A visual proof that neural nets can compute any function

http://neuralnetworksanddeeplearning.com/chap4.html

by jbarrow • 12 years ago

217 80 12 years ago

tomp• 12 years ago

As mentioned in the article, the formal statement is actually "neural nets can approximate (arbitrarily well, using the supremum metric) any continuous function". For other norms, it can also approximate non-continuous functions.

voidlogic4• 12 years ago

It would have been pretty interesting if this had NOT held. It would have meant that even though "neural nets can NOT approximate (arbitrarily well, using the supremum metric) any continuous function", a neural network (the humans involved) was able to discover this limitation.

I find the idea of a neural net finding a limitation of a neural net, to be interesting.

gphil2• 12 years ago

More generally, logic can be used to demonstrate the limitations of logic:

https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_th...

You could say that a neural network found this limitation of neural networks, to the extent that neural networks could be defined in terms mathematical logic. However, it's not guaranteed that the neural networks in our brain could be explained in these terms--the physical processes underlying them are not fully understood.

mjw3• 12 years ago

Humans are not neural networks in the formal sense used here, not even close.

maaku3• 12 years ago

A "neural net" is what a computer scientist decided to call something that he thought behaved somewhat like a now dated abstract model of what individual neurons worked like, from a period when neuroscience was really in its infancy.

Your brain is not a neural net.

gphil2• 12 years ago

Yep, my first thought upon seeing the title was: wait, what about non-computable functions?

mturmon6• 12 years ago

You also need the qualifier "...any continuous function, on a compact set."

Once you add all three qualifiers in (approximate/continuous/compact) it starts to sound more like math and less like a miracle.

Incidentally, one thing of great interest is, how does the number of hidden units required behave as a function of dimensionality of the input domain. In dramatic language, "Can neural networks get around the curse of dimensionality?"

The Cybenko proof does not give enlightenment about that question. Andrew Barron (http://www.stat.yale.edu/~arb4/) had some results that seemed to indicate the dependence was moderate (not exponential). I'm not aware what the state of the art currently is.

rcfox• 12 years ago

how does the number of hidden units required behave as a function of dimensionality of the input domain

If I recall correctly, a non-linear problem can be solved as a linear problem if you consider more dimensions. The hidden layer add dimensions. So, it's not a function of the input domain but of the problem domain, which usually isn't explicitly known.

gohrt• 12 years ago

it starts to sound more like math and less like a miracle.

that applies to all of math. There are no miracles.

mtdewcmu6• 12 years ago

> Once you add all three qualifiers in (approximate/continuous/compact) it starts to sound more like math and less like a miracle.

I agree. The problem with attempting to make subtle technical points digestible by untrained people is that the larger meaning of the exercise is likely to be lost along with the details. This would probably be a good demonstration to show to investors in your NN-based startup, because it seems impressive. But if I had to state the take-home message of this that the average person should care about, I'm not sure there is one. There are simpler systems with the same property that are taught to undergraduate engineers (and NNs don't represent a novel path to having it, even); there's a great distance between approximating functions and solving practical ML problems.

Edit: To me, personally, I would have been more interested had this been an argument about back propagation for training NNs. I suspected NNs could do this because of what's in them, but training is as much of a constraint on what they can actually do.

mikeash• 12 years ago

I liked this article a lot, but I found it extremely confusing how some of the diagrams were interactive and some weren't. Why not make them all interactive? Barring that, an obvious visual indicator when one is interactive would be handy. As it was, I clicked on a lot of static images.

lucio• 12 years ago

https://www.youtube.com/watch?v=KpUNA2nutbk

jokoon3• 12 years ago

who's to doubt neural networks are completely awesome ?

Any more news about those chips that were optimized for neural networks ? Was it IBM or Samsung ?

j2kun3• 12 years ago

We have very few theoretical results about neural networks. So not completely awesome.

jokoon3• 12 years ago

I mean there is a big future for neural networks

jbarrow2• 12 years ago

It was the IBM SyNAPSE Chip [1]. It's actually been known since 1988/89 that neural networks can approximate any continuous function [2], but this chapter explains how in a much more intuitive sense.

[1] http://www.research.ibm.com/cognitive-computing/neurosynapti... [2] http://dl.acm.org/citation.cfm?id=70408

mooneater• 12 years ago

Who cares if they can compute any function. The important question is, can they learn any function, and can they learn in a way that can generalize? (And clearly they can for many useful domains).

akuma73• 12 years ago

What about XOR?

From Wikipedia: In 1969 in a famous monograph entitled Perceptrons, Marvin Minsky and Seymour Papert showed that it was impossible for a single-layer perceptron network to learn an XOR function.

JoeAltmaier• 12 years ago

Neural nets now use non-linear functions so they can do things like xor.

pesenti2• 12 years ago

http://en.wikipedia.org/wiki/Feedforward_neural_network#medi...

Dn_Ab2• 12 years ago

If you join a bunch of perceptrons together that limitation goes away. Another path is to make the problem effectively linear again by transforming into higher dimensions, kernels do this with one clever trick that allows them to avoid the computational cost of doing so explicitly.

jbarrow2• 12 years ago

The simple answer is that we no longer use the standard linear perceptrons that McCulloch and Pitts put forth in 1943. The limitations of perceptrons and their training algorithms are well-known and well-studied, and in the last 60 years we've come a long way.

A book came out a little bit ago which covers basic neural networks, and culminates in showing a network that can estimate XOR [1]. I strongly recommend reading not only the chapter, but the whole book.

[1] http://natureofcode.com/book/chapter-10-neural-networks/

dragonwriter• 12 years ago

"Single-layer" is an important restriction there, however.

jo_5• 12 years ago

'Single-layer' is an easily overlooked and absolutely essential modifier. Very few networks these days are single layers. (Three is the minimum for infinite dimensional functional approximation, given nonlinear activations.) Deep Networks you see in new research papers these days have, at minimum, perhaps three layers. Some by LeCun et al. will go as high as nine or twelve layers, with four or sixteen layers in breadth.

mjw3• 12 years ago

Quite. It's not hard to come up with models or families of functions which share this property.

What matters is not only whether they can learn it but how much data they need to learn it to a given degree of accuracy. This is the kind of question addressed by nonparametric statistics and statistical learning theory.

wall_words2• 12 years ago

This is an important statement and should be upvoted more. Case in point: "the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function."

skarayan• 12 years ago

It computes because it is detached, there is no feeling. To learn, we would need to first create something that is conscious, an ever-changing self. It needs to be in reference to itself, not some notion of compute in a detached paradigm.

robrenaud• 12 years ago

This is what is meant by "learn" in the context of machine learning. From wikipedia, http://en.wikipedia.org/wiki/Machine_learning

"A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E". This definition is notable for its defining machine learning in fundamentally operational rather than cognitive terms, thus following Alan Turing's proposal in Turing's paper "Computing Machinery and Intelligence" that the question "Can machines think?" be replaced with the question "Can machines do what we (as thinking entities) can do?"

It's important, because we don't need to care about cognition or consciousness, and we can still write programs that solve problems well by making inferences from patterns in data.

wuliwong• 12 years ago

This is a chapter from a textbook, one that seems pretty introductory at that. I'd say it is a pretty important piece of information to convey and prove, this ability of neural networks to represent any function.

Seems to me that jumping to whether they can "learn any function" as you asked, would certainly have come after this is first established.

Houshalter4• 12 years ago

Sort of. It depends on the learning algorithm used. Typically some form of local optimization is used that gets it to a good local optima. This is usually good enough and fast. However you can use methods of global optimizations or whatever optimization algorithm you want to use.

NNs themselves are sort of agnostic to the algorithm you use to set their weights. It's sort of like saying "computers are Turing complete, but can we ever prove that human programmers are smart enough to program any function."

mostly_harmless2• 12 years ago

I think this ability to compute any function is important, and useful in understanding why neural nets work. Neural nets are essentially a blank computation matrix that, through training, best approximates a function.

comparing neural net neurons to logic gates in a computer circuit or FPGA, gives a nice intuitive understanding of why the training works in the first place. The training essentially sets up a custom virtual circuit for whatever function you need.

Given this property, it seems intuitive that they would be able to learn any function, since any function is representable as logic gates.

samizdatum• 12 years ago

The biologically-inspired nature of neural nets was a steady, mesmerizing flame for machine learning research. A chance to work on an extremely simple, provably universal system that had embarrassingly obvious rhetorical implications for profound-sounding problems like the computability of consciousness, or the computational complexity of the human brain, proved too seductive for the legions of research-moths that flocked to bask in that alluring flame.

The initial flurry of research activity slowed to a background simmer, however. As a steady march of implementation problems was knocked down, it became clear that the limitations of neural nets were inherent to neural nets themselves, rather than any particular implementation detail. For example, the Lagrangians calculated by SVMs are provably convex, in contrast to the lack of a convexity guarantee for neural nets, which happily gradient descent into local minima. There were workarounds, of course, with regularization techniques, stochastic descent, etc. providing some measure of relief, but a migration of research interest away from neural nets seemed increasingly promising, and today, the migration seems largely complete.

wall_words2• 12 years ago

but a migration of research interest away from neural nets seemed increasingly promising, and today, the migration seems largely complete.

What are you talking about? Deep learning is one of the hottest areas of research today, and a lot of it has to do with neural networks. NN's are the state of the art in several domains. Case in point: http://image-net.org/challenges/LSVRC/2014/results. All of the top entries use convolutional networks; in fact, almost all of the entries do.

The fact that the loss function represented by a neural network can be highly nonconvex is what makes them so effective in the domains in which they are used. See this presentation by Yann LeCun for more info: http://www.cs.nyu.edu/~yann/talks/lecun-20071207-nonconvex.p...

"ML theory has essentially never moved beyond convex models, the same way control theory has not really moved beyond linear systems. Often, the price we pay for insisting on convexity is an unbearable increase in the size of the model, or the scaling properties of the optimization algorithm ... This is not by choice: nonconvex models simply work better. Have you tried acoustic modeling in speech with a convex loss? ... To learn hierarchical representations (low-level features, mid- level representations, high-level concepts....), we need “deep architectures”. These inevitably lead to non-convex loss functions."

This isn't to say that NN's are going to solve all our problems, but to say that there has been a shift in interest away from NN's is absurd.

sotte• 12 years ago

If you're interested in the generalization ability of neural networks I can recommend the following paper: "Intriguing properties of neural networks" http://arxiv.org/pdf/1312.6199v4.pdf

TLDR: The authors create adversarial examples, i.e., they slightly modify the original images, which look exactly the same to humans but neural networks can't classify the images correctly anymore. What does that imply on the ability to generalize? :)

On a more general note: NN are always treated as something magic. I think a "sober view" is that NN are a special way to parameterize non-liner functions. This is neither good nor bad but it's easy to see when you look at, for example, a 2 layer NN:

$f(x) = w_2^T \sigma(W_1 \sigma (W_0 x))$

pmelendez• 12 years ago

"which look exactly the same to humans but neural networks can't classify the images correctly anymore. What does that imply on the ability to generalize?"

Usually that means that the model is experimenting overfitting, and that's actually one challenge on any machine learning model.

pesenti2• 12 years ago

Does anybody know if this is true for other machine learning techniques?

mturmon6• 12 years ago

The same arguments as in the original Cybenko paper, or the Stone-Weierstrass theorem, lend support to the idea that SVMs are universal approximators (with most typical kernels). This has been proven by a couple of authors.

I'm not aware of universal approximation results for random forests, but since they have the same general construction, this would not be surprising.

Dn_Ab2• 12 years ago

This is a wonderful post. One minor aspect which nags me is that when I read "any function", I think any "effectively calculable method". But regular feedforward MLPs are not Turing Complete (will you be going over recurrent or recursive networks?). If so, it would be useful to note this distinction as I've never seen that point for confusion dealt with properly in one place.

sz4kerto• 12 years ago

Feedforward NN's are only useful to a very narrow* set of problems (*-> compared to 'all' problems out there).

Recurrent networks are needed for stateful operation, i.e. where some kind of memory is needed (in any case where the input is spread across some time or the sequence of data is important). And learning in recurrent nets is in very early stages unfortunately.

Houshalter4• 12 years ago

They are universal function approximators, which means they can map any set of input values to any set of output values. Of course to do this sometimes requires rote memorization of every possible input and it's output, rather than generalizing the function with a few parameters.

Adding more layers improves on this and allows you to make functions that compose multiple smaller functions. The problem with this is the nonlinearities cause the gradients to explode or vanish after a few layers. So the amount of computing power required to train them is huge.

Recurrent NNs had the same problem since they are equivalent to a very deep feed forward network; where every layer is a time step and the weights between every layer are the same.

But the invention of Long Short Term Memory has made training RNNs practical. Basically, as I understand it, some connections do not use nonlinearities so the gradients don't explode or vanish.

bmease• 12 years ago

I loved reading that and interacting with the plots. It totally changes the dynamic of learning when you can interact and play with it in real time.

numlocked• 12 years ago

This is also why Neural Nets are susceptible to overfitting and fell out of vogue in the 90s :) They will merrily fit themselves, very precisely, to your noisy, wiggly data. Obviously there are ways to combat this, but it seems like an corollary to their 'universality'.

Tloewald• 12 years ago

Can't the same be said for Fourier series, which make no claims to be some kind of AI? And likewise humble polynomials:

http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theor...

mtdewcmu6• 12 years ago

Yes (I am not expert in neural nets, but that appears to be exactly what this is saying). If you look at what goes into a neural net and compare it to what goes into a Fourier transform, it should be obvious that neural nets have even more than they actually need to do this task.

j2kun3• 12 years ago

This statement doesn't make sense to me. A neural network literally can't produce anything besides a continuous function, and the universality theorem says that there is no continuous function they can't (approximately) produce.

So what could you possibly mean when you say neural networks have "more" than they need to do something which characterizes exactly what they can and can't do?

signa11• 12 years ago

may i also humbly suggest the two vol. series called "parallel distributed processing" (rumelhart et al) which provides an excellent overview of early NN research.

mostly_harmless2• 12 years ago

I wrote about this a few weeks ago:

http://serialprog.blogspot.ca/2014/07/neural-networks-like-c...

My point of view was that neurons in neural nets are essentially analogue logic gates. Given that combinations of logic gates are Turing complete, combinations of neural net neurons should be also.

My writing is not quite as nice or rigorous as the parent post, but the whole point of the blog is to get better at self-expression and explaining things.

coherentpony• 12 years ago

Computing a function and approximating a function are two different disciplines.

When approximating a function, one must also talk about the sense in which the approximation is being made. That is, L^2? H^1? Pointwise?

mkoryak• 12 years ago

For me "visual proof" was a java program I wrote in 2005 that used genetic algorithms to evolve a neural network checker AI. It beat me every time:

https://github.com/mkoryak/Evolutionary-Neural-Net-Checker-A...

theophrastus• 12 years ago

If neural nets can compute any function (as seems neatly proven here) then can they compute any function in more than a single way? If so, then upon applying the novel input (which was our goal following training) how can we know that the particular way which was computed via training set is 'right' for our novel input? If this is all true then it would seem to make neural nets perfectly unreliable as a means to modeling..?

A visual proof that neural nets can compute any function

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