Universal Approximation Theorem

In the mathematical theory of artificial neural networks, universal approximation theorems are results that establish the density of an algorithmically generated class of functions within a given function space of interest. Typically, these results concern the approximation capabilities of the feedforward architecture on the space of continuous functions between two Euclidean spaces, and the approximation is with respect to the compact convergence topology.

However, there are also a variety of results between non-Euclidean spaces and other commonly used architectures and, more generally, algorithmically generated sets of functions, such as the convolutional neural network (CNN) architecture, radial basis-functions, or neural networks with specific properties. Most universal approximation theorems can be parsed into two classes. The first quantifies the approximation capabilities of neural networks with an arbitrary number of artificial neurons ("''arbitrary width''" case) and the second focuses on the case with an arbitrary number of hidden layers, each containing a limited number of artificial neurons ("''arbitrary depth''" case). In addition to these two classes, there are also universal approximation theorems for neural networks with bounded number of hidden layers and a limited number of neurons in each layer ("''bounded depth and bounded width''" case).

Universal approximation theorems imply that neural networks can ''represent'' a wide variety of interesting functions when given appropriate weights. On the other hand, they typically do not provide a construction for the weights, but merely state that such a construction is possible.

History

One of the first versions of the ''arbitrary width'' case was proven by George Cybenko in 1989 for sigmoid activation functions. Kurt Hornik, Maxwell Stinchcombe, and Halbert White showed in 1989 that multilayer feed-forward networks with as few as one hidden layer are universal approximators. Hornik also showed in 1991 that it is not the specific choice of the activation function but rather the multilayer feed-forward architecture itself that gives neural networks the potential of being universal approximators. Moshe Leshno ''et al'' in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function.

The ''arbitrary depth'' case was also studied by a number of authors, such as Gustaf Gripenberg in 2003, Dmitry Yarotsky, Zhou Lu ''et al'' in 2017, Boris Hanin and Mark Sellke in 2018, and Patrick Kidger and Terry Lyons in 2020. The result minimal width per layer was refined in 2020 for residual networks.

The bounded depth and bounded width case was first studied by Maiorov and Pinkus in 1999. They showed that there exists an analytic sigmoidal activation function such that two hidden layer neural networks with bounded number of units in hidden layers are universal approximators.
Using algorithmic and computer programming techniques, Guliyev and Ismailov constructed a smooth sigmoidal activation function providing universal approximation property for two hidden layer feedforward neural networks with less units in hidden layers. It was constructively proved in 2018 paper that single hidden layer networks with bounded width are still universal approximators for univariate functions, but this property is no longer true for multivariable functions.

Several extensions of the theorem exist, such as to discontinuous activation functions, noncompact domains, certifiable networks,
random neural networks, and alternative network architectures and topologies.

Arbitrary-width case

The classical form of the universal approximation theorem for arbitrary width and bounded depth is as follows. It extends the classical results of George Cybenko and Kurt Hornik.

Universal approximation theorem: Let $C(X, Y)$ denote the set of continuous functions from $X$ to $Y$. Let $\sigma \in C(\mathbb, \mathbb)$. Note that $(\sigma \circ x)_i = \sigma(x_i)$, so $\sigma \circ x$ denotes $\sigma$ applied to each component of $x$.

Then $\sigma$ is not polynomial if and only if for every $n \in \mathbb$, $m \in \mathbb$, compact $K \subseteq \mathbb^n$, $f \in C(K, \mathbb^m), \varepsilon > 0$ there exist $k \in \mathbb$, $A \in \mathbb^$, $b \in \mathbb^k$, $C \in \mathbb^$ such that
$\sup_ \, f(x) - g(x) \, < \varepsilon$
where
$g(x) = C \cdot ( \sigma \circ (A \cdot x + b) )$

Such an $f$ can also be approximated by a network of greater depth by using the same construction for the first layer and approximating the identity function with later layers.

Arbitrary-depth case

The 'dual' versions of the theorem consider networks of bounded width and arbitrary depth. A variant of the universal approximation theorem was proved for the arbitrary depth case by Zhou Lu et al. in 2017. They showed that networks of width ''n+4'' with ReLU activation functions can approximate any Lebesgue integrable function on ''n''-dimensional input space with respect to $L^$ distance if network depth is allowed to grow. It was also shown that if the width was less than or equal to ''n'', this general expressive power to approximate any Lebesgue integrable function was lost. In the same paper it was shown that ReLU networks with width ''n+1'' were sufficient to approximate any continuous function of ''n''-dimensional input variables. The following refinement, specifies the optimal minimum width for which such an approximation is possible and is due to.

Universal approximation theorem ''(L1 distance, ReLU activation, arbitrary depth, minimal width).'' For any Bochner–Lebesgue p-integrable function $f : \mathbb ^ \rightarrow \mathbb ^$ and any $\epsilon > 0$, there exists a fully-connected ReLU network $F$ of width exactly $d _ = \max\$, satisfying

:$\int _ \left\, f ( x ) - F _ ( x ) \right\, ^p \mathrm x < \epsilon$.
Moreover, there exists a function $f \in L^p(\mathbb^n,\mathbb^m)$ and some $\epsilon >0$, for which there is no fully-connected ReLU network of width less than $d _ = \max\$ satisfying the above approximation bound.

''Quantitative Refinement:'' In the case where, when \mathcal= ,1d and $D=1$ and where $\sigma$ is the ReLU activation function then, the exact depth and width for a ReLU network to achive $\varepsilon$ error is also known. If, moreover, the target function $f$ is smooth then the required number of layer and their width can be exponentially smaller. Even if $f$ is not smooth, the curse of dimensionality can be broken if f admits additional "compositional structure".

Together, the central result of yields the following universal approximation theorem for networks with bounded width (cf. also for the first result of this kind).

Universal approximation theorem (Uniform non-affine activation, arbitrary depth, constrained width). Let $\mathcal$ be a compact subset of $\mathbb^d$. Let $\sigma:\mathbb\to\mathbb$ be any non-affine continuous function which is continuously differentiable at at least one point, with nonzero derivative at that point. Let $\mathcal_^$ denote the space of feed-forward neural networks with $d$ input neurons, $D$ output neurons, and an arbitrary number of hidden layers each with $d + D + 2$ neurons, such that every hidden neuron has activation function $\sigma$ and every output neuron has the identity as its activation function, with input layer $\phi$, and output layer $\rho$. Then given any $\varepsilon>0$ and any $f\in C(\mathcal,\mathbb^D)$, there exists $\hat\in \mathcal_^$ such that

: $\sup_\,\left\, \hat(x)-f(x)\right\, < \varepsilon.$

In other words, $\mathcal$ is dense in $C(\mathcal; \mathbb^D)$ with respect to the topology of uniform convergence.

''Quantitative Refinement:'' The number of layers and the width of each layer required to approximate f to $\varepsilon$ precision known; moreover, the result hold true when $\mathcal$ and $\mathbb^D$are replaced with any non-positively curved Riemannian manifold.

Certain necessary conditions for the bounded width, arbitrary depth case have been established, but there is still a gap between the known sufficient and necessary conditions.

Bounded depth and bounded width case

The first result on approximation capabilities of neural networks with bounded number of layers, each containing a limited number of artificial neurons was obtained by Maiorov and Pinkus. Their remarkable result revealed that such networks can be universal approximators and for achieving this property two hidden layers are enough.

Universal approximation theorem: There exists an activation function $\sigma$ which is analytic, strictly increasing and
sigmoidal and has the following property: For any f\in C ,1 and $\varepsilon >0$ there exist constants $d_, c_, \theta _, \gamma _$, and vectors $\mathbf^\in \mathbb^$ for which

$\left\vert f(\mathbf)-\sum_^d_\sigma \left( \sum_^c_\sigma (\mathbf^\cdot \mathbf\theta _)-\gamma _\right) \right\vert <\varepsilon$

for all \mathbf=(x_,...,x_)\in ,1.

This is an existence result. It says that activation functions providing universal approximation property for bounded depth bounded width networks exist. Using certain algorithmic and computer programming techniques, Guliyev and Ismailov efficiently constructed such activation functions depending on a numerical parameter. The developed algorithm allows one to compute the activation functions at any point of the real axis instantly. For the algorithm and the corresponding computer code see. The theoretical result can be formulated as follows.

Universal approximation theorem: Let ,b/math> be a finite segment of the real line, $s=b-a$ and $\lambda$ be any positive number. Then one can algorithmically construct a computable sigmoidal activation function $\sigma \colon \mathbb \to \mathbb$, which is infinitely differentiable, strictly increasing on $(-\infty, s)$, $\lambda$ -strictly increasing on ,+\infty) , and satisfies the following properties:

1) For any $f \in C[a,b$ and $\varepsilon > 0$ there exist numbers $c_1,c_2,\theta_1$ and $\theta_2$ such that for all $x \in [a,b]$
$, f(x) - c_1 \sigma(x - \theta_1) - c_2 \sigma(x - \theta_2), < \varepsilon$

2) For any continuous function $F$ on the $d$-dimensional box ,b and $\varepsilon >0$, there exist constants $e_p$, $c_$, $\theta_$ and $\zeta_p$ such that the inequality
$\left, F(\mathbf) - \sum_^ e_p \sigma \left( \sum_^ c_ \sigma(\mathbf^ \cdot \mathbf - \theta_) - \zeta_p \right) \ < \varepsilon$
holds for all $\mathbf = (x_1, \ldots, x_d) \in$, b. Here the weights $\mathbf^$, $q = 1, \ldots, d$, are fixed as follows:
$\mathbf^ = (1, 0, \ldots, 0), \quad \mathbf^ = (0, 1, \ldots, 0), \quad \ldots, \quad \mathbf^ = (0, 0, \ldots, 1).$
In addition, all the coefficients $e_p$, except one, are equal.

Here “$\sigma \colon \mathbb \to \mathbb$ is $\lambda$-strictly increasing on some set $X$” means that there exists a strictly increasing function $u \colon X \to \mathbb$ such that $, \sigma(x) - u(x), \le \lambda$ for all $x \in X$. Clearly, a $\lambda$-increasing function behaves like a usual increasing function as $\lambda$ gets small.
In the "''depth-width''" terminology, the above theorem says that for certain activation functions depth-$2$ width-$2$ networks are universal approximators for univariate functions and depth-$3$ width-$(2d+2)$ networks are universal approximators for $d$-variable functions ($d>1$).

Graph input

Achieving useful universal function approximation on graphs (or rather on graph isomorphism classes) has been a longstanding problem. The popular graph convolutional neural networks (GCNs or GNNs) can be made as discriminative as the Weisfeiler–Leman graph isomorphism test. In 2020, a universal approximation theorem result was established by Brüel-Gabrielsson, showing that graph representation with certain injective properties is sufficient for universal function approximation on bounded graphs and restricted universal function approximation on unbounded graphs, with an accompanying $O($#edges$\times$#nodes$)$-runtime method that performed at state of the art on a collection of benchmarks.

See also

* Kolmogorov–Arnold representation theorem
* Representer theorem
* No free lunch theorem
* Stone–Weierstrass theorem
* Fourier series

References

{{Differentiable computing

Theorems in analysis
Artificial neural networks
Network architecture
Networks