Large-width functional asymptotics for deep Gaussian neural networks

The interplay between infinitely-wide deep neural networks and classes of Gaussian Processes has its origins in the seminal work of Neal (1995), and it has been the subject of several theoretical studies. See, e.g., Der & Lee (2006), Lee et al. (2018), Matthews et al. (2018a; b), Yang (2019) and references therein. Let consider a fully-connected feed-forward neural network with re-scaled weights composed of $L\geq 1$ layers of widths $n_{1},\dots,n_{L}$, i.e.

where $\phi$ is a non-linearity and $x\in\mathbb{R}^{I}$ is a real-valued input of dimension $I\in{\mathbb{N}}$. Neal (1995) considered the case $L=2$, a finite number $k\in{\mathbb{N}}$ of fixed distinct inputs $(x^{(1)},\ldots,x^{(k)})$, with each $x^{(r)}\in\mathbb{R}^{I}$, and weights $w_{i,j}^{(l)}$ and biases $b_{i}^{(l)}$ independently and identically distributed (iid) as Gaussian distributions. Under appropriate assumptions on the activation $\phi$ Neal (1995) showed that: i) for a fixed unit $i$, the $k$-dimensional random vector $(f^{(2)}_{i}(x^{(1)}),\dots,f^{(2)}_{i}(x^{(k)}))$ converges in distribution, as the width $n_{1}$ goes to infinity, to a $k$-dimensional Gaussian random vector; ii) the large-width convergence holds jointly over finite collections of $i$’s and the limiting $k$-dimensional Gaussian random vectors are independent across the index $i$. These results concerns neural networks with a single hidden layer, but Neal (1995) also includes preliminary considerations on infinitely-wide deep neural networks. More recent works, such as Lee et al. (2018), established convergence results corresponding to Neal (1995) results i) and ii) for deep neural networks under the assumption that widths $n_{1},\dots,n_{L}$ go to infinity sequentially over network layers. Matthews et al. (2018a; b) extended the work of Neal (1995); Lee et al. (2018) by assuming that the width $n$ grows to infinity jointly over network layers, instead of sequentially, and by establishing joint convergence over all $i$ and countable distinct inputs. The joint growth over the layers is certainly more realistic than the sequential growth, since the infinite Gaussian limit is considered as an approximation of a very wide network. We operate in the same setting of Matthews et al. (2018b), hence from here onward $n\geq 1$ denotes the common layer width, i.e. $n_{1},\dots,n_{L}=n$. Finally, similar large-width limits have been established for a great variety of neural network architectures, see for instance Yang (2019).

The assumption of a countable number of fixed distinct inputs is the common trait of the literature on large-width asymptotics for deep neural networks. Under this assumption, the large-width limit of a network boils down to the study of the large-width asymptotic behavior of the $k$-dimensional random vector $(f^{(l)}_{i}(x^{(1)}),\dots,f^{(l)}_{i}(x^{(k)}))$ over $i\geq 1$ for finite $k$. Such limiting finite-dimensional distributions describe the large-width distribution of a neural network a priori over any dataset, which is finite by definition. When the limiting distribution is Gaussian, as it often is, this immediately paves the way to Bayesian inference for the limiting network. Such an approach is competitive with the more standard stochastic gradient descent training for the fully-connected architectures object of our study (Lee et al., 2020). However, knowledge of the limiting finite-dimensional distributions is not enough to infer properties of the limiting neural network which are inherently uncountable such as the continuity of the limiting neural network, or the distribution of its maximum over a bounded interval. Results in this direction give a more complete understanding of the assumptions being made a priori, and hence whether a given model is appropriate for a specific application. For instance, Van Der Vaart & Van Zanten (2011) shows that for Gaussian Processes the function smoothness under the prior should match the smoothness of the target function for satisfactory inference performance.

In this paper we thus consider a novel, and more natural, perspective to the study of large-width limits of deep neural networks. This is an infinite-dimensional perspective where, instead of fixing a countable number of distinct inputs, we look at $f^{(l)}_{i}(x,n)$ as a stochastic process over the input space ${\mathbb{R}}^{I}$. Under this perspective, establishing large-width limits requires considerable care and, in addition, it requires to show the existence of both the stochastic process induced by the neural network and its large-width limit. We start by proving the existence of i) a continuous stochastic process, indexed by the network width $n$, corresponding to the fully-connected feed-forward deep neural network; ii) a continuous Gaussian Process corresponding to the infinitely-wide limit of the deep neural network. Then, we prove that the stochastic process i) converges weakly, as the width $n$ goes to infinity, to the Gaussian Process ii) jointly over all units $i$. As a by-product of our results, we show that the limiting Gaussian Process has almost surely locally $\gamma$-Hölder continuous paths, for $0<\gamma<1$. To make the exposition self-contained we include an alternative proof of the main result of Matthews et al. (2018a; b), i.e. the finite-dimensional limit for full-connected neural networks. The major difference between our proof and that of Matthews et al. (2018b) is due to the use of the characteristic function to establish convergence in distribution, instead of relying on a CLT (Blum et al., 1958) for exchangeable sequences.

The paper is structured as follows. In Section 2 we introduce the setting under which we operate, whereas in Section 3 we present a high-level overview of the approach taken to establish our results. Section 4 contains the core arguments of the proof of our large-width functional limit for deep Gaussian neural networks, which are spelled out in detail in the supplementary material (SM). We conclude in Section 5.

Let $(\Omega,\mathcal{H},\mathbb{P})$ be the probability space on which all random elements of interest are defined. Furthermore, let $N(\mu,\sigma^{2})$ denote a Gaussian distribution with mean $\mu\in{\mathbb{R}}$ and strictly positive variance $\sigma^{2}\in{\mathbb{R}}_{+}$, and let $N_{k}(\mathbf{m},\Sigma)$ be a $k$-dimensional Gaussian distribution with mean $\mathbf{m}\in{\mathbb{R}}^{k}$ and covariance matrix $\Sigma\in{\mathbb{R}}^{k\times k}$. In particular, ${\mathbb{R}}^{k}$ is equipped with $\|\cdot\|_{{\mathbb{R}}^{k}}$, the euclidean norm induced by the inner product $\langle\cdot,\cdot\rangle_{{\mathbb{R}}^{k}}$, and ${\mathbb{R}}^{\infty}=\times_{i=1}^{\infty}{\mathbb{R}}$ is equipped with $\|\cdot\|_{{\mathbb{R}}^{\infty}}$, the norm induced by the distance $d(\textbf{a},\textbf{b})_{\infty}=\sum_{i\geq 1}\xi(|a_{i}-b_{i}|)/2^{i}$ for $\textbf{a},\textbf{b}\in{\mathbb{R}}^{\infty}$ (Theorem 3.38 of Aliprantis & Border (2006)), where $\xi(t)=t/(1+t)$ for all real values $t\geq 0$. Note that $({\mathbb{R}},|\cdot|)$ and $({\mathbb{R}}^{\infty},\|\cdot\|_{{\mathbb{R}}^{\infty}})$ are Polish spaces, i.e. separable and complete metric spaces (Corollary 3.39 of Aliprantis & Border (2006)). We choose $d_{\infty}$ since it generates a topology that coincides with the product topology (line 5 of the proof of Theorem 3.36 of Aliprantis & Border (2006)). The space $(S,d)$ will indicate a generic Polish space such as ${\mathbb{R}}$ or ${\mathbb{R}}^{\infty}$ with the associated distance. We indicate with $S^{{\mathbb{R}}^{I}}$ the space of functions from ${\mathbb{R}}^{I}$ into $S$ and $C({\mathbb{R}}^{I};S)\subset S^{{\mathbb{R}}^{I}}$ the space of continuous functions from ${\mathbb{R}}^{I}$ into $S$. Let $\omega_{i,j}^{(l)}$ be the random weights of the $l$-th layer, and assume that they are iid as $N(0,\sigma_{\omega}^{2})$, i.e.

is the characteristic function of $\omega_{i,j}^{(l)}$, for $i\geq 1$, $j=1,\dots,n$ and $l\geq 1$. Let $b^{(l)}_{i}$ be the random biases of the $l$-th layer, and assume that they are iid as $N(0,\sigma_{b}^{2})$, i.e.

is the characteristic function of $b_{i}^{(l)}$, for $i\geq 1$ and $l\geq 1$. Weights $\omega^{(l)}_{i,j}$ are independent of biases $b_{i}^{(l)}$, for any $i\geq 1,j=1,\dots,n$ and $l\geq 1$. Let $\phi:{\mathbb{R}}\to\ {\mathbb{R}}$ denote a continuous non-linearity. For the finite-dimensional limit we will assume the polynomial envelop condition

for any $s\in{\mathbb{R}}$ and some real values $a,b>0$ and $m\geq 1$. For the functional limit we will use a stronger assumption on $\phi$, assuming $\phi$ to be Lipschitz on ${\mathbb{R}}$ with Lipschitz constant $L_{\phi}$.

Let $Z$ be a stochastic process on $\mathbb{R}^{I}$, i.e. for each $x\in\mathbb{R}^{I}$, $Z(x)$ is defined on $(\Omega,\mathcal{H},\mathbb{P})$ and it takes values in $S$. For any $k\in{\mathbb{N}}$ and $x_{1},\ldots,x_{k}\in\mathbb{R}^{I}$, let $P^{Z}_{x_{1},\ldots,x_{k}}={\mathbb{P}}(Z(x_{1})\in A_{1},\ldots,Z(x_{k})\in A_{k})$, with $A_{1},\ldots,A_{k}\in\mathcal{B}(S)$. Then, the family of finite-dimensional distributions of $Z(x)$ is defined as the family of distributions $\{P^{Z}_{x_{1},\ldots,x_{k}}\text{ : }x_{1},\ldots,x_{k}\in\mathbb{R}^{I}\text{ and }k\in{\mathbb{N}}\}$. See, e.g., Billingsley (1995). In Definition 1 and Definition 2 we look at the deep neural network (1) as a stochastic process on input space $\mathbb{R}^{I}$, that is a stochastic process whose finite-dimensional distributions are determined by a finite number $k\in{\mathbb{N}}$ of fixed distinct inputs $(x^{(1)},\ldots,x^{(k)})$, with each $x^{(r)}\in\mathbb{R}^{I}$. The existence of the stochastic processes of Definition 1 and Definition 2 will be thoroughly discussed in Section 3.

Remark: in contrast to (1), we have defined (5)-(6) over an infinite number of units $i\geq 1$ over each layer $l$, but the dependency on each previous layer $l-1$ remains limited to the first $n$ components.

Remark: for $k$ inputs, the vector $\textbf{F}^{(l)}(\textbf{X},n)$ is an $\infty\times k$ array, and for a single input $x^{(r)}$, $\textbf{F}^{(l)}(x^{(r)},n)$ can be written as $[f^{(l)}_{1}(x^{(r)},n),f^{(l)}_{2}(x^{(r)},n),\dots]^{T}\in{\mathbb{R}}^{\infty\times 1}$. We define $\textbf{F}_{r}^{(l)}(\textbf{X},n)=\textbf{F}^{(l)}(x^{(r)},n)$ the $r$-th column of $\textbf{F}^{(l)}(\textbf{X},n)$. When we write $\langle\textbf{F}^{(l-1)}(x,n),\textbf{F}^{(l-1)}(y,n)\rangle_{{\mathbb{R}}^{n}}$ (see (11)) we treat $\textbf{F}^{(l)}(x,n)$ and $\textbf{F}^{(l)}(y,n)$ as elements in ${\mathbb{R}}^{n}$ and not in ${\mathbb{R}}^{\infty}$, i.e. we consider only the first $n$ components of $\textbf{F}^{(l)}(x,n)$ and $\textbf{F}^{(l)}(y,n)$.

We start by recalling the notion of convergence in law, also referred to as convergence in distribution or weak convergence, for a sequence of stochastic processes. See Billingsley (1995) for a comprehensive account.

In this paper, we deal with continuous and real-valued stochastic processes. More precisely, we consider random elements defined on $C({\mathbb{R}}^{I};S)$, with $(S,d)$ Polish space. Our aim is to study in $C({\mathbb{R}}^{I};S)$ the convergence in distribution as the width $n$ goes to infinity for:

1. i)

   the sequence $(f_{i}^{(l)}(n))_{n\geq 1}$ for a fixed $l\geq 2$ and $i\geq 1$ with $(S,d)=({\mathbb{R}},|\cdot|)$, i.e. the neural network process for a single unit;

2. ii)

   the sequence $(\textbf{F}^{(l)}(n))_{n\geq 1}$ for a fixed $l\geq 2$ with $(S,d)=({\mathbb{R}}^{\infty},\|\cdot\|_{\infty})$, i.e. the neural network process for all units.

Since applying Definition 3 in a function space is not easy, we need, proved in ??, the following proposition.

We denoted with $\stackrel{{\scriptstyle f_{d}}}{{\rightarrow}}$ the convergence in law of the finite-dimensional distributions of a sequence of stochastic processes. The notion of tightness formalizes the concept that the probability mass is not allowed to “escape at infinity”: a single random element $f$ in a topological space $C$ is said to be tight if for each $\epsilon>0$ there exists a compact $T\subset C$ such that $\mathbb{P}[f\in C\setminus T]<\epsilon$. If a metric space $(C,\rho)$ is Polish any random element on the Borel $\sigma$-algebra of $C$ is tight. A sequence of random elements $(f(n))_{n\geq 1}$ in a topological space $C$ is said to be uniformly tight¹¹1Kallenberg (2002) uses the same term “tightness” for both cases of a single random element and of sequences of random elements; we find that the introduction of “uniform tightness” brings more clarity. if for every $\epsilon>0$ there exists a compact $T\subset C$ such that $\mathbb{P}[f(n)\in C\setminus T]<\epsilon$ for all $n$.

According to Proposition 1, to achieve convergence in distribution in function spaces we need the following Steps A-D:

Step A) to establish the existence of the finite-dimensional weak-limit $f$ on ${\mathbb{R}}^{I}$. We will rely on Theorem 5.3 of Kallenberg (2002), known as Levy theorem.

Step B) to establish the existence of the stochastic processes $f$ and $(f(n))_{n\geq 1}$ as elements in $S^{{\mathbb{R}}^{I}}$ the space of function from ${\mathbb{R}}^{I}$ into $S$. We make use of Daniell-Kolmogorov criterion (Kallenberg, 2002, Theorem 6.16): given a family of multivariate distributions $\{P_{\mathcal{I}}\text{ probability measure on }{\mathbb{R}}^{\dim({\mathcal{I}})}\mid{\mathcal{I}}\subset\{x^{(1)},\dots,x^{(k)}\}_{x^{(z)}\in{\mathbb{R}}^{I},k\in{\mathbb{N}}}\}$ there exists a stochastic process with $\{P_{\mathcal{I}}\}$ as finite-dimensional distributions if $\{P_{\mathcal{I}}\}$ satisfies the projective property: $P_{J}(\cdot\times{\mathbb{R}}_{J\setminus{\mathcal{I}}})=P_{\mathcal{I}}(\cdot),{\mathcal{I}}\subset J\subset\{x^{(1)},\dots,x^{(k)}\}_{x^{(z)}\in{\mathbb{R}}^{I},k\in{\mathbb{N}}}$. That is, it is required consistency with respect to the marginalization over arbitrary components. In this step we also suppose, for a moment, that the stochastic processes $(f(n))_{n\geq 1}$ and $f$ belong to $C({\mathbb{R}}^{I};S)$ and we establish the existence of such stochastic processes in $C({\mathbb{R}}^{I};S)$ endowed with a $\sigma$-algebra and a probability measure that will be defined.

Step C) to show that the stochastic processes $(f(n))_{n\geq 1}$ and $f$ belong to $C({\mathbb{R}}^{I};S)\subset S^{{\mathbb{R}}^{I}}$. With regards to $(f(n))_{n\geq 1}$ this is a direct consequence of (5)-(6) and the continuity of $\phi$. With regards to the limiting process $f$, with an additional Lipschitz assumption on $\phi$, we rely on the following Kolmogorov-Chentsov criterion (Kallenberg, 2002, Theorem 3.23):

Step D) the uniform tightness of $(f(n))_{n\geq 1}$ in $C({\mathbb{R}}^{I};S)$. We rely on an extension of the Kolmogorov-Chentsov criterion (Kallenberg, 2002, Corollary 16.9), which is stated in the following proposition.