Deep residual architectures, such as ResNet and the Transformer, have enabled models of unprecedented depth, yet a formal understanding of why depth is so effective remains an open question. A popular intuition, following Veit et al. (2016), is that these residual networks behave like ensembles of many shallower models. Our key finding is an explicit analytical formula that verifies this ensemble perspective, proving that increasing network depth is mathematically equivalent to expanding the size of this implicit ensemble. Furthermore, our expansion reveals a hierarchical ensemble structure in which the combinatorial growth of computation paths leads to an explosion in the output signal, explaining the historical necessity of normalization layers in training deep models. This insight offers a first principles explanation for the historical dependence on normalization layers and sheds new light on a family of successful normalization-free techniques like SkipInit and Fixup. However, while these previous approaches infer scaling factors through optimizer analysis or a heuristic analogy to Batch Normalization, our work offers the first explanation derived directly from the network's inherent functional structure. Specifically, our Residual Expansion Theorem reveals that scaling each residual module provides a principled solution to taming the combinatorial explosion inherent to these architectures. We further show that this scaling acts as a capacity controls that also implicitly regularizes the model's complexity.
On the Depth Transfer of Neural Optimizers via L-Smoothness
University of Chicago（芝加哥大学）；Peking University（北京大学）；University of British Columbia（不列颠哥伦比亚大学）；Sauder School of Business（尚德商学院）；Beijing International Center for Mathematical Research（北京国际数学研究中心）
CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）
The training success, training speed and generalization ability of neural networks rely crucially on the choice of random parameter initialization. It has been shown for multiple architectures that initial dynamical isometry is particularly advantageous. Known initialization schemes for residual blocks, however, miss this property and suffer from degrading separability of different inputs for increasing depth and instability without Batch Normalization or lack feature diversity. We propose a random initialization scheme, RISOTTO, that achieves perfect dynamical isometry for residual networks w…
Deep Linear Network Training Dynamics from Random Initialization: Data, Width, Depth, and Hyperparameter Transfer
John Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·保罗森工程与应用科学学校）；Center for Brain Sciences（脑科学研究中心）；Kempner Institute（凯普纳研究所）
PLA Strategic Support Force Information Engineering University（中国人民解放军战略支援部队信息工程大学）
We present an efficient multi-view stereo (MVS) network for 3D reconstruction from multiview images. While previous learning based reconstruction approaches performed quite well, most of them estimate depth maps at a fixed resolution using plane sweep volumes with a fixed depth hypothesis at each plane, which requires densely sampled planes for desired accuracy and therefore is difficult to achieve high resolution depth maps. In this paper we introduce a coarseto-fine depth inference strategy to achieve high resolution depth. This strategy estimates the depth map at coarsest level, while the d…
We propose quadratic residual networks (QRes) as a new type of parameter-efficient neural network architecture, by adding a quadratic residual term to the weighted sum of inputs before applying activation functions. With sufficiently high functional capacity (or expressive power), we show that it is especially powerful for solving forward and inverse physics problems involving partial differential equations (PDEs). Using tools from algebraic geometry, we theoretically demonstrate that, in contrast to plain neural networks, QRes shows better parameter efficiency in terms of network width and de…
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…
We derive finite width and depth corrections for the Neural Tangent Kernel (NTK) of ResNets and DenseNets. Our analysis reveals that finite size residual architectures are initialized much closer to the "kernel regime" than their vanilla counterparts: while in networks that do not use skip connections, convergence to the NTK requires one to fix the depth, while increasing the layers' width. Our findings show that in ResNets, convergence to the NTK may occur when depth and width simultaneously tend to infinity, provided with a proper initialization. In DenseNets, however, convergence of the NTK…
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.
Batch normalization dramatically increases the largest trainable depth of residual networks, and this benefit has been crucial to the empirical success of deep residual networks on a wide range of benchmarks. We show that this key benefit arises because, at initialization, batch normalization downscales the residual branch relative to the skip connection, by a normalizing factor on the order of the square root of the network depth. This ensures that, early in training, the function computed by normalized residual blocks in deep networks is close to the identity function (on average). We use th…
LAIR-Net: Leaky Alignment-Impulse Residual Networks for Tabular Regression
Indian Institute of Technology Guwahati（印度技术学院加瓦哈蒂分校）；SAFIR, Sorbonne University Abu Dhabi（阿布扎比索邦大学 SAFIR）；Indian Institute of Technology Bombay（印度技术学院孟买分校）
The skip-connections used in residual networks have become a standard architecture choice in deep learning due to the increased training stability and generalization performance with this architecture, although there has been limited theoretical understanding for this improvement. In this work, we analyze overparameterized deep residual networks trained by gradient descent following random initialization, and demonstrate that (i) the class of networks learned by gradient descent constitutes a small subset of the entire neural network function class, and (ii) this subclass of networks is suffic…
The University of Western Australia（西澳大利亚大学）；The University of New South Wales（新南威尔士大学）
In this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a single hidden layer. The transformations of both deep rectifier nets and deep residual nets are conducted to demonstrate the advantages of the residual nets over the conventional neural nets and the advantages of the deep neural nets over the shallow neural nets. In summary, for …
We analyze the global convergence of gradient descent for deep linear residual networks by proposing a new initialization: zero-asymmetric (ZAS) initialization. It is motivated by avoiding stable manifolds of saddle points. We prove that under the ZAS initialization, for an arbitrary target matrix, gradient descent converges to an $\varepsilon$-optimal point in $O(L^3 \log(1/\varepsilon))$ iterations, which scales polynomially with the network depth $L$. Our result and the $\exp(Ω(L))$ convergence time for the standard initialization (Xavier or near-identity) [Shamir, 2018] together demonstrat…
Eindhoven University of Technology（埃因霍温理工大学）；Cooperatieve Rabobank U.A.（拉博银行）
Effectively detecting anomalous nodes in attributed networks is crucial for the success of many real-world applications such as fraud and intrusion detection. Existing approaches have difficulties with three major issues: sparsity and nonlinearity capturing, residual modeling, and network smoothing. We propose Residual Graph Convolutional Network (ResGCN), an attention-based deep residual modeling approach that can tackle these issues: modeling the attributed networks with GCN allows to capture the sparsity and nonlinearity; utilizing a deep neural network allows to directly learn residual fro…
The University of Manchester（曼彻斯特大学）；Beihang University（北京航空航天大学）；Qufu Normal University（曲阜师范大学）；North University of China（中北大学）
Eddy current testing (ECT) is an effective technique in the evaluation of the depth of metal surface defects. However, in practice, the evaluation primarily relies on the experience of an operator and is often carried out by manual inspection. In this paper, we address the challenges of automatic depth evaluation of metal surface defects by virtual of state-of-the-art deep learning (DL) techniques. The main contributions are three-fold. Firstly, a highly-integrated portable ECT device is developed, which takes advantage of an advanced field programmable gate array (Zynq-7020 system on chip) an…
Although ordinary differential equations (ODEs) provide insights for designing network architectures, its relationship with the non-residual convolutional neural networks (CNNs) is still unclear. In this paper, we present a novel ODE model by adding a damping term. It can be shown that the proposed model can recover both a ResNet and a CNN by adjusting an interpolation coefficient. Therefore, the damped ODE model provides a unified framework for the interpretation of residual and non-residual networks. The Lyapunov analysis reveals better stability of the proposed model, and thus yields robust…
We conduct mathematical analysis on the effect of batch normalization (BN) on gradient backpropogation in residual network training, which is believed to play a critical role in addressing the gradient vanishing/explosion problem, in this work. By analyzing the mean and variance behavior of the input and the gradient in the forward and backward passes through the BN and residual branches, respectively, we show that they work together to confine the gradient variance to a certain range across residual blocks in backpropagation. As a result, the gradient vanishing/explosion problem is avoided. W…
ELLIS Institute Tübingen（埃利斯研究所蒂宾根分所）；Max Planck Institute for Intelligent Systems, Tübingen（马克斯·普朗克智能系统研究所（蒂宾根））；Tübingen AI Center（蒂宾根人工智能中心）；ETH Zürich（苏黎世联邦理工学院）；Liquid AI（Liquid AI公司）
Residual Network (ResNet) is the state-of-the-art architecture that realizes successful training of really deep neural network. It is also known that good weight initialization of neural network avoids problem of vanishing/exploding gradients. In this paper, simplified models of ResNets are analyzed. We argue that goodness of ResNet is correlated with the fact that ResNets are relatively insensitive to choice of initial weights. We also demonstrate how batch normalization improves backpropagation of deep ResNets without tuning initial values of weights.
Training deep recurrent neural network (RNN) architectures is complicated due to the increased network complexity. This disrupts the learning of higher order abstracts using deep RNN. In case of feed-forward networks training deep structures is simple and faster while learning long-term temporal information is not possible. In this paper we propose a residual memory neural network (RMN) architecture to model short-time dependencies using deep feed-forward layers having residual and time delayed connections. The residual connection paves way to construct deeper networks by enabling unhindered f…
Deep networks often suffer from vanishing or exploding gradients due to inefficient signal propagation, leading to long training times or convergence difficulties. Various architecture designs, sophisticated residual-style networks, and initialization schemes have been shown to improve deep signal propagation. Recently, Pennington et al. used free probability theory to show that dynamical isometry plays an integral role in efficient deep learning. We show that the simplest architecture change of gating each residual connection using a single zero-initialized parameter satisfies initial dynamic…
Training deep neural networks with stochastic gradient descent (SGD) can often achieve zero training loss on real-world tasks although the optimization landscape is known to be highly non-convex. To understand the success of SGD for training deep neural networks, this work presents a mean-field analysis of deep residual networks, based on a line of works that interpret the continuum limit of the deep residual network as an ordinary differential equation when the network capacity tends to infinity. Specifically, we propose a new continuum limit of deep residual networks, which enjoys a good lan…
PCANet and its variants provided good accuracy results for classification tasks. However, despite the importance of network depth in achieving good classification accuracy, these networks were trained with a maximum of nine layers. In this paper, we introduce a residual compensation convolutional network, which is the first PCANet-like network trained with hundreds of layers while improving classification accuracy. The design of the proposed network consists of several convolutional layers, each followed by post-processing steps and a classifier. To correct the classification errors and signif…
Michigan State University（密歇根州立大学）；Sun Yat-sen University（中山大学）
Efforts to improve the adversarial robustness of convolutional neural networks have primarily focused on developing more effective adversarial training methods. In contrast, little attention was devoted to analyzing the role of architectural elements (such as topology, depth, and width) on adversarial robustness. This paper seeks to bridge this gap and present a holistic study on the impact of architectural design on adversarial robustness. We focus on residual networks and consider architecture design at the block level, i.e., topology, kernel size, activation, and normalization, as well as a…
University of California, Los Angeles（加州大学洛杉矶分校）；Amazon Web Services（亚马逊云服务）；Chung-Ang University（中央大学）
We present a method to infer a dense depth map from a color image and associated sparse depth measurements. Our main contribution lies in the design of an annealing process for determining co-visibility (occlusions, disocclusions) and the degree of regularization to impose on the model. We show that regularization and co-visibility are related via the fitness (residual) of model to data and both can be unified into a single framework to improve the learning process. Our method is an adaptive weighting scheme that guides optimization by measuring the residual at each pixel location over each tr…
Euler Institute, Università della Svizzera Italiana（瑞士意大利语区大学欧拉研究所）
We train deep residual networks with a stochastic variant of the nonlinear multigrid method MG/OPT. To build the multilevel hierarchy, we use the dynamical systems viewpoint specific to residual networks. We report significant speed-ups and additional robustness for training MNIST on deep residual networks. Our numerical experiments also indicate that multilevel training can be used as a pruning technique, as many of the auxiliary networks have accuracies comparable to the original network.
