A Unified and Constructive Framework for the Universality of Neural Networks

Human brain consists of networks of billions of neurons, each of which, roughly speaking, receives information—electrical pulses —from other neurons via dendrites, processes the information using soma, is activated by difference of electrical potential, and passes the output along its axon to other neurons through synapse. Attempt to understand the extraordinary ability of the brain in categorizing, classifying, regressing, processing, etc information has inspired numerous scientists to develop computational models to mimic brain functionalities. Most well-known is perhaps the McCulloch-Pitts model [1], which is also called perceptron by Rosenblatt who extended McCulloch-Pitts model to networks of artificial neurons capable of learning from data [2]. The question is if such a network could mimic some of the brain capability, such as learning to classify. The answer lies in the fact that perceptron networks can represent Boolean logic functions exactly. From a mathematical point of view, perceptron networks with Heaviside activation functions compute step functions, linear combinations of which form the space of simple functions which in turn is dense in the space of measurable functions [3, 4]. That is, linear combination of perceptions can approximate a measurable function to any desired accuracy [5, 6]: the first universal approximation for neural networks. The universal approximation capability partially “explains” why human brains can be trained to virtually learn any tasks.

For training one-hidden layer networks, Widrow-Hoff rule [7] can be used for supervised learning. For many-layer ones, the most popular approach is back-propagation [8] which requires the derivative of activation functions. Heaviside function is, however, not differentiable in the classical sense. A popular smooth approximation is the standard logistic (also called sigmoidal) activation function which is infinitely differentiable. The question is now: are neural networks of sigmoidal function universal? An answer to this question was first presented by Cybenko [9] and Funahashi [10]. The former, though non-constructive (a more constructive proof was then provided in [11] and revisited in [12]), elegantly used the Hahn-Banach theorem to show that sigmoidal neural networks (NNs) with one hidden layer is dense in the space of continuous functions on compacta. The latter used Fourier transform and an integral formulation for integrable function with bounded variation [13] to show the same results for NNs with continuous, bounded, monotone increasing sigmoidal activation function. Recognizing NN output as an approximate back-projection operator, [14] employed inverse Radon transformation to show the universal approximation property in space of squared integrable functions. Making use of the classical Stone-Weierstrass approximation theorem [15] successfully proved NNs with non-decreasing sigmoidal function are dense in space of continuous functions over compacta and dense in space of measurable functions. Using the separability of space of continuous functions over compacta [16] constructed a strictly increasing analytic sigmoidal function that is universal. The work in [17], based on a Kolmogorov theorem, showed that any continuous function on hypercubes can be approximated well with two-hidden layer NNs of sigmoidal functions.

Though sigmoidal functions are natural continuous approximation to Heaviside function, and hence mimicking the activation mechanism of neurons, they are not the only ones having universal approximation property. Indeed, [18] showed, using distributional theory, that NNs with $k$th degree sigmoidal function are dense in space of continuous functions over compacta. Meanwhile, [19] designed a cosine squasher activation function so that the output of one-hidden layer neural network is truncated Fourier series and thus can approximate square integrable functions to any desired accuracy. Following and extending [9], [20] showed that NN is dense in $\mathcal{L}^{p}$ with bounded non-constant activations function and in space of continuous functions over compacta with continuous bounded nonconstant activation functions. Using the Stone-Weierstrass theorem, [6] showed that any network with activations (e.g. exponential, cosine squasher, modified sigma-pi, modified logistic, and step functions) that can transform product of functions into sum of functions is universal in the space of bounded measurable functions over compacta. The work in [21] provided universal approximation for bounded weights (and biases) with piecewise polynomial and superanalytic activation functions (e.g. sine, cosine, logistic functions and piecewise continuous functions are superanalytic at some point with positive radius of convergence).

One hidden-layer NN is in fact dense in space of continuous functions and $\mathcal{L}^{p}$ if the activation function is not polynomial almost everywhere [22, 23]. Using Taylor expansion and Vandermonde determinant, [24] provided an elementary proof of universal approximation for NNs with $C^{\infty}$ activation functions. Recently, [25] provides universal approximation theorem for multilayer NNs with ReLU activation functions using partition of unity and Taylor expansion for functions in Sobolev spaces. Universality of multilayer ReLU NNs can also be obtained using approximate identity and rectangular quadrature rule [26]. Restricting in Barron spaces [27, 28], the universality of ReLU NNs is a straightforward application of the law of large numbers [29]. The universality of ReLU NNs can also be achieved by emulating finite element approximations [30, 31].

Unlike others, [32] introduced $\mathcal{B}$ellshape function (as derivative of squash-type activation function, for example) as a means to explicitly construct one-hidden layer NNs to approximate continuous function over compacta in multiple dimensions. More detailed analysis was then carried out in [33, 34]. The idea was revisited and extended to establish universal approximation in the uniform norm for tensor product sigmoidal and hyperbolic tangent activations [35, 36, 37, 38, 39]. Similar approach was also taken in [40] using cardinal B-spline and in [41] using the distribution function as sigmoid but for a family of sigmoids with a certain of decaying-tail conditions.

While it is sufficient for most universal approximation results to hold when each weight (and bias) varying over the whole real line $\mathbb{R}$, this is not necessary. In fact, universal approximation results for continuous function on compacta can also be obtained using finite set of weights [42, 43, 44]. It is quite striking that one-hidden layer with only one neuron is enough for universality: [45] constructs a smooth, sigmoidal, almost monotone activation function so that one-hidden layer with one neuron can approximate any continuous function over any compact subset in $\mathbb{R}$ to any desired accuracy.

Universal theorems with convergence rate for sigmoidal and others NNs have also been established. Modifying the proofs in [11, 18], [46] showed that, in one dimension, the error incurred by sigmoidal NNs with $N$ hidden units scales as $\mathcal{O}\left(N^{-1}\right)$ in the uniform norm over compacta. The result was then extended to $\mathbb{R}$ for bounded continuous function [47]. Similar results were obtained for functions with bounded variations [48] and with bounded $\phi$-variations [49]. For multiple dimensions, [27] provided universal approximation for sigmoidal NNs in the space of functions with bounded first moment of the magnitude distribution of their Fourier transform with rate $\mathcal{O}\left(N^{-1/2}\right)$ in the $\mathcal{L}^{2}$-norm, independent of dimensions. This result can be generalized to Lipschitz functions (with additional assumptions) [50]. Using explicit NN construction in [32], [35, 36, 37, 38, 39, 33, 34, 41] provided convergence rate of $\mathcal{O}\left(N^{-\alpha}\right)$, $0<\alpha<1$ in the uniform norm for Hölder continuous functions with exponent $\alpha$. Recently, [51] has revisited universal approximation theory with rate $\mathcal{O}\left(N^{-1/2}\right)$ in Sobolev norms for smooth activation functions with polynomial decay condition on all derivatives. This improves/extends the previous similar work for sigmoidal functions in [27] and exponentially decaying activation functions in [52]. The setting in [51] is valid for sigmoidal, arctan, hyperbolic tangent, softplus, ReLU, Leaky ReLU, and $k$th power of ReLU, as their central differences satisfy polynomial decaying condition. However, the result is only valid for smooth functions in high-order Barron spaces. For activation functions without decay but essentially bounded and having bounded Fourier transform on some interval (or having bounded variation), the universal approximation results with slower rate $\mathcal{O}\left(N^{-1/4}\right)$ in the $\mathcal{L}^{2}$-norm can be obtained for first order Barron space. These rates can be further improved using stratified sampling [53, 28]. Convergence rates for ReLU NNs in Sobolev spaces has been recently established using finite elements [30, 31].

The main objective of this paper is to provide the first constructive and unified frameworks for a large class of activation functions including most of existing ones. At the heart of this new framework is the introduction of the neural network approximate identity (nAI) concept. The important consequence is: any nAI activation function is universal. The following are the main contributions: i) Unlike existing works, our framework is constructive using elementary means from functional analysis, probability theory, non-asymptotic analysis, and numerical analysis. ii) While existing approaches are either technical or specialized for particular activation functions, our framework is the first unified attempt that is valid for most of the existing activation functions and beyond; iii) The framework provides the first universality proof for some of the existing activation functions including Mish, SiLU, ELU, GELU, and etc; iv) The framework provides new proofs for all activation functions; v) The framework discovers and facilitates the discovery of new activation functions with guaranteed universality property. Indeed, any activation—whose $k$th derivative, with $k$ being an integer, is integrable and essentially bounded—is universal. In that case, the activation function and all of its $j$th derivative, $j=1,\ldots,k$ are not only a valid activation but also universal. vi) For a given activation and error tolerance, the framework provides precisely the architecture of the corresponding one-hidden neural network with a predetermined number of neurons, and the values of weights/biases; vii) The framework is the first that allows for abstractly presenting the universal approximation with the favorable non-asymptotic rate of $N^{-{\frac{1}{2}}}$, where $N$ is the number of neurons. This provides not only theoretical insights into the required number of neurons but also guidance on how to choose the number of neurons to achieve a certain accuracy with controllable successful probability. Perhaps more importantly, it shows that neural network may fail, with non-zero probability, to provide the desired testing error in practice when an architecture is selected; viii) Our framework also provides insights into the developments, and hence providing constructive derivations, of some of the existing approaches.

The paper is organized as follows. Section 2 introduces conventions and notations used in the paper. Elementary facts about convolution and approximate identities that are useful for our purposes are presented in section 3. Section 4 recalls quadrature rules in terms of Riemann sums for continuous functions on compacta and their error analysis using moduli of continuity. This follows by a unified abstract framework for universality in section 6. The key to achieve this is to introduce the concept of neural network approximate identity (nAI), which immediately provides an abstract universality result in Lemma 3. This abstract framework reduces universality proof to nAI proof. Section 7 shows that most of existing activations are nAI. This includes the family of rectified polynomial units (RePU) in which the parametric and leaky ReLUs are a member, a family of generalized sigmoidal functions in which the standard sigmoidal, hyperbolic tangent and softplus are a member, the exponential linear unit (ELU), the Gaussian error linear unit (GELU), the sigmoid linear unit (SiLU), and the Mish. It is the nAI proof of the Mish activation function that guides us to devise a general framework for a large class of nAI functions (including all activations in this paper and beyond) in section 8. Abstract universal approximation result with non-asymptotic rates is presented in section 9. Section 10 concludes the paper.

This section describes notations used in the paper. We reserve lower case roman letters for scalars or scalar-valued function. Boldface lower case roman letters are for vectors with components denoted by subscripts. We denote by $\mathbb{R}$ the set of real numbers and by $*$ the convolution operator. For ${\boldsymbol{{x}}}\in\mathbb{R}^{n}$, where $n\in\mathbb{N}$ is the ambient dimension, $\left|{\boldsymbol{{x}}}\right|_{\_}p:=\left(\sum_{\_}{i=1}^{n}\left|{\boldsymbol{{x}}}_{\_}i\right|^{p}\right)^{1/p}$ denotes the standard $\ell^{p}$ norm in $\mathbb{R}^{n}$. We conventionally write $f\left({\boldsymbol{{x}}}\right):=f\left({\boldsymbol{{x}}}_{\_}1,\ldots,{\boldsymbol{{x}}}_{\_}n\right)$ and $\left\|f\right\|_{\_}p:=\left(\int_{\mathbb{R}^{n}}\left|f\left({\boldsymbol{{x}}}\right)\right|^{p}\,d\mu\left({\boldsymbol{{x}}}\right)\right)^{1/p}$ for $1\leq p<\infty$ and $\mu$ is the Lebesgue measure in $\mathbb{R}^{n}$. For $p=\infty$, $\left\|f\right\|_{\_}\infty:=\text{ess}\sup_{\_}{\mathbb{R}^{n}}\left|f\right|:=\left\{M>0:\mu\left\{{\boldsymbol{{x}}}:\left|f(x)\right|>M\right\}=0\right\}$. For simplicity, we use $d{\boldsymbol{{x}}}$ in place of $d\mu\left({\boldsymbol{{x}}}\right)$. Note that we also use $\left\|f\right\|_{\_}\infty$ to denote the uniform norm of continuous function $f$. We define $\mathcal{L}^{p}:=\mathcal{L}^{p}\left(\mathbb{R}^{n}\right):=\left\{f:\left\|f\right\|_{\_}p<\infty\right\}$ for $1\leq p\leq\infty$, $\mathcal{C}\left(\mathcal{K}\right)$ as the space of continuous functions on $\mathcal{K}\subseteq\mathbb{R}^{n}$, $\mathcal{C}_{\_}0\left(\mathbb{R}^{n}\right)$ as the space of continuous functions vanishing at infinity, $\mathcal{C}_{\_}b\left(\mathbb{R}^{n}\right)$ as the space of bounded functions in $\mathcal{C}\left(\mathbb{R}^{n}\right)$, and $\mathcal{C}_{\_}c\left(\mathcal{K}\right)$ as the space of functions in $\mathcal{C}\left(\mathcal{K}\right)$ with compact support.

The mathematical foundation for our framework is approximate identity, which relies on convolution. Convolution has been used by many authors including [23, 40, 22] to assist in proving universal approximation theorems. Approximate identity generated by ReLU function has been used in [26] for showing ReLU universality in $\mathcal{L}^{p}$. In this section we collect some important results from convolution and approximate identity that are useful for our developments (see, e.g., [54] for a comprehensive treatment). Let $\tau_{\_}{\boldsymbol{{y}}}f\left({\boldsymbol{{x}}}\right):=f\left({\boldsymbol{{x}}}-{\boldsymbol{{y}}}\right)$ be the translation operator. The following is standard.

Let $f,g:\mathbb{R}^{n}\to\mathbb{R}$ be two measurable functions in $\mathbb{R}^{n}$, their convolution is defined as

when the integral exists. We are interested in the conditions under which $f\left({\boldsymbol{{x}}}-{\boldsymbol{{y}}}\right)g\left({\boldsymbol{{y}}}\right)$ (or $f\left({\boldsymbol{{y}}}\right)g\left({\boldsymbol{{x}}}-{\boldsymbol{{y}}}\right)$) is integrable for almost every (a.e.) ${\boldsymbol{{x}}}$, as our approach requires a discretization of the convolution integral. Below is a relevant case.

We are interested in $g$ which is an approximate identity in the following sense.

An important class of approximate identity is obtained by rescaling $\mathcal{L}^{1}$ functions.

Recall that for a bounded function on a compact set, it is Riemann integrable if and only if it is continuous almost everywhere. In that case, Riemann sums converge to Riemann integrals as the corresponding partition size approaches 0. It is well-known (see, e.g. [55, 56], and references therein) that most common numerical quadrature rules—including trapezoidal, some Newton-Cotes formulas, and Gauss-Legendre quadrature formulas—are Riemann sums. In this section, we use the Riemann sum interpretation of quadrature rule to approximate integrals of bounded functions. The goal is to characterize quadrature error in terms of modulus of continuity. We first discuss quadrature error in one dimension and then extend the result to $n$ dimensions.

We assume the domain of interest is $\left[-1,1\right]^{n}$. Let $f\in{\mathcal{C}}\left[-1,1\right]^{n}:={\mathcal{C}}\left(\left[-1,1\right]^{n}\right)$, $\mathcal{P}^{m}:=\left\{\boldsymbol{{z}}^{1},\ldots,\boldsymbol{{z}}^{m+1}\right\}$ a partition of $\left[-1,1\right]$, and $\mathcal{Q}_{\_}m:=\left\{\xi^{1},\ldots,\xi^{m}\right\}$ the collection of all “quadrature points” such that $-1\leq\boldsymbol{{z}}^{j}\leq\xi^{j}\leq\boldsymbol{{z}}^{j+1}\leq 1$ for $j=1,\ldots,m$. We assume that

be a valid Riemann sum, e.g. trapezoidal rule, and thus converging to $\int_{\_}{-1}^{1}g\left(\boldsymbol{{z}}\right)\,d\boldsymbol{{z}}$ as $m$ approaches $\infty$. We define a quadrature rule for $\left[-1,1\right]^{n}$ as the tensor product of the aforementioned one dimensional quadrature rule, and thus

is a valid Riemann sum for $\int_{\_}{\left[-1,1\right]^{n}}f\left({\boldsymbol{{x}}}\right)\,d{\boldsymbol{{x}}}$.

Recall the modulus of continuity $\omega\left(f,h\right)$ of a function $f$:

$\omega\left(f,0\right)=0$, and $\omega\left(f,h\right)$ is continuous with respect to $h$ at $0$ (due to the uniform continuity of $f$ on $\left[-1,1\right]^{n}$). The following error bound for the tensor product quadrature rule is an extension of the standard one dimensional case [55].

We are interested in approximating functions in $\mathcal{C}_{\_}c\left[-1,1\right]^{n}$ using neural networks. How to extend the results to $\mathcal{C}\left[-1,1\right]^{n}$ is given in A. At the heart of our framework is to first approximate any function $f$ in $\mathcal{C}_{\_}c\left[-1,1\right]^{n}$ with an approximate identity $\mathcal{B}_{\theta}$ and then numerically integrate the convolution integral $f*\mathcal{B}_{\theta}$ with quadrature (later with Monte Carlo sampling in Section 9). This extends the similar approach in [26] for ReLU activation functions in several directions: 1) we rigorously account for the approximate identity and quadrature errors, 2) our unified framework holds for most activation functions (see Section 7) using the network approximate identity (nAI) concept introduced in Section 6, 4) we identify sufficient conditions under which an activation is nAI (see Section 8), and 5) we provide non-asymptotic rate of convergence (see Section 9). Moreover, this procedure is the key to our unification of neural network universality.

Let $f\in\mathcal{C}_{\_}c\left[-1,1\right]^{n}$ and $\mathcal{B}_{\theta}$ be an approximate identity. From Lemma 3 we know that $f*\mathcal{B}_{\theta}$ converges to $f$ uniformly as $\theta\to 0$. We, however, are interested in estimating the error $\left\|f*\mathcal{B}_{\theta}-f\right\|_{\_}\infty$ for a given $\theta$. From the proof of Lemma 3, for any $\delta>0$, we have

where $\mathcal{T}\left(\mathcal{B}_{\theta},\delta\right):=\int_{\_}{\left\|{\boldsymbol{{y}}}\right\|>\delta}\left|\mathcal{B}_{\theta}\left({\boldsymbol{{y}}}\right)\right|\,d{\boldsymbol{{y}}}$ is the tail mass of $\mathcal{B}_{\theta}$.

Next, for any ${\boldsymbol{{x}}}\in\left[-1,1\right]^{n}$, we approximate

with the Riemann sum defined in (1). The following result is obvious by triangle inequality, continuity of $\omega\left(f,h\right)$ at $h=0$, the third condition of the definition of $\mathcal{B}_{\theta}$, and Lemma 4.

Thus, up to a scaling, $g\left({x}\right)$ is an approximate identity.

We have explicitly constructed a one-hidden layer neural network $\mathcal{S}\left(m,f*\mathcal{B}_{\theta}\right)\left({x}\right)$ with an arbitrary nAI activation $\sigma$ in (5) to approximate well any continuous function $f\left({x}\right)$ with compact support in $\left[-1,1\right]$. A few observations are in order: i) if we know the modulus of continuity of $f\left({x}\right)$ and the tail behavior of $\mathcal{B}_{\theta}$ (from property of $\sigma$), we can precisely determine the total number of quadrature points $m$, the scaling $\theta$, and the cut-off radius $\delta$ in terms of $\varepsilon$ (see Lemma 5). That is, the topology of the network is completely determined; ii) The weights and biases of the network are also readily available from the nAI property of $\sigma$, the quadrature points, and $\theta$; iii) the coefficients of the output layer is also pre-determined by nAI (i.e. $\alpha_{\_}\ell$) and the values of the unknown function $f$ at the quadrature points; and iv) any nAI activation function is universal in $\mathcal{C}_{\_}c\left[-1,1\right]$.

Clearly the Gaussian activation function $e^{-{x}^{2}}$ is an nAI with $k=1$, $\alpha_{\_}1=1$, ${w}_{\_}1=1$ and ${b}_{\_}1=0$. The interesting fact is that, as we shall prove in Section 7, most existing activation functions, though not integrable by themselves, are nAI.

We now exploit properties of each activation (and its family, if any) under consideration to show that they are nAI for $n=1$. That is, these activations generate one-dimensional approximate identity, which in turn shows that they are universal by Lemma 6. To show that they are also universal in $n$ dimensions²²2Note that by Proposition 3.3 of [23], with an additional assumption on the denseness of the set of ridge functions in $\mathcal{C}\left(\mathbb{R}^{n}\right)$, we can conclude that they are also universal in $n$ dimensions. This approach, however, does not provide an explicit construction of the corresponding neural networks. we extend a constructive and explicit approach from [26] that was done for ReLU activation function. We start with the family of rectified polynomial units (RePU)—also parametric and leaky ReLUs—with many properties, then family of sigmoidal functions, then the exponential linear unit (ELU), then the Gaussian error linear unit (GELU), then the sigmoid linear unit (SiLU), and we conclude with the Mish activation with least properties. The proofs for all results in this section, together with figures demonstrating the $\mathcal{B}$ functions for each activation, can be found in Appendix B and Appendix C.