Sharp Lower Bounds on Interpolation by Deep ReLU Neural Networks at
  Irregularly Spaced Data
We study the interpolation power of deep ReLU neural networks. Specifically,
we consider the question of how efficiently, in terms of the number of
parameters, deep ReLU networks can interpolate values at $N$ datapoints in the
unit ball which are separated by a distance $\delta$. We show that $\Omega(N)$
parameters are required in the regime where $\delta$ is exponentially small in
$N$, which gives the sharp result in this regime since $O(N)$ parameters are
always sufficient. This also shows that the bit-extraction technique used to
prove lower bounds on the VC dimension cannot be applied to irregularly spaced
datapoints. Finally, as an application we give a lower bound on the
approximation rates that deep ReLU neural networks can achieve for Sobolev
spaces at the embedding endpoint.
Jonathan W. Siegel
# Sharp Lower Bounds on Interpolation by Deep ReLU Neural Networks at Irregularly Spaced Data

###### Abstract

We study the interpolation, or memorization, power of deep ReLU neural networks. Specifically, we consider the question of how efficiently, in terms of the number of parameters, deep ReLU networks can interpolate values at \(N\) datapoints in the unit ball which are separated by a distance \(\delta\). We show that \(\Omega(N)\) parameters are required in the regime where \(\delta\) is exponentially small in \(N\), which gives the sharp result in this regime since \(O(N)\) parameters are always sufficient. This also shows that the bit-extraction technique used to prove lower bounds on the VC dimension cannot be applied to irregularly spaced datapoints.

## 1 Introduction

We study the problem of how efficiently, in terms of the number of parameters, deep neural networks can interpolate prescribed values at \(N\) datapoints. Let \(x_{1},...,x_{N}\in\mathbb{R}^{d}\) be distinct points and suppose that we are given values \(y_{1},...,y_{N}\in C\subset\mathbb{R}\). Here the set \(C\) is the set of classes and the \(y_{i}\) are the class labels corresponding to the data point \(x_{i}\). We wish to find a neural network \(f\) with as few parameters as possible such that \(f(x_{i})=y_{i}\) for \(i=1,...,N\).

To be specific, let us gives some notation describing the class of deep (feedforward) neural networks [11] we consider. Let \(\sigma\) be a piecewise polynomial activation function. We will mainly consider the important special case of \(\sigma=\max(0,x)\), i.e. the rectified linear unit (ReLU) activation function [16]. Let \(L\geq 1\) be an integer denoting the depth of the network and denote by \(\mathbf{W}=(w_{1},...,w_{L})\) a vector with positive integer values which contains the widths of the intermediate layers. We write \(A_{\mathbf{W},b}\) to denote the affine map with weight matrix \(\mathbf{M}\) and offset, or bias, \(b\), i.e.

\[A_{\mathbf{M},b}(x)=\mathbf{M}x+b. \tag{1.1}\]

When the weight matrix \(\mathbf{M}\in\mathbb{R}^{k\times n}\) the bias \(b\in\mathbb{R}^{k}\), the function \(A_{\mathbf{M},b}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}\) maps \(\mathbb{R}^{n}\) to \(\mathbb{R}^{k}\). We denote the set of deep neural networks with activation function \(\sigma\) and architecture defined by the vector \(\mathbf{W}\) by

\[\Upsilon^{\mathbf{W}}(\mathbb{R}^{d}):=\{A_{\mathbf{M},b_{L}}\circ\sigma\circ A _{\mathbf{M},-1,b_{L-1}}\circ\sigma\circ\cdots\circ\sigma\circ A_{\mathbf{M},b _{1}}\circ\sigma\circ A_{\mathbf{M},b_{0}}\}, \tag{1.2}\]

where the weight matrices satisfy \(\mathbf{M}_{L}\in\mathbb{R}^{1\times w_{L}}\), \(\mathbf{M}_{0}\in\mathbb{R}^{w_{1}\times d}\), and \(\mathbf{M}_{i}\in\mathbb{R}^{w_{i+1}\times w_{i}}\) for \(i=1,...,L\), and the biases satisfy \(b_{i}\in\mathbb{R}^{w_{i+1}}\) for \(i=0,...,L-1\) and \(b_{L}\in\mathbb{R}\). For such a neural network architecture, the total number of free parameters is

\[P=P(\mathbf{W}):=w_{L}+dw_{1}+1+\sum_{i=1}^{L-1}w_{i+1}w_{i}+\sum_{i=1}^{L}w_{ i}. \tag{1.3}\]

Let

\[\mathcal{X}_{N}\subset(\mathbb{R}^{d})^{\neq N}:=\{(x_{1},...,x_{N})\in( \mathbb{R}^{d})^{N},\;x_{i}\neq x_{j}\;\text{for}\;i\neq j\}\]

be a set of possible configurations of distinct data points and let \(C\subset\mathbb{R}\) be a set of labels. We wish to find a neural network architecture \(\mathbf{W}\) with the minimal number of parameters \(P(\mathbf{W})\) which can interpolate labels from the set
at any datapoints in \(\mathcal{X}_{N}\), i.e. such that for any \((x_{1},...,x_{N})\in\mathcal{X}_{N}\) and \(y_{1},...,y_{n}\in C\) there exists an \(f\in\Upsilon^{\mathbf{W}}(\mathbb{R}^{d})\) with \(f(x_{i})=y_{i}\).

This has been a central and well-studied problem in the theory of the expressive power of neural networks (see, for instance [3, 7, 8, 12, 13, 18, 19, 20, 30]) alongside other problems related to expressiveness, such as universal approximation [5, 14] and the benefits of depth [6, 9, 21, 22, 27]. The problem of interpolation, also called memorization, has recently been connected to the phenomenon of double descent [4, 17].

We remark that it is easy to see that \(O(N)\) parameters suffice to interpolate arbitrary values at an arbitrary set of \(N\) distinct points, i.e. with \(O(N)\) parameters one can interpolate with \(\mathcal{X}_{N}=(\mathbb{R}^{d})^{\neq N}\) and \(C=\mathbb{R}\). Indeed, there exists a direction \(v\in\mathbb{R}^{d}\) such that \(v\cdot x_{i}\) are all distinct, and any piecewise linear function with \(N\) pieces can be represented using a (deep or shallow) ReLU neural network with \(O(N)\) parameters. Note that by throwing away some of the coordinates, we can always assume that \(d<N\) since the pairwise differences \(x_{i}-x_{j}\) lie in an \(N-1\) dimensional subspace, so that we can restrict to some set of \(N-1\) coordinates while keeping the datapoints distinct.

Using the VC-dimension [28], it can be proved that if the set of possible labels \(C\) is infinite and the architecture \(\Upsilon^{\mathbf{W}}(\mathbb{R}^{d})\) can interpolate, then \(P(\mathbf{W})\geq cN\) for a constant \(c\), i.e. if the label set is infinite, then a neural network with piecewise polynomial activation function requires \(\Omega(N)\) parameters to interpolate at \(N\) values (see [29], Lemma 4.1 or [25], Theorem 5). In addition, it has been shown that if \(\mathcal{X}_{N}=(\mathbb{R}^{d})^{\neq N}\) and \(|C|>1\), then if \(\Upsilon^{\mathbf{W}}(\mathbb{R}^{d})\) interpolates it must hold that \(P(\mathbf{W})\geq(N-1)/2\), i.e. that a neural network with piecewise polynomial activation function requires \(\Omega(N)\) parameters to interpolate at _arbitrary_ distinct datapoints [26].

Of particular interest are situations where interpolation is possible with \(P(\mathbf{W})=o(N)\) parameters. By the preceding remarks, this requires additional assumptions on the allowed datasets \(\mathcal{X}_{N}\) and the label set \(C\). A typical assumption is that the label set \(C\) is finite (independent of \(N\)) and that the datapoints are well separated [19, 29, 30]. Specifically, we consider datasets of the form

\[\mathcal{X}_{N}=\{(x_{1},...,x_{N})\subset(\mathbb{R}^{d})^{N},\ |x_{i}|\leq 1,\ |x_{i}-x_{j}|\geq\delta(N)\text{ if }i\neq j\}. \tag{1.4}\]

The assumption (1.4) means that the datapoints lie in the unit ball and are separated by at least a distance \(\delta(N)\) (clearly the separation distance \(\delta\) must depend upon \(N\)). This situation was recently analyzed in [19, 29], and in [29] (building upon the work in [19]) it was proved that there exists a deep ReLU architecture \(\mathbf{W}\) which can interpolate labels from a finite set \(C\) at datasets \(\mathcal{X}_{N}\) satisfying the separation condition (1.4) using only

\[P(\mathbf{W})=O\left(\sqrt{N\log N}+\sqrt{\frac{N}{\log N}}\max(\log(R),\log( |C|))\right) \tag{1.5}\]

parameters, where \(R=10N^{2}\delta^{-1}\sqrt{\pi d}\). This remarkable result shows that if \(\delta\) is polynomial in \(N\), then deep ReLU neural networks can interpolate binary labels using only \(\sqrt{N}\) parameters (up to logarithmic factors). This is also tight up to logarithmic factors, which follows from bounds on the VC-dimension of deep neural networks with polynomial activation function [1, 2, 10].

An interesting open question is to determine the precise dependence of the number of parameters \(P(\mathbf{W})\) on both the number of datapoints \(N\) and the separation distance \(\delta(N)\). The result (1.5) from [29] gives a nearly complete solution to this problem when \(\delta(N)\) depends polynomially on \(N\), only a logarithmic gap in \(N\) remains. In this work, we consider the opposite regime, where \(\delta(N)\) is exponentially small in \(N\), and prove that in this case \(\Omega(N)\) parameters are required. Since, as remarked above, \(O(N)\) parameters are always sufficient, this provides a complete solution in the regime where \(\delta(N)\) is exponentially small in \(N\).

To explain this result, we first recall the notions of shattering and VC-dimension [28]. Let \(X=\{x_{1},...,x_{N}\}\subset\mathbb{R}^{d}\) be a finite set of points and \(\mathcal{F}\) a class of real-valued functions on \(\mathbb{R}^{d}\). The class \(\mathcal{F}\) is said to shatter [23, 24] the point set \(X\) if given any signs \(\epsilon_{1},...\epsilon_{N}\in\{\pm 1\}\) there exists an \(f\in\mathcal{F}\) such that \(\operatorname{sgn}(f(x_{i}))=\epsilon_{i}\). Here we are using the convention that

\[\operatorname{sgn}(x)=\begin{cases}-1&x<0\\ 1&x\geq 0.\end{cases}\]

The VC-dimension of the class \(\mathcal{F}\) is the size of the largest set of points that \(\mathcal{F}\) shatters, i.e.

\[\text{VC-dim}(\mathcal{F})=\sup\{N:\ \exists X=\{x_{1},...,x_{N}\}\subset \mathbb{R}^{d},\ X\text{ is shattered by }\mathcal{F}\}. \tag{1.6}\]

In [10], the VC-dimension of a neural network architecture \(\Upsilon^{\mathbf{W}}(\mathbb{R}^{d})\) is shown to be bounded by \(P(\mathbf{W})^{2}\) for any piecewise polynomial activation function \(\sigma\). Using the 'bit-extraction' technique, this bound is shown to be optimal for very deep ReLU neural networks [1, 2].
Note that the VC-dimension only requires that a single set of \(N\) points be shattered by the class \(\mathcal{F}\). If we instead require that _all_ sets of \(N\) distinct points be shattered by a neural network architecture \(\Upsilon^{\mathbf{W}}(\mathbb{R}^{d})\), then it is shown in [26] that the number of parameters must be at least \(P(\mathbf{W})\geq(N-1)/2\). Indeed, Theorem 1 in [26] implies that if \(P(\mathbf{W})<(N-1)/2\), then the set of shattered point sets is not dense in \((\mathbb{R}^{d})^{N}\). This implies that if \(\mathcal{R}_{N}\) is a dense subset of \((\mathbb{R}^{d})^{N}\) and a neural network architecture \(\Upsilon^{\mathbf{W}}(\mathbb{R}^{d})\) can interpolate binary class labels for all datasets in \(\mathcal{X}_{N}\), then \(P(\mathbf{W})\geq(N-1)/2\). However, note that the collection of \(\delta\)-separated datasets defined by (1.4) are not dense in \((\mathbb{R}^{d})^{N}\) for any \(\delta>0\), and so this result gives no indication how small \(\delta(N)\) needs to be to rule out interpolation with \(o(N)\) parameters. Our main result is a quantitative version of Theorem 1 in [26] which answers precisely this question and shows that \(\delta(N)\) need only be exponentially small in \(N\) to rule out interpolation with \(o(N)\) parameters using deep ReLU neural networks.

**Theorem 1**.: _Let \(\sigma\) be a piecewise polynomial activation function and \(\mathbf{W}=(w_{1},...,w_{L})\) be a vector representing the widths of each layer of a deep neural network architecture with activation function \(\sigma\). Let \(\delta>0\) and suppose that \(\Upsilon^{\mathbf{W}}(\mathbb{R}^{d})\) shatters **every** subset \(X=\{x_{1},...,x_{N}\}\subset\mathbb{R}^{d}\) which satisfies \(|x_{i}|\leq 1\) for all \(i\) and_

\[|x_{i}-x_{j}|\geq\delta\text{ for }i\neq j.\]

_There exists a constant \(c\) depending only on \(\sigma\) such that if_

\[\delta<\begin{cases}e^{-cN^{2}}&\sigma\text{ is a general piecewise polynomial function}\\ e^{-cN}&\sigma\text{ is a piecewise linear function},\end{cases} \tag{1.7}\]

_we must have_

\[P(\mathbf{W})\geq N/6. \tag{1.8}\]

We apply this to the case of interpolation with deep ReLU neural networks to get the following Corollary, which shows that if the separation distance is exponentially small in \(N\), then deep ReLU neural networks require \(\Omega(N)\) parameters to interpolate.

**Corollary 1**.: _Let \(\mathcal{X}_{N}\) be defined by (1.4) for a separation distance \(\delta(N)\) and suppose that the number of classes \(|C|>1\). Let \(\mathbf{W}=(w_{1},...,w_{L})\) describe a deep neural network architecture with ReLU activation function._

_There exists a constant \(c<\infty\) such that if \(\delta(N)<e^{-cN}\) and for any \((x_{1},...,x_{N})\in\mathcal{X}_{N}\) and \(y_{1},...,y_{n}\in C\) there exists an \(f\in\Upsilon^{\mathbf{W}}(\mathbb{R}^{d})\) which satisfies_

\[f(x_{i})=y_{i}, \tag{1.9}\]

_then necessarily \(P(\mathbf{W})\geq N/6\)._

Proof.: By translating the output, we may assume there exists \(c_{1},c_{2}\in C\) with \(c_{1}<0<c_{2}\). If arbitrary values from \(C\) can be interpolated, then every dataset in \(\mathcal{X}_{N}\) can be shattered. We now apply Theorem 1 with \(\sigma=\max(0,x)\), which is piecewise linear, to get the desired result. 

## 2 Proof of the Main Result

In this section we prove Theorem 1. Observe that it suffices to prove the Theorem in the case \(d=1\). This follows by restricting the input of the network to the \(x\)-axis, which can only reduce the number of parameters. If every subset of \(\mathbb{R}^{d}\) satisfying the assumptions of Theorem 1 is shattered, then so is every subset of the \(x\)-axis, and the one-dimensional result gives the desired lower bound.

The first ingredient is a Lemma which restricts the number of sign changes that a function \(f\in\Upsilon^{\mathbf{W}}(\mathbb{R})\) can have. The Lemma and proof follow closely the ideas of [27] and of Theorem 14 in [2].

**Lemma 1**.: _Let \(\sigma\) be a piecewise polynomial activation function with \(p\) pieces and degree at most \(d\) in each piece. Suppose that \(\mathbf{W}=(w_{1},...,w_{L})\) is a vector representing the widths of each layer of a deep neural network architecture. Let \(f\in\Upsilon^{\mathbf{W}}(\mathbb{R})\) and \(x_{0}<x_{1}<\cdots<x_{M}\) be a sequence of real numbers such that \(\operatorname{sgn}(x_{i})\neq\operatorname{sgn}(x_{i+1})\) for \(i=0,...,M-1\). Then_

\[M\leq M^{*}(\mathbf{W},\sigma):=((d^{L}+2)p^{L}d^{L(L+1)/2}{\prod_{i=1}^{L}}w_{ i}. \tag{2.1}\]

Proof.: The proof essentially follows from repeated application of Lemma 13 in [2]. Specifically, consider the output of the network after the \(k\)-th layer, which is given by

\[f_{k}(x):=A_{\mathbf{M}_{k},h_{k}}\circ\sigma\circ A_{\mathbf{M}_{k-1},h_{k-1}} \circ\sigma\circ\cdots\circ\sigma\circ A_{\mathbf{M}_{1},b_{1}}\circ\sigma \circ A_{\mathbf{M}_{0},b_{0}}(x). \tag{2.2}\]
We prove by induction that each component of \(f_{k}\) is a piecewise polynomial function with at most \(P_{k}\) pieces, each of degree at most \(D_{k}\), where

\[P_{k}=p^{k}d^{k(k+1)/2}\prod_{i=1}^{k}w_{i},\ D_{k}=d^{k}. \tag{2.3}\]

This evidently holds when \(k=0\), since each component of \(f_{0}\) is an affine linear function, which has one piece of degree one. For the inductive step, we first note that if the \(i\)-th component \(f_{k}^{i}\) has \(\leq P_{k}\) pieces of degree \(\leq D_{k}\), then \(\sigma(f_{k}^{i})\) has at most \(pP_{k}D_{k}\) pieces of degree \(\leq dD_{k}\).

Indeed, let \(b_{1},...,b_{p-1}\) denote the breakpoints of \(\sigma\). On each if its at most \(P_{k}\) pieces, \(f_{k}^{i}\) is a polynomial of degree at most \(D_{k}\). This means that there can be at most \(D_{k}\) solutions to \(f_{k}^{i}(x)=b_{i}\) for \(i=1,...,p-1\) on each of these pieces. Hence applying \(\sigma\) divides each piece of \(f_{k}^{i}\) into at most \(pD_{k}\) further sub-pieces. On each of these sub-pieces \(\sigma(f_{k}^{i})\) is the composition of a polynomial of degree \(\leq D_{k}\) with a polynomial of degree at most \(d\) and so has degree at most \(dD_{k}\).

Thus each component of \(f_{k+1}\) is a linear combination of at most \(w_{k}\) piecewise polynomials with \(\leq pP_{k}D_{k}\) pieces of degree \(\leq dD_{k}\). Taking the union of their breakpoints, we see that this can have at most \(w_{k}pP_{k}D_{k}\) pieces of degree \(\leq dD_{k}\). This completes the inductive step and implies that \(f\) has at most \(P\) pieces of degree at most \(D\), where

\[P=p^{L}d^{L(L+1)/2}\prod_{i=1}^{L}w_{i},\ D=d^{L}. \tag{2.4}\]

On each of these pieces, the function \(f\) is a polynomial of degree \(D\) and thus can switch sign at most \(D+1\) times. In addition, moving from one piece to the next the sign can switch at most an additional \(P\) times (note that this only needs to be counted if \(\sigma\) is not continuous). This gives a total number of sign changes which is at most

\[M\leq P(D+1)+P=(d^{L}+2)p^{L}d^{L(L+1)/2}\prod_{i=1}^{L}w_{i}. \tag{2.5}\]

Note that in the important special case where \(\sigma=\max(0,x)\) is the ReLU activation function [16] the bound in Lemma 1 reduces to

\[M^{*}(\mathbf{W},\sigma)\leq 3\prod_{i=1}^{L}(2w_{i}),\]

which has been shown to be essentially sharp in [27]. In particular, in [27] a network with small fixed width \(w\) and depth \(L=O(m)\) is constructed which represents the function \(f(x)=x\pmod{2}\) for \(x=0,...,2^{m}-1\). Thus, for this family of networks we have

\[\log(M)\geq c\log(M^{*}(\mathbf{W},\sigma))\]

for a fixed constant \(c\).

Proof of Theorem 1.: Assume without loss of generality that \(N\) is even. Let \(T\geq 2N\) be an even integer and consider the set of points

\[X_{T}=\{i/T,\ i=0,...,T-1\}\subset[0,1]. \tag{2.6}\]

Note that every \(N\)-element subset of \(X_{T}\) satisfies the assumptions of Theorem 1 with \(\delta=T^{-1}\).

Let \(S(\mathbf{W},T)\) denote the set of sign patters that \(\Upsilon^{\mathbf{W}}(\mathbb{R})\) achieves on the large set \(X_{T}\), i.e.

\[S(\mathbf{W},T):=\{(\epsilon_{i})_{i=0}^{T-1},\ \exists f\in\Upsilon^{ \mathbf{W}}(\mathbb{R})\ \text{s.t.}\ \operatorname{sgn}(f(i/T))=\epsilon_{i}\}. \tag{2.7}\]

Suppose that \(\Upsilon^{\mathbf{W}}(\mathbb{R})\) shatters every subset \(X\subset X_{T}\) with \(|X|=N\). We claim that this implies

\[|S(\mathbf{W},T)|\geq\left(\frac{T}{4M^{*}(\mathbf{W},\sigma)}\right)^{N/2}. \tag{2.8}\]

Indeed, let \(I\subset\{0,...,T/2-1\}\) be an arbitrary subset of size \(|I|=N/2\) and consider the set

\[X_{I}=\left\{\frac{2i}{T}\right\}_{i\in I}\cup\left\{\frac{2i+1}{T}\right\}_{i \in I}\]

and the sign pattern on \(X_{I}\) given by

\[\epsilon_{I}\left(\frac{2i}{T}\right)=1,\ \epsilon_{I}\left(\frac{2i+1}{T}\right)=-1 \tag{2.9}\]
for \(i\in I\). Since the set \(X_{I}\) is shattered by \(\Upsilon^{\mathbf{W}}(\mathbb{R})\), the sign pattern \(\varepsilon_{I}\) can be matched by an \(f\in\Upsilon^{\mathbf{W}}(\mathbb{R})\), so there must exist an \(\varepsilon\in S(\mathbf{W},T)\) such that

\[\varepsilon(2i)=1,\ \varepsilon(2i+1)=-1 \tag{2.10}\]

for all \(i\in I\).

For each fixed sign pattern \(\varepsilon\in S(\mathbf{W},T)\) let \(J_{\varepsilon}\) denote the set of indices for which \(\varepsilon(2i)=1\) and \(\varepsilon(2i+1)=-1\), i.e.

\[J_{\varepsilon}=\{i:\ \varepsilon(2i)=1\ \text{and}\ \varepsilon(2i+1)=-1\} \subset\{0,...,2T-1\}. \tag{2.11}\]

Note that any subset \(I\subset\{0,...,T/2-1\}\) for which \(\varepsilon\) satisfies (2.10) must satisfy \(I\subset J_{\varepsilon}\). On the other hand, Lemma 1 implies that \(|J_{\varepsilon}|\leq M^{*}(\mathbf{W},\sigma)\). This means that the number of subsets \(I\) of size \(|I|=N/2\) for which \(\varepsilon\) satisfies (2.10) is bounded by

\[\binom{M^{*}(\mathbf{W},\sigma)}{N/2}\leq\frac{M^{*}(\mathbf{W},\sigma)^{N/2} }{(N/2)!}. \tag{2.12}\]

On the other hand, the total number of subsets \(I\subset\{0,...,T/2-1\}\) of size \(|I|=N/2\) is

\[\binom{T/2}{N/2}\geq\frac{T^{N/2}}{4^{N/2}(N/2)!}, \tag{2.13}\]

since \(T\geq 2N\). Since by assumption \(\varepsilon_{I}\) is matched by an \(f\in\Upsilon^{\mathbf{W}}(\mathbb{R})\) for every subset \(I\subset\{0,...,T/2-1\}\) of size \(N/2\) and the number of \(\varepsilon_{I}\) that each \(\varepsilon\in S(\mathbf{W},T)\) can match is bounded by (2.12), we get the lower bound (2.8).

Next, we upper bound \(|S(\mathbf{W},T)|\) using Warren's Theorem [31]. Let \(b_{1}<b_{2}<\cdots<b_{p-1}\) denote the break points of the activation function \(\sigma\) and \(q_{1},...,q_{p}\) be the polynomials in each piece. Specifically, setting \(b_{0}=-\infty\) and \(b_{p}=\infty\), \(\sigma\) is given by

\[\sigma(x)=\big{\{}q_{i}(x)\quad b_{i-1}\leq x<b_{i}. \tag{2.14}\]

Note that here we assume that \(\sigma\) is left continuous, but changing the value of \(\sigma\) at the break points \(b_{i}\) (if it is discontinuous) doesn't significantly change the proof.

For \(i=1,...,L\), let \(\mathbf{t}_{i}\in\{1,...,p\}^{w_{i}}\) and define a function \(\mathbf{t}_{i}:\mathbb{R}^{w_{i}}\rightarrow\mathbb{R}^{w_{i}}\) by

\[\mathbf{t}_{i}(\mathbf{x})_{j}=q_{(\mathbf{t}_{i})_{j}}(\mathbf{x}_{j}) \tag{2.15}\]

for \(j=1,...,w_{i}\). For indices \(\mathbf{t}_{i}\), the function \(\mathbf{t}_{i}\) applies the pieces of \(\sigma\) indicated by \(\mathbf{t}_{i}\) to the corresponding entries of the input \(\mathbf{x}\in\mathbb{R}^{w_{i}}\).

Given an input \(x\in X_{T}\), consider the signs of the following quantities

\[\begin{split}(A_{\mathbf{M}_{0},b_{0}}(x))_{j}-b_{q},\,j=1,...,w_ {1},\ q=1,...,p-1\\ (A_{\mathbf{M}_{1},b_{1}}\circ\mathbf{t}_{1}\circ A_{\mathbf{M}_{ 0},b_{0}}(x))_{j}-b_{q},\,j=1,...,w_{2},\ q=1,...,p-1\\ (A_{\mathbf{M}_{2},b_{2}}\circ\mathbf{t}_{2}\circ A_{\mathbf{M}_{ 1},b_{1}}\circ\mathbf{t}_{1}\circ A_{\mathbf{M}_{0},b_{0}}(x))_{j}-b_{q},\,j=1,...,w_{3},\ q=1,...,p-1\\ \vdots\\ (A_{\mathbf{M}_{L-1},b_{L-1}}\circ\mathbf{t}_{L-1}\circ\cdots\circ \mathbf{t}_{2}\circ A_{\mathbf{M}_{1},b_{1}}\circ\mathbf{t}_{1}\circ A_{ \mathbf{M}_{0},b_{0}}(x))_{j}-b_{q},\,j=1,...,w_{L},\ q=1,...,p-1\\ A_{\mathbf{M}_{L},b_{L}}\circ\mathbf{t}_{L}\circ A_{\mathbf{M}_{L-1},b _{L-1}}\circ\mathbf{t}_{L-1}\circ\cdots\circ\mathbf{t}_{2}\circ A_{\mathbf{M} _{1},b_{1}}\circ\mathbf{t}_{1}\circ A_{\mathbf{M}_{0},b_{0}}(x),\end{split} \tag{2.16}\]

where the \(\mathbf{t}_{i}\) range over all elements of \(\{1,...,p\}^{w_{i}}\) for \(i=1,...,L\).

Given \(x\in X_{T}\) and parameters \(\mathbf{P}:=\{\mathbf{M}_{0},...,\mathbf{M}_{L},b_{1},...,b_{L}\}\) the sign of \(f_{\mathbf{P}}(x)\), where \(f_{\mathbf{P}}\) is the neural network function defined by the parameters \(\mathbf{P}\), is uniquely determine by the signs of the quantities in (2.16). Indeed, we recursively set \(\mathbf{t}_{i}\) to index the pieces of \(\sigma\) which contain the outputs of the neurons at layer \(i\), i.e. we set (recursively in \(i\))

\[(\mathbf{t}_{i})_{j}=\min\{q:\text{sgn}((A_{\mathbf{M}_{i-1},b_{i-1}}\circ \mathbf{t}_{i-1}\circ\cdots\circ\mathbf{t}_{2}\circ A_{\mathbf{M}_{1},b_{1}} \circ\mathbf{t}_{1}\circ A_{\mathbf{M}_{0},b_{0}}(x))_{j}-b_{q})=-1\}\cup\{p\} \tag{2.17}\]

for \(j=1,...,w_{i}\). Then

\[\text{sgn}(f_{\mathbf{P}}(x))=\text{sgn}(A_{\mathbf{M}_{L},b_{L}}\circ\mathbf{t }_{L}\circ A_{\mathbf{M}_{L-1},b_{L-1}}\circ\mathbf{t}_{L-1}\circ\cdots\circ \mathbf{t}_{2}\circ A_{\mathbf{M}_{1},b_{1}}\circ\mathbf{t}_{1}\circ A_{\mathbf{ M}_{0},b_{0}}(x)). \tag{2.18}\]

Now, for each fixed \(x\in X_{T}\) and set of \(\mathbf{t}_{i}\in\{1,...,p\}^{w_{i}}\), each of the quantities in (2.16) is a polynomial of degree at most \(d^{L}\), where \(d=\max\{\text{deg}(q_{i})\}_{i=1}^{p}\), in the parameters \(\mathbf{P}\). Ranging over all \(x\in X_{T}\), \(\mathbf{t}_{i}\in\{1,...,p\}^{w_{i}}\), and all quantities in (2.16), we see that the sign pattern \(\text{sgn}(f_{\mathbf{P}}(x))_{x\in X_{T}}\) is uniquely determined by the signs of (setting \(V=\sum_{i=1}^{L}w_{i}\))

\[Tp^{V}\left(1+(p-1)V\right)\]
polynomials of degree \(d^{L}\) in the \(P=P(\mathbf{W})\) parameters \(\mathbf{P}\). We now apply Warren's Theorem (Theorem 3 in [31]) to see that the number of such sign patterns, and thus \(|S(\mathbf{W},T)|\), is bounded by

\[|S(\mathbf{W},T)|\leq\left(\frac{4ed^{L}Tp^{V}\left(1+(p-1)V\right)}{P}\right)^ {P}\leq\left(4ed^{L}Tp^{V}\left(1+(p-1)V\right)\right)^{P}. \tag{2.19}\]

We take logarithms and compare with the lower bound (2.8) to get

\[\frac{N}{2}(\log(T)-\log(4M^{*}(\mathbf{W},\sigma)))\leq P(\log(T)+\log(4ed^{L }p^{V}(1+(p-1)V)). \tag{2.20}\]

We complete the proof of Theorem 1 by means of a contradiction. Suppose that \(P<N/6\) and observe that by Lemma 1

\[\log(4M^{*}(\mathbf{W},\sigma))\leq C\left(\sum_{i=1}^{L}w_{i}+L^{2}\log(d)+L \log(p)\right)\leq C(\sigma)P^{2}, \tag{2.21}\]

since we clearly have that \(\sum_{i=1}^{L}w_{i}\leq P\) and \(L\leq P\). In addition, we easily calculate that

\[\log(4ed^{L}p^{V}(1+(p-1)V))\leq C(\sigma)P. \tag{2.22}\]

This means that if we choose \(T>e^{cN^{2}}>e^{cP^{2}}\) (recall that by assumption \(P<N/6\)) for a constant \(c\) depending upon \(\sigma\), then we can ensure that

\[\log(T)\geq 2\max(\log(4M^{*}(\mathbf{W},\sigma)),\ \log(4ed^{L}p^{V}(1+(p-1)V))). \tag{2.23}\]

But then (2.20) becomes

\[\frac{N}{4}\log(T)\leq\frac{3P}{2}\log(T), \tag{2.24}\]

which implies \(P\geq N/6\), contradicting our assumption.

If \(\sigma\) is piecewise linear, then the bound (2.21) can be improved to (since \(d=1\))

\[\log(4M^{*}(\mathbf{W},\sigma))\leq C(\sigma)P, \tag{2.25}\]

so that we can take \(T>e^{cN}>e^{cP}\) in the above argument. Since \(\delta=T^{-1}\), this completes the proof.

## 3 Acknowledgements

We would like to thank Ronald DeVore, Jinchao Xu, and Juncai He for helpful discussions. This work was supported by the National Science Foundation (DMS-2111387 and CCF-2205004).
