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What other algorithms can do this and which situations would they be more useful than neural networks?

gnyeki 2y ago
This area is covered by non-parametric statistics more generally. There are many other methods to non-parametrically estimate functions (that satisfy some regularity conditions). Tree-based methods are one family of such methods, and the consensus still seems to be that they perform better than neural networks on tabular data. For example: https://arxiv.org/abs/2106.03253 https://arxiv.org/abs/2106.03253
someoneontenet 2y ago
Newtons Method approximates square roots. Its useful if you want to approximate something like that without pulling in the computational power required of NN.
- kristjansson 2y ago
  Newton's method related to universal function approximation in the same way a natural oil seep is related to a modern IC engine...
- astrobe_ 2y ago
  I think the problem to solve is more like : given a set of inputs and outputs, find a function that gives the expected output for each input [1]. This is like Newton's method on a higher order ;-). One can find such a tool in Squeak or Pharo Smalltalk, IIRC. [1] https://stackoverflow.com/questions/1539286/create-a-function-for-given-input-and-ouput https://stackoverflow.com/questions/1539286/create-a-functio...
- jb1991 2y ago
  By definition, that’s not a “universal“ function approximator.
jb1991 2y ago
https://www.quora.com/Is-there-any-universal-function-approximators-other-than-a-neural-network https://www.quora.com/Is-there-any-universal-function-approx...
dontlikeyoueith 2y ago
The Taylor Series dates to 1715. Fourier Series dates to the 1820s. Both are universal function approximators and both can be learned via gradient descent. For the case where the function you want to learn actually is polynomial or periodic (respectively), these are better than neural networks.
- camjw 2y ago
  For your interest, Taylor Series are not universal function approximators - the Taylor Series around 0 for f(x) = e^(-1/x^2) if x != 0 else 0 is identically zero (all partial derivatives are 0 at 0) but the function is clearly not identically zero. So the radius of convergence for this Taylor series is infinite but it only equals the approximated function at one point. I'm sure there are some conditions you can put on f to make the Taylor Series a UFA but it's been quite a while since I did any real analysis so I have forgotten! Doesn't detract from the overall point though that there are UFAs that are not neural nets. I should say that I don't know what the precise definition of a UFA really is, but I assume you have to have more than equality at one point.
  - Horffupolde 2y ago
    Taylor series work on differentiable intervals. You specifically chose a function and interval where this is not true. Of course it will not be a good approximation.
  - dontlikeyoueith 2y ago
    I'm pretty sure the UFA theorems for neural networks wouldn't apply to that function either: https://en.wikipedia.org/wiki/Universal_approximation_theorem https://en.wikipedia.org/wiki/Universal_approximation_theore... Generally, they assume the function to be approximated is continuous.

This area is covered by non-parametric statistics more generally. There are many other methods to non-parametrically estimate functions (that satisfy some regularity conditions). Tree-based methods are one family of such methods, and the consensus still seems to be that they perform better than neural networks on tabular data. For example: https://arxiv.org/abs/2106.03253 https://arxiv.org/abs/2106.03253

someoneontenet 2y ago

Newtons Method approximates square roots. Its useful if you want to approximate something like that without pulling in the computational power required of NN.

kristjansson 2y ago
Newton's method related to universal function approximation in the same way a natural oil seep is related to a modern IC engine...
astrobe_ 2y ago
I think the problem to solve is more like : given a set of inputs and outputs, find a function that gives the expected output for each input [1]. This is like Newton's method on a higher order ;-). One can find such a tool in Squeak or Pharo Smalltalk, IIRC. [1] https://stackoverflow.com/questions/1539286/create-a-function-for-given-input-and-ouput https://stackoverflow.com/questions/1539286/create-a-functio...
jb1991 2y ago
By definition, that’s not a “universal“ function approximator.

jb1991 2y ago

https://www.quora.com/Is-there-any-universal-function-approximators-other-than-a-neural-network https://www.quora.com/Is-there-any-universal-function-approx...

dontlikeyoueith 2y ago

The Taylor Series dates to 1715. Fourier Series dates to the 1820s. Both are universal function approximators and both can be learned via gradient descent. For the case where the function you want to learn actually is polynomial or periodic (respectively), these are better than neural networks.

camjw 2y ago
For your interest, Taylor Series are not universal function approximators - the Taylor Series around 0 for f(x) = e^(-1/x^2) if x != 0 else 0 is identically zero (all partial derivatives are 0 at 0) but the function is clearly not identically zero. So the radius of convergence for this Taylor series is infinite but it only equals the approximated function at one point. I'm sure there are some conditions you can put on f to make the Taylor Series a UFA but it's been quite a while since I did any real analysis so I have forgotten! Doesn't detract from the overall point though that there are UFAs that are not neural nets. I should say that I don't know what the precise definition of a UFA really is, but I assume you have to have more than equality at one point.
- Horffupolde 2y ago
  Taylor series work on differentiable intervals. You specifically chose a function and interval where this is not true. Of course it will not be a good approximation.
- dontlikeyoueith 2y ago
  I'm pretty sure the UFA theorems for neural networks wouldn't apply to that function either: https://en.wikipedia.org/wiki/Universal_approximation_theorem https://en.wikipedia.org/wiki/Universal_approximation_theore... Generally, they assume the function to be approximated is continuous.