View source for Universal approximation theorem

{{Short description|Feed-forward neural network with a 1 hidden layer can approximate continuous functions}}
{{Technical|date=July 2023}}

In the [[mathematics|mathematical]] theory of [[artificial neural networks]], '''universal approximation theorems''' are theorems<ref name="MLP-UA">{{cite journal |last1=Hornik |first1=Kurt |last2=Stinchcombe |first2=Maxwell |last3=White |first3=Halbert |title=Multilayer feedforward networks are universal approximators |journal=Neural Networks |date=January 1989 |volume=2 |issue=5 |pages=359–366 |doi=10.1016/0893-6080(89)90020-8 }}</ref><ref>Balázs Csanád Csáji (2001) Approximation with Artificial Neural Networks; Faculty of Sciences; Eötvös Loránd University, Hungary</ref> of the following form: Given a family of neural networks, for each function <math>f</math> from a certain [[function space]], there exists a sequence of neural networks <math>\phi_1, \phi_2, \dots</math> from the family, such that <math>\phi_n \to f</math> according to some criterion. That is, the family of neural networks is [[Dense set|dense]] in the function space.

The most popular version states that [[feedforward neural network|feedforward networks]] with non-[[polynomial]] [[activation function]]s are dense in the space of [[continuous function]]s between two [[Euclidean space]]s, with respect to the [[compact convergence]] [[topology]].

Universal approximation theorems are existence theorems: They simply state that there ''exists'' such a sequence <math>\phi_1, \phi_2, \dots \to f</math>, and do not provide any way to actually find such a sequence. They also do not guarantee any method, such as [[backpropagation]], might actually find such a sequence. Any method for searching the space of neural networks, including backpropagation, might find a converging sequence, or not (i.e. the backpropagation might get stuck in a local optimum). 

Universal approximation theorems are limit theorems: They simply state that for any <math>f</math> and a criterion of closeness <math>\epsilon > 0</math>, if there are ''enough'' neurons in a neural network, then there exists a neural network with that many neurons that does approximate <math>f</math> to within <math>\epsilon</math>. There is no guarantee that any finite size, say, 10000 neurons, is enough.

== Setup ==
[[Artificial neural networks]] are combinations of multiple simple mathematical functions that implement more complicated functions from (typically) real-valued [[vector (mathematics and physics)|vectors]] to real-valued [[vector (mathematics and physics)|vectors]].  The spaces of multivariate functions that can be implemented by a network are determined by the structure of the network, the set of simple functions, and its multiplicative parameters. A great deal of theoretical work has gone into characterizing these function spaces.

Most universal approximation theorems are in one of 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).

== History ==

=== Arbitrary width ===
The first examples were the ''arbitrary width'' case. [[George Cybenko]] in 1989 proved it for [[sigmoid function|sigmoid]] activation functions.<ref name="cyb">{{cite journal |citeseerx=10.1.1.441.7873 |doi=10.1007/BF02551274|title=Approximation by superpositions of a sigmoidal function|year=1989|last1=Cybenko|first1=G.|journal=Mathematics of Control, Signals, and Systems|volume=2|issue=4|pages=303–314|bibcode=1989MCSS....2..303C |s2cid=3958369}}</ref> {{ill|Kurt Hornik|de}}, Maxwell Stinchcombe, and [[Halbert White]] showed in 1989 that multilayer [[feed-forward network]]s with as few as one hidden layer are universal approximators.<ref name="MLP-UA" /> Hornik also showed in 1991<ref name="horn">{{Cite journal|doi=10.1016/0893-6080(91)90009-T|title=Approximation capabilities of multilayer feedforward networks|year=1991|last1=Hornik|first1=Kurt|journal=Neural Networks|volume=4|issue=2|pages=251–257|s2cid=7343126 }}</ref> 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<ref name="leshno">{{Cite journal|last1=Leshno|first1=Moshe|last2=Lin|first2=Vladimir Ya.|last3=Pinkus|first3=Allan|last4=Schocken|first4=Shimon|date=January 1993|title=Multilayer feedforward networks with a nonpolynomial activation function can approximate any function|journal=Neural Networks|volume=6|issue=6|pages=861–867|doi=10.1016/S0893-6080(05)80131-5|s2cid=206089312|url=http://archive.nyu.edu/handle/2451/14329 }}</ref> and later Allan Pinkus in 1999<ref name="pinkus">{{Cite journal|last=Pinkus|first=Allan|date=January 1999|title=Approximation theory of the MLP model in neural networks|journal=Acta Numerica|volume=8|pages=143–195|doi=10.1017/S0962492900002919|bibcode=1999AcNum...8..143P|s2cid=16800260 }}</ref> showed that the universal approximation property is equivalent to having a nonpolynomial activation function.

=== Arbitrary depth ===
The ''arbitrary depth'' case was also studied by a number of authors such as Gustaf Gripenberg in 2003,<ref name= gripenberg >{{Cite journal|last1=Gripenberg|first1=Gustaf|date=June 2003|title= Approximation by neural networks with a bounded number of nodes at each level|journal= Journal of Approximation Theory |volume=122|issue=2|pages=260–266|doi= 10.1016/S0021-9045(03)00078-9 |doi-access=}}</ref> Dmitry Yarotsky,<ref>{{cite journal |last1=Yarotsky |first1=Dmitry |title=Error bounds for approximations with deep ReLU networks |journal=Neural Networks |date=October 2017 |volume=94 |pages=103–114 |doi=10.1016/j.neunet.2017.07.002 |pmid=28756334 |arxiv=1610.01145 |s2cid=426133 }}</ref> Zhou Lu ''et al'' in 2017,<ref name="ZhouLu">{{cite journal |last1=Lu |first1=Zhou |last2=Pu |first2=Hongming |last3=Wang |first3=Feicheng |last4=Hu |first4=Zhiqiang |last5=Wang |first5=Liwei |title=The Expressive Power of Neural Networks: A View from the Width |journal=Advances in Neural Information Processing Systems |volume=30 |year=2017 |pages=6231–6239 |url=http://papers.nips.cc/paper/7203-the-expressive-power-of-neural-networks-a-view-from-the-width |publisher=Curran Associates |arxiv=1709.02540 }}</ref> Boris Hanin and Mark Sellke in 2018<ref name=hanin>{{cite arXiv |last1=Hanin|first1=Boris|last2=Sellke|first2=Mark|title=Approximating Continuous Functions by ReLU Nets of Minimal Width|eprint=1710.11278|class=stat.ML|date=2018}}</ref> who focused on neural networks with ReLU activation function.  In 2020, Patrick Kidger and Terry Lyons<ref name=kidger>{{Cite conference|last1=Kidger|first1=Patrick|last2=Lyons|first2=Terry|date=July 2020|title=Universal Approximation with Deep Narrow Networks|arxiv=1905.08539|conference=Conference on Learning Theory}}</ref> extended those results to neural networks with ''general activation functions'' such, e.g. tanh or GeLU.

One special case of arbitrary depth is that each composition component comes from a finite set of mappings. In 2024, Cai <ref name= cai2024 >{{Cite journal|last1=Yongqiang|first1=Cai|date=2024|title= Vocabulary for Universal Approximation: A Linguistic Perspective of Mapping Compositions|journal= ICML|pages=5189–5208 |arxiv=2305.12205 |url= https://proceedings.mlr.press/v235/cai24a.html}}</ref> constructed a finite set of mappings, named a vocabulary, such that any continuous function can be approximated by compositing a sequence from the vocabulary. This is similar to the concept of compositionality in linguistics, which is the idea that a finite vocabulary of basic elements can be combined via grammar to express an infinite range of meanings.

=== Bounded depth and bounded width ===
The bounded depth and bounded width case was first studied by Maiorov and Pinkus in 1999.<ref name="maiorov">{{Cite journal|last1=Maiorov|first1=Vitaly|last2=Pinkus|first2=Allan|date=April 1999|title=Lower bounds for approximation by MLP neural networks|journal=Neurocomputing|volume=25|issue=1–3|pages=81–91|doi=10.1016/S0925-2312(98)00111-8}}</ref> 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.

In 2018, Guliyev and Ismailov<ref name="guliyev1">{{Cite journal |last1=Guliyev |first1=Namig |last2=Ismailov |first2=Vugar |date=November 2018 |title=Approximation capability of two hidden layer feedforward neural networks with fixed weights |journal=Neurocomputing |volume=316 |pages=262–269 |arxiv=2101.09181 |doi=10.1016/j.neucom.2018.07.075 |s2cid=52285996}}</ref> constructed a smooth sigmoidal activation function providing universal approximation property for two hidden layer feedforward neural networks with less units in hidden layers. In 2018, they also constructed<ref name="guliyev2">{{Cite journal|last1=Guliyev|first1=Namig|last2=Ismailov|first2=Vugar|date=February 2018|title=On the approximation by single hidden layer feedforward neural networks with fixed weights|journal=Neural Networks|volume=98| pages=296–304|doi=10.1016/j.neunet.2017.12.007|pmid=29301110 |arxiv=1708.06219 |s2cid=4932839 }}</ref> single hidden layer networks with bounded width that are still universal approximators for univariate functions. However, this does not apply for multivariable functions.

In 2022, Shen ''et al.''<ref>{{cite journal |last1=Shen |first1=Zuowei |last2=Yang |first2=Haizhao |last3=Zhang |first3=Shijun |date=January 2022 |title=Optimal approximation rate of ReLU networks in terms of width and depth |journal=Journal de Mathématiques Pures et Appliquées |volume=157 |pages=101–135 |arxiv=2103.00502 |doi=10.1016/j.matpur.2021.07.009 |s2cid=232075797}}</ref> obtained precise quantitative information on the depth and width required to approximate a target function by deep and wide ReLU neural networks.

=== Quantitative bounds ===
The question of minimal possible width for universality was first studied in 2021, Park et al obtained the minimum width required for the universal approximation of ''[[Lp space|L<sup>p</sup>]]'' functions using feed-forward neural networks with [[Rectifier (neural networks)|ReLU]] as activation functions.<ref name="park">{{Cite conference |last1=Park |first1=Sejun |last2=Yun |first2=Chulhee |last3=Lee |first3=Jaeho |last4=Shin |first4=Jinwoo |date=2021 |title=Minimum Width for Universal Approximation |conference=International Conference on Learning Representations |arxiv=2006.08859}}</ref> Similar results that can be directly applied to [[residual neural network]]s were also obtained in the same year by Paulo Tabuada and Bahman Gharesifard using [[Control theory|control-theoretic]] arguments.<ref>{{Cite conference |last1=Tabuada |first1=Paulo |last2=Gharesifard |first2=Bahman |date=2021 |title=Universal approximation power of deep residual neural networks via nonlinear control theory |conference=International Conference on Learning Representations |arxiv=2007.06007}}</ref><ref>{{cite journal |last1=Tabuada |first1=Paulo |last2=Gharesifard |first2=Bahman |date=May 2023 |title=Universal Approximation Power of Deep Residual Neural Networks Through the Lens of Control |journal=IEEE Transactions on Automatic Control |volume=68 |issue=5 |pages=2715–2728 |doi=10.1109/TAC.2022.3190051 |s2cid=250512115}}{{Erratum|doi=10.1109/TAC.2024.3390099|checked=yes}}</ref> In 2023, Cai obtained the optimal minimum width bound for the universal approximation.<ref name=":1">{{Cite journal |last=Cai |first=Yongqiang |date=2023-02-01 |title=Achieve the Minimum Width of Neural Networks for Universal Approximation |url=https://openreview.net/forum?id=hfUJ4ShyDEU |journal=ICLR |language=en |arxiv=2209.11395}}</ref>

For the arbitrary depth case, Leonie Papon and Anastasis Kratsios derived explicit depth estimates depending on the regularity of the target function and of the activation function.<ref name="jmlr.org">{{Cite journal |last1=Kratsios |first1=Anastasis |last2=Papon |first2=Léonie |date=2022 |title=Universal Approximation Theorems for Differentiable Geometric Deep Learning |url=http://jmlr.org/papers/v23/21-0716.html |journal=Journal of Machine Learning Research |volume=23 |issue=196 |pages=1–73 |arxiv=2101.05390}}</ref>

=== Kolmogorov network ===
The [[Kolmogorov–Arnold representation theorem]] is similar in spirit. Indeed, certain neural network families can directly apply the Kolmogorov–Arnold theorem to yield a universal approximation theorem. [[Robert Hecht-Nielsen]] showed that a three-layer neural network can approximate any continuous multivariate function.<ref>{{Cite journal |last=Hecht-Nielsen |first=Robert |date=1987 |title=Kolmogorov's mapping neural network existence theorem |url=https://cir.nii.ac.jp/crid/1572543025788928512 |journal=Proceedings of International Conference on Neural Networks, 1987 |volume=3 |pages=11–13}}</ref> This was extended to the discontinuous case by Vugar Ismailov.<ref>{{cite journal |last1=Ismailov |first1=Vugar E. |date=July 2023 |title=A three layer neural network can represent any multivariate function |journal=Journal of Mathematical Analysis and Applications |volume=523 |issue=1 |pages=127096 |arxiv=2012.03016 |doi=10.1016/j.jmaa.2023.127096 |s2cid=265100963}}</ref> In 2024, Ziming Liu and co-authors showed a practical application.<ref>{{cite arXiv |last1=Liu |first1=Ziming |title=KAN: Kolmogorov-Arnold Networks |date=2024-05-24 |eprint=2404.19756 |last2=Wang |first2=Yixuan |last3=Vaidya |first3=Sachin |last4=Ruehle |first4=Fabian |last5=Halverson |first5=James |last6=Soljačić |first6=Marin |last7=Hou |first7=Thomas Y. |last8=Tegmark |first8=Max|class=cs.LG }}</ref>

=== Reservoir computing and quantum reservoir computing===
In reservoir computing a sparse recurrent neural network with fixed weights equipped of fading memory and echo state property is followed by a trainable output layer. Its universality has been demonstrated separately for what concerns networks of rate neurons <ref>{{Cite journal |last1=Grigoryeva |first1=L. |last2=Ortega |first2=J.-P. |date=2018 |title=Echo state networks are universal |journal=Neural Networks |volume=108 |issue=1 |pages=495–508 |arxiv=1806.00797 |doi=10.1016/j.neunet.2018.08.025}}</ref> and spiking neurons, respectively. <ref>{{Cite journal |last=Maass |first=Wolfgang |last2=Markram |first2=Henry  |date=2004 |title=On the computational power of circuits of spiking neurons |url=http://www.igi.tugraz.at/maass/psfiles/135.pdf |journal=Journal of computer and system sciences |volume=69 |pages=593–616}}</ref> In 2024, the framework has been generalized and extended to quantum reservoirs where the reservoir is based on qubits defined over Hilbert spaces. <ref>{{cite arXiv |last1=Monzani |first1=Francesco |title=Universality conditions of unified classical and quantum reservoir computing |date=2024|eprint=2401.15067 |last2=Prati |first2=Enrico |class=quant-ph }}</ref>

=== Variants ===
Discontinuous activation functions,<ref name="leshno" /> noncompact domains,<ref name="kidger" /><ref>{{Cite journal |last1=van Nuland |first1=Teun |year=2024 |title=Noncompact uniform universal approximation |url=https://doi.org/10.1016/j.neunet.2024.106181 |journal=Neural Networks |volume=173|doi=10.1016/j.neunet.2024.106181 |pmid=38412737 |arxiv=2308.03812 }}</ref> certifiable networks,<ref>{{cite conference |last1=Baader |first1=Maximilian |last2=Mirman |first2=Matthew |last3=Vechev |first3=Martin |date=2020 |title=Universal Approximation with Certified Networks |url=https://openreview.net/forum?id=B1gX8kBtPr |conference=ICLR}}</ref> 
random neural networks,<ref>{{Cite journal |last1=Gelenbe |first1=Erol |last2=Mao |first2=Zhi Hong |last3=Li |first3=Yan D. |year=1999 |title=Function approximation with spiked random networks |url=https://zenodo.org/record/6817275 |journal=IEEE Transactions on Neural Networks |volume=10 |issue=1 |pages=3–9 |doi=10.1109/72.737488 |pmid=18252498}}</ref> and alternative network architectures and topologies.<ref name="kidger" /><ref>{{Cite conference |last1=Lin |first1=Hongzhou |last2=Jegelka |first2=Stefanie|author2-link=Stefanie Jegelka |date=2018 |title=ResNet with one-neuron hidden layers is a Universal Approximator |url=https://papers.nips.cc/paper/7855-resnet-with-one-neuron-hidden-layers-is-a-universal-approximator |publisher=Curran Associates |volume=30 |pages=6169–6178 |journal=Advances in Neural Information Processing Systems}}</ref>

The universal approximation property of width-bounded networks has been studied as a ''dual'' of classical universal approximation results on depth-bounded networks. For input dimension dx and output dimension dy the minimum width required for the universal approximation of the ''[[Lp space|L<sup>p</sup>]]'' functions is exactly max{dx + 1, dy} (for a ReLU network). <!-- ReLU alone is not sufficient in general "In light of Theorem 2, is it impossible to approximate C(K, R dy) in general while maintaining width max{dx + 1, dy}? Theorem 3 shows that an additional activation comes to rescue." --> More generally this also holds if ''both'' ReLU and a [[step function|threshold activation function]] are used.<ref name="park" />

Universal function approximation on graphs (or rather on [[Graph isomorphism|graph isomorphism classes]]) by popular [[Graph neural network|graph convolutional neural networks]] (GCNs or GNNs) can be made as discriminative as the Weisfeiler–Leman graph isomorphism test.<ref name="PowerGNNs">{{Cite conference |last1=Xu |first1=Keyulu |last2=Hu |first2=Weihua |last3=Leskovec |first3=Jure |last4=Jegelka |first4=Stefanie|author4-link=Stefanie Jegelka |date=2019 |title=How Powerful are Graph Neural Networks? |url=https://openreview.net/forum?id=ryGs6iA5Km |journal=International Conference on Learning Representations}}</ref> In 2020,<ref name="UniversalGraphs">{{Cite conference |last1=Brüel-Gabrielsson |first1=Rickard |date=2020 |title=Universal Function Approximation on Graphs |url=https://proceedings.neurips.cc//paper/2020/hash/e4acb4c86de9d2d9a41364f93951028d-Abstract.html |publisher=Curran Associates |volume=33 |journal=Advances in Neural Information Processing Systems}}</ref> 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 <math>\mathcal O(\left|V\right| \cdot \left|E\right|)</math>-runtime method that performed at state of the art on a collection of benchmarks (where <math>V</math> and <math>E</math> are the sets of nodes and edges of the graph respectively).

There are also a variety of results between [[non-Euclidean space]]s<ref name="NonEuclidean">{{Cite conference |last1=Kratsios |first1=Anastasis |last2=Bilokopytov |first2=Eugene |date=2020 |title=Non-Euclidean Universal Approximation |url=https://papers.nips.cc/paper/2020/file/786ab8c4d7ee758f80d57e65582e609d-Paper.pdf |publisher=Curran Associates |volume=33 |journal=Advances in Neural Information Processing Systems}}</ref> and other commonly used architectures and, more generally, algorithmically generated sets of functions, such as the [[convolutional neural network]] (CNN) architecture,<ref>{{cite journal |last1=Zhou |first1=Ding-Xuan |year=2020 |title=Universality of deep convolutional neural networks |journal=[[Applied and Computational Harmonic Analysis]] |volume=48 |issue=2 |pages=787–794 |arxiv=1805.10769 |doi=10.1016/j.acha.2019.06.004 |s2cid=44113176}}</ref><ref>{{Cite journal |last1=Heinecke |first1=Andreas |last2=Ho |first2=Jinn |last3=Hwang |first3=Wen-Liang |year=2020 |title=Refinement and Universal Approximation via Sparsely Connected ReLU Convolution Nets |journal=IEEE Signal Processing Letters |volume=27 |pages=1175–1179 |bibcode=2020ISPL...27.1175H |doi=10.1109/LSP.2020.3005051 |s2cid=220669183}}</ref> [[radial basis functions]],<ref>{{Cite journal |last1=Park |first1=J. |last2=Sandberg |first2=I. W. |year=1991 |title=Universal Approximation Using Radial-Basis-Function Networks |journal=Neural Computation |volume=3 |issue=2 |pages=246–257 |doi=10.1162/neco.1991.3.2.246 |pmid=31167308 |s2cid=34868087}}</ref> or neural networks with specific properties.<ref>{{cite journal |last1=Yarotsky |first1=Dmitry |year=2021 |title=Universal Approximations of Invariant Maps by Neural Networks |journal=Constructive Approximation |volume=55 |pages=407–474 |arxiv=1804.10306 |doi=10.1007/s00365-021-09546-1 |s2cid=13745401}}</ref><ref>{{cite journal |last1=Zakwan |first1=Muhammad |last2=d’Angelo |first2=Massimiliano |last3=Ferrari-Trecate |first3=Giancarlo |date=2023 |title=Universal Approximation Property of Hamiltonian Deep Neural Networks |journal=IEEE Control Systems Letters |page=1 |arxiv=2303.12147 |doi=10.1109/LCSYS.2023.3288350 |s2cid=257663609}}</ref>

== Arbitrary-width case ==
A spate of papers in the 1980s—1990s, from [[George Cybenko]] and {{ill|Kurt Hornik|de}} etc, established several universal approximation theorems for arbitrary width and bounded depth.<ref>{{cite journal |last1=Funahashi |first1=Ken-Ichi |title=On the approximate realization of continuous mappings by neural networks |journal=Neural Networks |date=January 1989 |volume=2 |issue=3 |pages=183–192 |doi=10.1016/0893-6080(89)90003-8 }}</ref><ref name=cyb /><ref name=":0">{{cite journal |last1=Hornik |first1=Kurt |last2=Stinchcombe |first2=Maxwell |last3=White |first3=Halbert |title=Multilayer feedforward networks are universal approximators |journal=Neural Networks |date=January 1989 |volume=2 |issue=5 |pages=359–366 |doi=10.1016/0893-6080(89)90020-8 }}</ref><ref name=horn /> See<ref>Haykin, Simon (1998). ''Neural Networks: A Comprehensive Foundation'', Volume 2, Prentice Hall. {{isbn|0-13-273350-1}}.</ref><ref>Hassoun, M. (1995) ''Fundamentals of Artificial Neural Networks'' MIT Press, p.&nbsp;48</ref><ref name="pinkus" /> for reviews. The following is the most often quoted:{{math_theorem
| name = Universal approximation theorem|Let <math>C(X, \mathbb{R}^m)</math> denote the set of [[continuous functions]] from a subset <math>X </math> of a Euclidean <math>\mathbb{R}^n</math> space to a Euclidean space <math>\mathbb{R}^m</math>. Let <math>\sigma \in C(\mathbb{R}, \mathbb{R})</math>. Note that <math>(\sigma \circ x)_i = \sigma(x_i)</math>, so <math>\sigma \circ x</math> denotes <math>\sigma</math> applied to each component of <math>x</math>.

Then <math>\sigma</math> is not [[polynomial]] [[if and only if]] for every <math>n \in \mathbb{N}</math>, <math>m \in \mathbb{N}</math>, [[compact subspace|compact]] <math>K \subseteq \mathbb{R}^n</math>, <math>f \in C(K, \mathbb{R}^m), \varepsilon > 0</math> there exist <math>k \in \mathbb{N}</math>, <math>A \in \mathbb{R}^{k \times n}</math>, <math>b \in \mathbb{R}^k</math>, <math>C \in \mathbb{R}^{m \times k}</math> such that
<math display='block'>
\sup_{x \in K} \| f(x) - g(x) \| < \varepsilon
</math>
where
<math> g(x) = C \cdot ( \sigma \circ (A \cdot x + b) )</math>
}}

Also, certain non-continuous activation functions can be used to approximate a sigmoid function, which then allows the above theorem to apply to those functions. For example, the [[step function]] works. In particular, this shows that a [[perceptron]] network with a single infinitely wide hidden layer can approximate arbitrary functions.

Such an <math>f</math> 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.

{{Math proof
| title = Proof sketch
| proof = It suffices to prove the case where <math>m = 1</math>, since uniform convergence in <math>\R^m</math> is just uniform convergence in each coordinate.

Let <math>F_\sigma</math> be the set of all one-hidden-layer neural networks constructed with <math>\sigma</math>. Let <math>C_0(\R^d, \R)</math> be the set of all <math>C(\R^d, \R)</math> with compact support.

If the function is a polynomial of degree <math>d</math>, then <math>F_\sigma</math> is contained in the closed subspace of all polynomials of degree <math>d</math>, so its closure is also contained in it, which is not all of <math>C_0(\R^d, \R)</math>.

Otherwise, we show that <math>F_\sigma</math>'s closure is all of <math>C_0(\R^d, \R)</math>. Suppose we can construct arbitrarily good approximations of the ramp function 
<math display="block">r(x) = \begin{cases}
-1 & \text{if } x < -1 \\
\phantom{+}x & \text{if } |x|\leq 1 \\
\phantom{+}1 & \text{if } x > 1 \\
\end{cases}</math>
then it can be combined to construct arbitrary compactly-supported continuous function to arbitrary precision. It remains to approximate the ramp function.

Any of the commonly used [[activation function]]s used in machine learning can obviously be used to approximate the ramp function, or first approximate the ReLU, then the ramp function.

if <math>\sigma</math> is "squashing", that is, it has limits <math>\sigma(-\infty) < \sigma(+\infty)</math>, then one can first affinely scale down its x-axis so that its graph looks like a step-function with two sharp "overshoots", then make a linear sum of enough of them to make a "staircase" approximation of the ramp function. With more steps of the staircase, the overshoots smooth out and we get arbitrarily good approximation of the ramp function.

The case where <math>\sigma</math> is a generic non-polynomial function is harder, and the reader is directed to.<ref name="pinkus" />}}

The above proof has not specified how one might use a ramp function to approximate arbitrary functions in <math>C_0(\R^n, \R)</math>. A sketch of the proof is that one can first construct flat bump functions, intersect them to obtain spherical bump functions that approximate the [[Dirac delta function]], then use those to approximate arbitrary functions in <math>C_0(\R^n, \R)</math>.<ref>{{Cite book |last=Nielsen |first=Michael A. |date=2015 |title=Neural Networks and Deep Learning |url=http://neuralnetworksanddeeplearning.com/ |language=en}}</ref> The original proofs, such as the one by Cybenko, use methods from functional analysis, including the [[Hahn–Banach theorem|Hahn-Banach]] and [[Riesz–Markov–Kakutani representation theorem|Riesz–Markov–Kakutani representation]] theorems. 

Notice also that the neural network is only required to approximate within a compact set <math>K</math>. The proof does not describe how the function would be extrapolated outside of the region. 

The problem with polynomials may be removed by allowing the outputs of the hidden layers to be multiplied together (the "pi-sigma networks"), yielding the generalization:<ref name=":0" />
{{math_theorem
| name = Universal approximation theorem for pi-sigma networks|With any nonconstant activation function, a one-hidden-layer pi-sigma network is a universal approximator.
}}

== 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.<ref name=ZhouLu />  They showed that networks of width ''n''&nbsp;+&nbsp;4 with [[ReLU]] activation functions can approximate any [[Lebesgue integration|Lebesgue-integrable function]] on ''n''-dimensional input space with respect to [[L1 distance|<math>L^1</math> 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<ref name=ZhouLu /> it was shown that [[ReLU]] networks with width ''n''&nbsp;+&nbsp;1 were sufficient to approximate any [[continuous function|continuous]] function of ''n''-dimensional input variables.<ref>Hanin, B. (2018). [[arxiv:1710.11278|Approximating Continuous Functions by ReLU Nets of Minimal Width]]. arXiv preprint arXiv:1710.11278.</ref>  The following refinement, specifies the optimal minimum width for which such an approximation is possible and is due to.<ref>{{Cite journal |last=Park, Yun, Lee, Shin |first=Sejun, Chulhee, Jaeho, Jinwoo |date=2020-09-28 |title=Minimum Width for Universal Approximation |url=https://openreview.net/forum?id=O-XJwyoIF-k |journal=ICLR |arxiv=2006.08859 |language=en}}</ref>

{{math theorem
| name = Universal approximation theorem ''(L1 distance, ReLU activation, arbitrary depth, minimal width)''
| math_statement = For any [[Bochner integral|Bochner–Lebesgue p-integrable]] function <math>f : \mathbb R^n \to \mathbb R^m</math> and any <math>\varepsilon > 0</math>, there exists a [[fully connected network|fully connected]] [[ReLU]] network <math>F</math> of width exactly <math>d_m = \max\{n + 1, m\}</math>, satisfying
<math display="block">\int_{\mathbb R^n} \|f(x) - F(x)\|^p \, \mathrm{d}x < \varepsilon.</math>
Moreover, there exists a function <math>f \in L^p(\mathbb{R}^n, \mathbb{R}^m)</math> and some <math>\varepsilon > 0</math>, for which there is no [[fully connected network|fully connected]] [[ReLU]] network of width less than <math>d_m = \max\{n + 1 ,m\}</math> satisfying the above approximation bound.

Remark: If the activation is replaced by leaky-ReLU, and the input is restricted in a compact domain, then the exact minimum width is<ref name=":1" /> <math>d_m = \max\{n, m, 2\}</math>.

''Quantitative refinement:'' In the case where <math>f:[0, 1]^n \rightarrow \mathbb{R} </math>, (i.e. <math> m = 1 </math>) and <math>\sigma</math> is the [[Rectifier (neural networks)|ReLU activation function]], the exact depth and width for a ReLU network to achieve <math>\varepsilon</math> error is also known.<ref>{{cite journal |last1=Shen |first1=Zuowei |last2=Yang |first2=Haizhao |last3=Zhang |first3=Shijun |title=Optimal approximation rate of ReLU networks in terms of width and depth |journal=Journal de Mathématiques Pures et Appliquées |date=January 2022 |volume=157 |pages=101–135 |doi=10.1016/j.matpur.2021.07.009 |arxiv=2103.00502 |s2cid = 232075797 }}</ref>  If, moreover, the target function <math>f</math> is smooth, then the required number of layer and their width can be exponentially smaller.<ref>{{cite journal |last1=Lu |first1=Jianfeng |last2=Shen |first2=Zuowei |last3=Yang |first3=Haizhao |last4=Zhang |first4=Shijun |title=Deep Network Approximation for Smooth Functions |journal = SIAM Journal on Mathematical Analysis |date=January 2021 |volume=53 |issue=5 |pages=5465–5506 |doi=10.1137/20M134695X |arxiv=2001.03040 |s2cid=210116459 }}</ref>  Even if <math>f</math> is not smooth, the curse of dimensionality can be broken if <math>f</math> admits additional "compositional structure".<ref>{{Cite journal |last1=Juditsky |first1=Anatoli B. |last2=Lepski |first2=Oleg V. |last3=Tsybakov |first3=Alexandre B. |date=2009-06-01 |title=Nonparametric estimation of composite functions |journal=The Annals of Statistics |volume=37 |issue=3 |doi=10.1214/08-aos611 |s2cid=2471890 |issn=0090-5364|doi-access=free }}</ref><ref>{{Cite journal |last1=Poggio |first1=Tomaso |last2=Mhaskar |first2=Hrushikesh |last3=Rosasco |first3=Lorenzo |last4=Miranda |first4=Brando |last5=Liao |first5=Qianli |date=2017-03-14 |title=Why and when can deep-but not shallow-networks avoid the curse of dimensionality: A review |journal=International Journal of Automation and Computing |volume=14 |issue=5 |pages=503–519 |doi=10.1007/s11633-017-1054-2 |s2cid=15562587 |issn=1476-8186|doi-access=free |arxiv=1611.00740 }}</ref>
}}

Together, the central result of<ref name=kidger /> yields the following universal approximation theorem for networks with bounded width (see also<ref name=gripenberg /> for the first result of this kind).

{{math theorem
| name = Universal approximation theorem (Uniform non-[[Affine transformation|affine]] activation, arbitrary [[Deep learning|depth]], constrained width).
| math_statement = Let <math>\mathcal{X}</math> be a [[Compact set|compact subset]] of <math>\mathbb{R}^d</math>.  Let <math>\sigma:\mathbb{R} \to \mathbb{R}</math> be any non-[[Affine transformation|affine]] [[Continuous function|continuous]] function which is [[Differentiable function#Differentiability classes|continuously differentiable]] at at least one point, with nonzero [[derivative]] at that point. Let <math>\mathcal{N}_{d,D:d+D+2}^\sigma</math> denote the space of feed-forward neural networks with <math>d</math> input neurons, <math>D</math> output neurons, and an arbitrary number of hidden layers each with <math>d + D + 2</math> neurons, such that every hidden neuron has activation function <math>\sigma</math> and every output neuron has the [[identity function|identity]] as its activation function, with input layer <math>\phi</math> and output layer <math>\rho</math>. Then given any <math>\varepsilon > 0</math> and any <math>f \in C(\mathcal{X}, \mathbb{R}^D)</math>, there exists <math>\hat{f} \in \mathcal{N}_{d,D:d+D+2}^\sigma</math> such that
<math display="block">
\sup_{x \in \mathcal{X}} \left\|\hat{f}(x) - f(x)\right\| < \varepsilon.
</math>

In other words, <math>\mathcal{N}</math> is [[dense set|dense]] in <math>C(\mathcal{X}; \mathbb{R}^D)</math> with respect to the topology of [[uniform convergence]].

''Quantitative refinement:'' The number of layers and the width of each layer required to approximate <math>f</math> to <math>\varepsilon</math> precision known;<ref name="jmlr.org"/> moreover, the result hold true when <math>\mathcal{X}</math> and <math>\mathbb{R}^D</math> 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.<ref name="ZhouLu" /><ref name=hanin /><ref name=johnson>{{cite conference |last=Johnson |first=Jesse |conference=International Conference on Learning Representations |date=2019 |url=https://openreview.net/forum?id=ryGgSsAcFQ |title=Deep, Skinny Neural Networks are not Universal Approximators}}</ref>

== 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.<ref name=maiorov /> Their remarkable result revealed that such networks can be universal approximators and for achieving this property two hidden layers are enough.
{{math theorem
| name = Universal approximation theorem:<ref name=maiorov />
| math_statement=  There exists an activation function <math>\sigma</math> which is analytic, strictly increasing and sigmoidal and has the following property: For any <math> f\in C[0,1]^{d}</math>  and <math>
\varepsilon >0</math> there exist constants <math>d_{i}, c_{ij}, \theta _{ij}, \gamma _{i}</math>, and vectors <math> \mathbf{w}^{ij}\in \mathbb{R}^{d}</math>  for which
<math display='block'> \left\vert f(\mathbf{x})-\sum_{i=1}^{6d+3} d_{i}\sigma \left(
\sum_{j=1}^{3d} c_{ij} \sigma (\mathbf{w}^{ij}\cdot \mathbf{x-}\theta_{ij}) - \gamma_{i}\right) \right\vert <\varepsilon </math>
for all <math> \mathbf{x}=(x_{1},...,x_{d})\in [0,1]^{d}</math>.
}}

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.<ref name=guliyev1 /> The theoretical result can be formulated as follows.
{{math theorem 
| name = Universal approximation theorem:<ref name=guliyev1 /><ref name=guliyev2 />
| math_statement = Let  <math> [a,b]</math>  be a finite segment of the real line, <math> s=b-a</math> and <math> \lambda</math>  be any positive number. Then one can algorithmically construct a computable sigmoidal activation function <math> \sigma \colon \mathbb{R} \to \mathbb{R}</math>, which is infinitely differentiable, strictly increasing on <math> (-\infty, s) </math>, <math> \lambda</math> -strictly increasing on <math> [s,+\infty) </math>, and satisfies the following properties:

# For any <math> f \in C[a,b] </math>  and <math> \varepsilon > 0</math>  there exist numbers <math> c_1,c_2,\theta_1</math>  and <math> \theta_2</math>  such that for all <math>x \in [a,b] </math> <math display='block'> |f(x) - c_1 \sigma(x - \theta_1) - c_2 \sigma(x - \theta_2)| < \varepsilon</math>
# For any continuous function <math>F</math> on the <math>d</math>-dimensional box <math>[a,b]^{d}</math> and <math>\varepsilon > 0</math>, there exist constants <math>e_p</math>, <math>c_{pq}</math>, <math>\theta_{pq}</math> and <math>\zeta_p</math> such that the inequality <math display='block'>  \left| F(\mathbf{x}) - \sum_{p=1}^{2d+2} e_p \sigma \left( \sum_{q=1}^{d} c_{pq} \sigma(\mathbf{w}^{q} \cdot \mathbf{x} - \theta_{pq}) - \zeta_p \right) \right| < \varepsilon</math> holds for all <math>\mathbf{x} = (x_1, \ldots, x_d) \in [a, b]^{d}</math>. Here the weights <math>\mathbf{w}^{q}</math>, <math>q = 1, \ldots, d</math>, are fixed as follows: <math display='block'>  \mathbf{w}^{1} = (1, 0, \ldots, 0), \quad \mathbf{w}^{2} = (0, 1, \ldots, 0), \quad \ldots, \quad \mathbf{w}^{d} = (0, 0, \ldots, 1). </math> In addition, all the coefficients <math>e_p</math>, except one, are equal.
}}

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

== See also ==
* [[Kolmogorov–Arnold representation theorem]]
* [[Representer theorem]]
* [[No free lunch theorem]]
* [[Stone–Weierstrass theorem]]
* [[Fourier series]]

== References ==
{{Reflist}}

{{Differentiable computing}}

[[Category:Theorems in analysis]]
[[Category:Artificial neural networks]]
[[Category:Network architecture]]
[[Category:Networks]]