For any positive integer $k$, there exist neural networks with $Θ(k^3)$ layers, $Θ(1)$ nodes per layer, and $Θ(1)$ distinct parameters which can not be approximated by networks with $\mathcal{O}(k)$ layers unless they are exponentially large --- they must possess $Ω(2^k)$ nodes. This result is proved here for a class of nodes termed "semi-algebraic gates" which includes the common choices of ReLU, maximum, indicator, and piecewise polynomial functions, therefore establishing benefits of depth against not just standard networks with ReLU gates, but also convolutional networks with ReLU and maximization gates, sum-product networks, and boosted decision trees (in this last case with a stronger separation: $Ω(2^{k^3})$ total tree nodes are required).
Bar Ilan University（巴伊兰大学）；Allen Institute for Artificial Intelligence（艾伦人工智能研究所）
We present a simple proof for the benefit of depth in multi-layer feedforward network with rectified activation ("depth separation"). Specifically we present a sequence of classification problems indexed by $m$ such that (a) for any fixed depth rectified network there exist an $m$ above which classifying problem $m$ correctly requires exponential number of parameters (in $m$); and (b) for any problem in the sequence, we present a concrete neural network with linear depth (in $m$) and small constant width ($\leq 4$) that classifies the problem with zero error. The constructive proof is based on…
We provide several new depth-based separation results for feed-forward neural networks, proving that various types of simple and natural functions can be better approximated using deeper networks than shallower ones, even if the shallower networks are much larger. This includes indicators of balls and ellipses; non-linear functions which are radial with respect to the $L_1$ norm; and smooth non-linear functions. We also show that these gaps can be observed experimentally: Increasing the depth indeed allows better learning than increasing width, when training neural networks to learn an indicat…
We show that there is a simple (approximately radial) function on $\reals^d$, expressible by a small 3-layer feedforward neural networks, which cannot be approximated by any 2-layer network, to more than a certain constant accuracy, unless its width is exponential in the dimension. The result holds for virtually all known activation functions, including rectified linear units, sigmoids and thresholds, and formally demonstrates that depth -- even if increased by 1 -- can be exponentially more valuable than width for standard feedforward neural networks. Moreover, compared to related results in …
We show that the number of unique function mappings in a neural network hypothesis space is inversely proportional to $\prod_lU_l!$, where $U_{l}$ is the number of neurons in the hidden layer $l$.
We solve an open question from Lu et al. (2017), by showing that any target network with inputs in $\mathbb{R}^d$ can be approximated by a width $O(d)$ network (independent of the target network's architecture), whose number of parameters is essentially larger only by a linear factor. In light of previous depth separation theorems, which imply that a similar result cannot hold when the roles of width and depth are interchanged, it follows that depth plays a more significant role than width in the expressive power of neural networks. We extend our results to constructing networks with bounded w…
Inference in deep Bayesian neural networks is only fully understood in the infinite-width limit, where the posterior flexibility afforded by increased depth washes out and the posterior predictive collapses to a shallow Gaussian process. Here, we interpret finite deep linear Bayesian neural networks as data-dependent scale mixtures of Gaussian process predictors across output channels. We leverage this observation to study representation learning in these networks, allowing us to connect limiting results obtained in previous studies within a unified framework. In total, these results advance o…
Hebrew University（希伯来大学）；Weizmann Institute of Science（魏茨曼科学研究所）
Several recent works have shown separation results between deep neural networks, and hypothesis classes with inferior approximation capacity such as shallow networks or kernel classes. On the other hand, the fact that deep networks can efficiently express a target function does not mean that this target function can be learned efficiently by deep neural networks. In this work we study the intricate connection between learnability and approximation capacity. We show that learnability with deep networks of a target function depends on the ability of simpler classes to approximate the target. Spe…
Freie Universität Berlin（柏林自由大学）；London School of Economics and Political Science（伦敦政治经济学院）；University of Twente（特文特大学）
We prove that the set of functions representable by ReLU neural networks with integer weights strictly increases with the network depth while allowing arbitrary width. More precisely, we show that $\lceil\log_2(n)\rceil$ hidden layers are indeed necessary to compute the maximum of $n$ numbers, matching known upper bounds. Our results are based on the known duality between neural networks and Newton polytopes via tropical geometry. The integrality assumption implies that these Newton polytopes are lattice polytopes. Then, our depth lower bounds follow from a parity argument on the normalized vo…
University of Cambridge（剑桥大学）；Microsoft Research（微软研究院）；The Alan Turing Institute（艾伦·图灵研究所）
Existing methods for estimating uncertainty in deep learning tend to require multiple forward passes, making them unsuitable for applications where computational resources are limited. To solve this, we perform probabilistic reasoning over the depth of neural networks. Different depths correspond to subnetworks which share weights and whose predictions are combined via marginalisation, yielding model uncertainty. By exploiting the sequential structure of feed-forward networks, we are able to both evaluate our training objective and make predictions with a single forward pass. We validate our a…
We introduce Graph Neural Processes (GNP), inspired by the recent work in conditional and latent neural processes. A Graph Neural Process is defined as a Conditional Neural Process that operates on arbitrary graph data. It takes features of sparsely observed context points as input, and outputs a distribution over target points. We demonstrate graph neural processes in edge imputation and discuss benefits and drawbacks of the method for other application areas. One major benefit of GNPs is the ability to quantify uncertainty in deep learning on graph structures. An additional benefit of this m…
In studying the expressiveness of neural networks, an important question is whether there are functions which can only be approximated by sufficiently deep networks, assuming their size is bounded. However, for constant depths, existing results are limited to depths $2$ and $3$, and achieving results for higher depths has been an important open question. In this paper, we focus on feedforward ReLU networks, and prove fundamental barriers to proving such results beyond depth $4$, by reduction to open problems and natural-proof barriers in circuit complexity. To show this, we study a seemingly u…
This paper studies the expressive power of graph neural networks falling within the message-passing framework (GNNmp). Two results are presented. First, GNNmp are shown to be Turing universal under sufficient conditions on their depth, width, node attributes, and layer expressiveness. Second, it is discovered that GNNmp can lose a significant portion of their power when their depth and width is restricted. The proposed impossibility statements stem from a new technique that enables the repurposing of seminal results from distributed computing and leads to lower bounds for an array of decision,…
RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）；The University of Tokyo（东京大学）；University of Tokyo（东京大学）
It has been recognized that heavily overparameterized deep neural networks (DNNs) exhibit surprisingly good generalization performance in various machine-learning tasks. Although benefits of depth have been investigated from different perspectives such as the approximation theory and the statistical learning theory, existing theories do not adequately explain the empirical success of overparameterized DNNs. In this work, we report a remarkable interplay between depth and locality of a target function. We introduce $k$-local and $k$-global functions, and find that depth is beneficial for learni…
In this paper, we study the infinite-depth limit of finite-width residual neural networks with random Gaussian weights. With proper scaling, we show that by fixing the width and taking the depth to infinity, the pre-activations converge in distribution to a zero-drift diffusion process. Unlike the infinite-width limit where the pre-activation converge weakly to a Gaussian random variable, we show that the infinite-depth limit yields different distributions depending on the choice of the activation function. We document two cases where these distributions have closed-form (different) expression…
We propose a neural machine translation architecture that models the surrounding text in addition to the source sentence. These models lead to better performance, both in terms of general translation quality and pronoun prediction, when trained on small corpora, although this improvement largely disappears when trained with a larger corpus. We also discover that attention-based neural machine translation is well suited for pronoun prediction and compares favorably with other approaches that were specifically designed for this task.
In practice, multi-task learning (through learning features shared among tasks) is an essential property of deep neural networks (NNs). While infinite-width limits of NNs can provide good intuition for their generalization behavior, the well-known infinite-width limits of NNs in the literature (e.g., neural tangent kernels) assume specific settings in which wide ReLU-NNs behave like shallow Gaussian Processes with a fixed kernel. Consequently, in such settings, these NNs lose their ability to benefit from multi-task learning in the infinite-width limit. In contrast, we prove that optimizing wi…
National University of Singapore（新加坡国立大学）；University of Oxford（牛津大学）
Recent work by Jacot et al. (2018) has shown that training a neural network using gradient descent in parameter space is related to kernel gradient descent in function space with respect to the Neural Tangent Kernel (NTK). Lee et al. (2019) built on this result by establishing that the output of a neural network trained using gradient descent can be approximated by a linear model when the network width is large. Indeed, under regularity conditions, the NTK converges to a time-independent kernel in the infinite-width limit. This regime is often called the NTK regime. In parallel, recent works o…
Massachusetts Institute of Technology（麻省理工学院）；Tel Aviv University（特拉维夫大学）；eBay Research（eBay研究院）
Recent results in the literature suggest that the penultimate (second-to-last) layer representations of neural networks that are trained for classification exhibit a clustering property called neural collapse (NC). We study the implicit bias of stochastic gradient descent (SGD) in favor of low-depth solutions when training deep neural networks. We characterize a notion of effective depth that measures the first layer for which sample embeddings are separable using the nearest-class center classifier. Furthermore, we hypothesize and empirically show that SGD implicitly selects neural networks o…
The selection of initial parameter values for gradient-based optimization of deep neural networks is one of the most impactful hyperparameter choices in deep learning systems, affecting both convergence times and model performance. Yet despite significant empirical and theoretical analysis, relatively little has been proved about the concrete effects of different initialization schemes. In this work, we analyze the effect of initialization in deep linear networks, and provide for the first time a rigorous proof that drawing the initial weights from the orthogonal group speeds up convergence re…
Neural Ordinary Differential Equations (Neural ODEs) are the continuous analog of Residual Neural Networks (ResNets). We investigate whether the discrete dynamics defined by a ResNet are close to the continuous one of a Neural ODE. We first quantify the distance between the ResNet's hidden state trajectory and the solution of its corresponding Neural ODE. Our bound is tight and, on the negative side, does not go to 0 with depth N if the residual functions are not smooth with depth. On the positive side, we show that this smoothness is preserved by gradient descent for a ResNet with linear resi…
Texas A&M University（得克萨斯农工大学）；University of Toronto（多伦多大学）
We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not deterministic if depth and width simultaneously tend to infinity. Moreover, we prove that for such deep and wide networks, the NTK has a non-trivial evolution during training by showing that the mean of its first SGD update is also exponential in the ratio of network depth to width. This is…
Graph Convolutional Networks (GCNs) are known to suffer from performance degradation as the number of layers increases, which is usually attributed to over-smoothing. Despite the apparent consensus, we observe that there exists a discrepancy between the theoretical understanding of over-smoothing and the practical capabilities of GCNs. Specifically, we argue that over-smoothing does not necessarily happen in practice, a deeper model is provably expressive, can converge to global optimum with linear convergence rate, and achieve very high training accuracy as long as properly trained. Despite b…
We introduce the unbounded depth neural network (UDN), an infinitely deep probabilistic model that adapts its complexity to the training data. The UDN contains an infinite sequence of hidden layers and places an unbounded prior on a truncation L, the layer from which it produces its data. Given a dataset of observations, the posterior UDN provides a conditional distribution of both the parameters of the infinite neural network and its truncation. We develop a novel variational inference algorithm to approximate this posterior, optimizing a distribution of the neural network weights and of the …
Let $f:\mathbb{S}^{d-1}\times \mathbb{S}^{d-1}\to\mathbb{S}$ be a function of the form $f(\mathbf{x},\mathbf{x}') = g(\langle\mathbf{x},\mathbf{x}'\rangle)$ for $g:[-1,1]\to \mathbb{R}$. We give a simple proof that shows that poly-size depth two neural networks with (exponentially) bounded weights cannot approximate $f$ whenever $g$ cannot be approximated by a low degree polynomial. Moreover, for many $g$'s, such as $g(x)=\sin(πd^3x)$, the number of neurons must be $2^{Ω\left(d\log(d)\right)}$. Furthermore, the result holds w.r.t.\ the uniform distribution on $\mathbb{S}^{d-1}\times \mathbb{S}…
KAIST（韩国科学技术院）；The University of Tokyo（东京大学）；Syntensor
We introduce the framework of continuous-depth graph neural networks (GNNs). Neural graph differential equations (Neural GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of GNN layers, blending discrete topological structures and differential equations. The proposed framework is shown to be compatible with static GNN models and is extended to dynamic and stochastic settings through hybrid dynamical system theory. Here, Neural GDEs improve performance by exploiting the underlying dynamics geometry, further introducing the ability t…
MILA, Université de Montréal（蒙特利尔大学 MILA）；University of Toronto（多伦多大学）；Institut des Hautes Études Scientifiques, France（法国高等科学研究所）；Carnegie Mellon University（卡内基梅隆大学）
In this paper, we systematically analyze the connecting architectures of recurrent neural networks (RNNs). Our main contribution is twofold: first, we present a rigorous graph-theoretic framework describing the connecting architectures of RNNs in general. Second, we propose three architecture complexity measures of RNNs: (a) the recurrent depth, which captures the RNN's over-time nonlinear complexity, (b) the feedforward depth, which captures the local input-output nonlinearity (similar to the "depth" in feedforward neural networks (FNNs)), and (c) the recurrent skip coefficient which captures…
Adversarial attacks against machine learning models are a rather hefty obstacle to our increasing reliance on these models. Due to this, provably robust (certified) machine learning models are a major topic of interest. Lipschitz continuous models present a promising approach to solving this problem. By leveraging the expressive power of a variant of neural networks which maintain low Lipschitz constants, we prove that three layer neural networks using the FullSort activation function are Universal Lipschitz function Approximators (ULAs). This both explains experimental results and paves the w…
National University of Singapore（新加坡国立大学）；University of Maryland, College Park（马里兰大学帕克分校）
This paper proposes a new neural network architecture by introducing an additional dimension called height beyond width and depth. Neural network architectures with height, width, and depth as hyper-parameters are called three-dimensional architectures. It is shown that neural networks with three-dimensional architectures are significantly more expressive than the ones with two-dimensional architectures (those with only width and depth as hyper-parameters), e.g., standard fully connected networks. The new network architecture is constructed recursively via a nested structure, and hence we call…
University of Toronto（多伦多大学）；Stanford Unviersity（斯坦福大学）
We perform scalable approximate inference in continuous-depth Bayesian neural networks. In this model class, uncertainty about separate weights in each layer gives hidden units that follow a stochastic differential equation. We demonstrate gradient-based stochastic variational inference in this infinite-parameter setting, producing arbitrarily-flexible approximate posteriors. We also derive a novel gradient estimator that approaches zero variance as the approximate posterior over weights approaches the true posterior. This approach brings continuous-depth Bayesian neural nets to a competitive …
While classic studies proved that wide networks allow universal approximation, recent research and successes of deep learning demonstrate the power of deep networks. Based on a symmetric consideration, we investigate if the design of artificial neural networks should have a directional preference, and what the mechanism of interaction is between the width and depth of a network. Inspired by the De Morgan law, we address this fundamental question by establishing a quasi-equivalence between the width and depth of ReLU networks in two aspects. First, we formulate two transforms for mapping an arb…
On the Depth of Monotone ReLU Neural Networks and ICNNs
Department of Computer Science, the University of Copenhagen（计算机科学系，哥本哈根大学）；University of Technology Nuremberg（纽伦堡技术大学）；Worcester Polytechnic Institute（沃思堡理工学院）；Department of Mathematics, Technion-IIT（数学系，技术学院-以色列理工学院）
We provide a unifying framework where artificial neural networks and their architectures can be formally described as particular cases of a general mathematical construction--machines of finite depth. Unlike neural networks, machines have a precise definition, from which several properties follow naturally. Machines of finite depth are modular (they can be combined), efficiently computable and differentiable. The backward pass of a machine is again a machine and can be computed without overhead using the same procedure as the forward pass. We prove this statement theoretically and practically,…
Recent years have witnessed an increasing interest in the correspondence between infinitely wide networks and Gaussian processes. Despite the effectiveness and elegance of the current neural network Gaussian process theory, to the best of our knowledge, all the neural network Gaussian processes are essentially induced by increasing width. However, in the era of deep learning, what concerns us more regarding a neural network is its depth as well as how depth impacts the behaviors of a network. Inspired by a width-depth symmetry consideration, we use a shortcut network to show that increasing th…
One of the central problems in the study of deep learning theory is to understand how the structure properties, such as depth, width and the number of nodes, affect the expressivity of deep neural networks. In this work, we show a new connection between the expressivity of deep neural networks and topological entropy from dynamical system, which can be used to characterize depth-width trade-offs of neural networks. We provide an upper bound on the topological entropy of neural networks with continuous semi-algebraic units by the structure parameters. Specifically, the topological entropy of Re…
Northwestern Polytechnical University（西北工业大学）；Australian National University（澳大利亚国立大学）
This paper proposes a new residual convolutional neural network (CNN) architecture for single image depth estimation. Compared with existing deep CNN based methods, our method achieves much better results with fewer training examples and model parameters. The advantages of our method come from the usage of dilated convolution, skip connection architecture and soft-weight-sum inference. Experimental evaluation on the NYU Depth V2 dataset shows that our method outperforms other state-of-the-art methods by a margin.
Depth-induced NTK: Bridging Over-parameterized Neural Networks and Deep Neural Kernels
State Key Laboratory of Novel Software Technology, Nanjing University, China（新型软件技术国家重点实验室，南京大学，中国）；School of Intelligent Science and Technology, Nanjing University, China（智能科学与技术学院，南京大学，中国）；Department of Data Science, City University of Hong Kong, Hong Kong（数据科学系，香港城市大学，香港）
Northwestern University（西北大学）；HSR University of Applied Science Rapperswil（瑞士东部应用科技大学拉珀斯维尔分校）
Capsule networks are a recently developed class of neural networks that potentially address some of the deficiencies with traditional convolutional neural networks. By replacing the standard scalar activations with vectors, and by connecting the artificial neurons in a new way, capsule networks aim to be the next great development for computer vision applications. However, in order to determine whether these networks truly operate differently than traditional networks, one must look at the differences in the capsule features. To this end, we perform several analyses with the purpose of elucida…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Zhejiang University（浙江大学）；Instituto Nacional de Matemática Pura e Aplicada（巴西国家纯粹与应用数学研究所）
Depth maps captured with commodity sensors often require super-resolution to be used in applications. In this work we study a super-resolution approach based on a variational problem statement with Tikhonov regularization where the regularizer is parametrized with a deep neural network. This approach was previously applied successfully in photoacoustic tomography. We experimentally show that its application to depth map super-resolution is difficult, and provide suggestions about the reasons for that.
The standard neural machine translation model can only decode with the same depth configuration as training. Restricted by this feature, we have to deploy models of various sizes to maintain the same translation latency, because the hardware conditions on different terminal devices (e.g., mobile phones) may vary greatly. Such individual training leads to increased model maintenance costs and slower model iterations, especially for the industry. In this work, we propose to use multi-task learning to train a flexible depth model that can adapt to different depth configurations during inference. …
Graph neural networks (GNNs) are widely used for modelling graph-structured data in numerous applications. However, with their inherently finite aggregation layers, existing GNN models may not be able to effectively capture long-range dependencies in the underlying graphs. Motivated by this limitation, we propose a GNN model with infinite depth, which we call Efficient Infinite-Depth Graph Neural Networks (EIGNN), to efficiently capture very long-range dependencies. We theoretically derive a closed-form solution of EIGNN which makes training an infinite-depth GNN model tractable. We then furth…
Stanford University（斯坦福大学）；Singapore University of Technology and Design（新加坡科技设计大学）
The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new connection with dynamical systems, using a generalized notion of fixed points of a continuous map $f$, called periodic points. In this work, we strengthen the connection with dynamical systems and we improve the existing width lower bounds along several aspects. Our first main result is period-specific width lower bounds that hold under the stronger notion of $L^1…
People believe that depth plays an important role in success of deep neural networks (DNN). However, this belief lacks solid theoretical justifications as far as we know. We investigate role of depth from perspective of margin bound. In margin bound, expected error is upper bounded by empirical margin error plus Rademacher Average (RA) based capacity term. First, we derive an upper bound for RA of DNN, and show that it increases with increasing depth. This indicates negative impact of depth on test performance. Second, we show that deeper networks tend to have larger representation power (meas…
Consider the problem: given the data pair $(\mathbf{x}, \mathbf{y})$ drawn from a population with $f_*(x) = \mathbf{E}[\mathbf{y} | \mathbf{x} = x]$, specify a neural network model and run gradient flow on the weights over time until reaching any stationarity. How does $f_t$, the function computed by the neural network at time $t$, relate to $f_*$, in terms of approximation and representation? What are the provable benefits of the adaptive representation by neural networks compared to the pre-specified fixed basis representation in the classical nonparametric literature? We answer the above qu…
The infinite-depth paradigm pioneered by Neural ODEs has launched a renaissance in the search for novel dynamical system-inspired deep learning primitives; however, their utilization in problems of non-trivial size has often proved impossible due to poor computational scalability. This work paves the way for scalable Neural ODEs with time-to-prediction comparable to traditional discrete networks. We introduce hypersolvers, neural networks designed to solve ODEs with low overhead and theoretical guarantees on accuracy. The synergistic combination of hypersolvers and Neural ODEs allows for cheap…
Google Research（谷歌研究院）；Northwestern University（西北大学）
We present polynomial time and sample efficient algorithms for learning an unknown depth-2 feedforward neural network with general ReLU activations, under mild non-degeneracy assumptions. In particular, we consider learning an unknown network of the form $f(x) = {a}^{\mathsf{T}}σ({W}^\mathsf{T}x+b)$, where $x$ is drawn from the Gaussian distribution, and $σ(t) := \max(t,0)$ is the ReLU activation. Prior works for learning networks with ReLU activations assume that the bias $b$ is zero. In order to deal with the presence of the bias terms, our proposed algorithm consists of robustly decomposing…
We present and review an algorithmic and theoretical framework for improving neural network architecture design via momentum. As case studies, we consider how momentum can improve the architecture design for recurrent neural networks (RNNs), neural ordinary differential equations (ODEs), and transformers. We show that integrating momentum into neural network architectures has several remarkable theoretical and empirical benefits, including 1) integrating momentum into RNNs and neural ODEs can overcome the vanishing gradient issues in training RNNs and neural ODEs, resulting in effective learni…
