can be hacked, even specifically by the RLHF tuning. Or even the tokenization. These things are hard to make real good judgements on if you don't know the limits of the evaluation method (i.e. the assumptions it makes).

[4] https://huggingface.co/blog/evaluating-mmlu-leaderboard

This is such a bullshit HN comment, lmao.

If anything inference will probably get worse. Typically in ML constraining your architecture (which is what you're doing when you make a deep narrow network out of a shallow wide network, or rnns or grus/lstms, or transformers out of fcnns) yields inference and training performance benefits.

> So any N-layer network with the relu _/ activation function

The paper talks only about models with additive operations among activations. It doesn't say anything about more complex networks like transformers.

In transformers there are multiplicative interactions between activations inside the attention matrix, it is unclear if they can be approximated with just a 3 layer ReLU network, or if such conversion would be practical at all.

> How well does a classic deep net architecture like AlexNet or VGG19 classify on a standard dataset such as CIFAR-10 when its “width”— namely, number of channels in convolutional layers, and number of nodes in fully-connected internal layers — is allowed to increase to infinity? Such questions have come to the forefront in the quest to theoretically understand deep learning and its mysteries about optimization and generalization. They also connect deep learning to notions such as Gaussian processes and kernels. A recent paper [Jacot et al., 2018] introduced the Neural Tangent Kernel (NTK) which captures the behavior of fully-connected deep nets in the infinite width limit trained by gradient descent; this object was implicit in some other recent papers. An attraction of such ideas is that a pure kernel-based method is used to capture the power of a fully-trained deep net of infinite width.

https://openreview.net/pdf?id=rkl4aESeUH, https://github.com/google/neural-tangents

> It has long been known that a single-layer fully-connected neural network with an i.i.d. prior over its parameters is equivalent to a Gaussian process (GP), in the limit of infinite network width.

https://arxiv.org/abs/1711.00165

And of course, one needs to look back at SVMs applying a kernel function and separating with a line, which looks a lot like an ANN with a single hidden layer followed by a linear mapping.

https://stats.stackexchange.com/questions/238635/kernel-meth...

My hunch is that you can really easily find (adversarial) counterexamples that may show that in the worst case you will always use more compute by shallow-ing. In my head it should be related to no free lunch theorem.

Besides pruning (which is what you describe) you might find the "lottery ticket hypothesis" interesting. The idea is essentially that in a randomly initialized large network there is a subset of weights that are already closeish for a given task and training helps tune that and suppress everything else, and that knowing this you can actually get better results by pruning before training.

Weight pruning is already a thing. See, for example: https://www.tensorflow.org/model_optimization/guide/pruning

That was my thought as well, it would be nice if someone more knowledgeable came and explained us why it's stupid/doesn't work

Could you point to an inductive proof of this?

See these comments on the same post if not already.

https://news.ycombinator.com/item?id=36453136

Short answer to your question: Not true in general.

It seems like it doesn’t hold since this relies on turning the whole thing into a piecewise linear function.