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).
Seems to me that jumping to whether they can "learn any function" as you asked, would certainly have come after this is first established.
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.
Minsky and Papert hate him! Use this one weird trick ….
(Sorry; I couldn't resist!)
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/
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.
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.
The next quantum leap is expected with the introduction of more specialized hardware such as neuromorphic chips: http://www.technologyreview.com/view/428235/intel-reveals-ne...
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:
$f(x) = w_2^T \sigma(W_1 \sigma (W_0 x))$
Usually that means that the model is experimenting overfitting, and that's actually one challenge on any machine learning model.
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.
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.
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."
> "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.