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Consider a neural network, e.g. as presented by Nielsen here. Abstractly, we just construct some function $f: \mathbb{R}^n \to [0,1]^m$ for some $n,m \in \mathbb{N}$ (i.e. the dimensions of the input and output space) that depends on a large set of parameters, $p_j$. We then just define the cost function $C$ and calculate $\nabla_p C$ and just map $p \to p - \epsilon \nabla_p C$ repeatedly.

The question is why do we choose $f$ to be what it is in standard neural networks, e.g. a bunch of linear combinations and sigmoids? One answer is that there a theorem saying any suitably nice function can be approximated using neural networks. But the same is true of other types of functions $f$. The Stone-Weierstrass theorem gives that we could use polynomials in $n$ variables: $$f(x) = c^0_0 + (c^1_1 x_1 + c^1_2 x_2 + \cdots + c^1_n x_n) + (c^2_{11}x_1 x_1 + c^2_{12} x_1x_2 + \cdots + c^2_{1n} x_1 x_2 + c^2_{21} x_2x_1 + c^2_{22} x_2x_2 + \cdots) + \cdots,$$

and still have a nice approximation theorem. Here the gradient would be even easier to calculate. Why not use polynomials?

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    $\begingroup$ Because polynomials are limited by their degree. The approximation ability of functions are generally given by a measure called VC dimension, and 2 degree polynomials have VC dimension 2/3 (I forget the exact number). So you basically have to take infinite degree polynomial combinations. NNs can be somewhat flexible in this regard that you don't have to manually choose functions, you can pretty much approximate a polynomial within an interval given sufficient nodes. This is the general theory, there are much more detailed nuances which goes against this aforementioned theory. $\endgroup$ – user9947 Aug 29 '20 at 21:38
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    $\begingroup$ Here is a related question What are the differences between artificial neural networks and other function approximators?. $\endgroup$ – nbro Aug 30 '20 at 2:34
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You can indeed fit a polynomial to your labelled data, which is known as polynomial regression (which can e.g. be done with the function numpy.polyfit). One apparent limitation of polynomial regression is that, in practice, you need to assume that your data follows some specific polynomial of some degree $n$, i.e. you assume that your data has the form of the polynomial that you choose, which may not be true.

When you use a neural network to solve a classification or regression problem, you also need to choose the activation functions, the number of neurons, how they are connected, etc., so you also need to limit the number and type of functions that you can learn with neural networks, i.e. the hypothesis space.

Now, it is not necessarily a bad thing to limit the hypothesis space. In fact, learning is generally an ill-posed problem, i.e. in simple terms, there could be multiple solutions or no solutions at all (and other problems), so, actually, you often need to limit the hypothesis space to find some useful solutions (e.g. solutions that generalise better to unseen data). Regularisations techniques are ways of constraining the learning problem, and the hypothesis space (i.e. the set of functions that your learning algorithm can choose from), and thus making the learning problem well-posed.

Neural networks are not preferred over polynomial regression because they are theoretically more powerful. In fact, both can approximate any continuous function [1], but these are just theoretical results, i.e. these results do not give you the magical formula to choose the most appropriate neural network or polynomial that best approximates the desired unknown function.

In practice, neural networks have been proven to effectively solve many tasks (e.g. translation of natural language, playing go or atari games, image classification, etc.), so I would say that this is the main reason they are widely studied and there is a lot of interest in them. However, neural networks typically require large datasets to approximate well the desired but unknown function, it can be computationally expensive to train or perform inference with them, and there are other limitations (see this), so neural networks are definitely not perfect tools, and there is the need to improve them to make them more efficient and useful in certain scenarios (e.g. scenarios where uncertainty estimation is required).

I am not really familiar with research on polynomial regression, but it is possible that this and other tools have been overlooked by the ML community. You may want to have a look at this paper, which states that NNs are essentially doing polynomial regression, though I have not read it, so I don't know the details about the main ideas and results in this paper.

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