One-shot neural architecture search allows joint learning of weights and network architecture, reducing computational cost. We limit our search space to the depth of residual networks and formulate an analytically tractable variational objective that allows for obtaining an unbiased approximate posterior over depths in one-shot. We propose a heuristic to prune our networks based on this distribution. We compare our proposed method against manual search over network depths on the MNIST, Fashion-MNIST, SVHN datasets. We find that pruned networks do not incur a loss in predictive performance, obtaining accuracies competitive with unpruned networks. Marginalising over depth allows us to obtain better-calibrated test-time uncertainty estimates than regular networks, in a single forward pass.
Massachusetts Institute of Technology（麻省理工学院）；University of Montreal（蒙特利尔大学）
In this paper, we prove that depth with nonlinearity creates no bad local minima in a type of arbitrarily deep ResNets with arbitrary nonlinear activation functions, in the sense that the values of all local minima are no worse than the global minimum value of corresponding classical machine-learning models, and are guaranteed to further improve via residual representations. As a result, this paper provides an affirmative answer to an open question stated in a paper in the conference on Neural Information Processing Systems 2018. This paper advances the optimization theory of deep learning onl…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；DeepMind（深度思维）
Training very deep neural networks is still an extremely challenging task. The common solution is to use shortcut connections and normalization layers, which are both crucial ingredients in the popular ResNet architecture. However, there is strong evidence to suggest that ResNets behave more like ensembles of shallower networks than truly deep ones. Recently, it was shown that deep vanilla networks (i.e. networks without normalization layers or shortcut connections) can be trained as fast as ResNets by applying certain transformations to their activation functions. However, this method (called…
Two main obstacles preventing the widespread adoption of variational Bayesian neural networks are the high parameter overhead that makes them infeasible on large networks, and the difficulty of implementation, which can be thought of as "programming overhead." MC dropout [Gal and Ghahramani, 2016] is popular because it sidesteps these obstacles. Nevertheless, dropout is often harmful to model performance when used in networks with batch normalization layers [Li et al., 2018], which are an indispensable part of modern neural networks. We construct a general variational family for ensemble-based…
Batch Normalization is a key component in almost all state-of-the-art image classifiers, but it also introduces practical challenges: it breaks the independence between training examples within a batch, can incur compute and memory overhead, and often results in unexpected bugs. Building on recent theoretical analyses of deep ResNets at initialization, we propose a simple set of analysis tools to characterize signal propagation on the forward pass, and leverage these tools to design highly performant ResNets without activation normalization layers. Crucial to our success is an adapted version …
Deep ResNet architectures have achieved state of the art performance on many tasks. While they solve the problem of gradient vanishing, they might suffer from gradient exploding as the depth becomes large (Yang et al. 2017). Moreover, recent results have shown that ResNet might lose expressivity as the depth goes to infinity (Yang et al. 2017, Hayou et al. 2019). To resolve these issues, we introduce a new class of ResNet architectures, called Stable ResNet, that have the property of stabilizing the gradient while ensuring expressivity in the infinite depth limit.
We demonstrate that in residual neural networks (ResNets) dynamical isometry is achievable irrespectively of the activation function used. We do that by deriving, with the help of Free Probability and Random Matrix Theories, a universal formula for the spectral density of the input-output Jacobian at initialization, in the large network width and depth limit. The resulting singular value spectrum depends on a single parameter, which we calculate for a variety of popular activation functions, by analyzing the signal propagation in the artificial neural network. We corroborate our results with n…
University of Oxford（牛津大学）；Huawei Technologies（华为技术有限公司）
Regularization plays a major role in modern deep learning. From classic techniques such as L1,L2 penalties to other noise-based methods such as Dropout, regularization often yields better generalization properties by avoiding overfitting. Recently, Stochastic Depth (SD) has emerged as an alternative regularization technique for residual neural networks (ResNets) and has proven to boost the performance of ResNet on many tasks [Huang et al., 2016]. Despite the recent success of SD, little is known about this technique from a theoretical perspective. This paper provides a hybrid analysis combinin…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；University of Guelph（圭尔夫大学）
Theoretical results show that neural networks can be approximated by Gaussian processes in the infinite-width limit. However, for fully connected networks, it has been previously shown that for any fixed network width, $n$, the Gaussian approximation gets worse as the network depth, $d$, increases. Given that modern networks are deep, this raises the question of how well modern architectures, like ResNets, are captured by the infinite-width limit. To provide a better approximation, we study ReLU ResNets in the infinite-depth-and-width limit, where both depth and width tend to infinity as their…
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）
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…
Sorbonne Université（索邦大学）；MINES ParisTech（巴黎高等矿业学院）；PSL Research University（巴黎文理研究大学）；Institut Curie（居里研究所）；INSERM（法国国家健康与医学研究院）；Institut universitaire de France（法国大学研究院）；Google Research（谷歌研究院）；EPFL（洛桑联邦理工学院）；Califrais（加利弗雷）；Owkin（奥金公司）
The Residual Network (ResNet), proposed in He et al. (2015), utilized shortcut connections to significantly reduce the difficulty of training, which resulted in great performance boosts in terms of both training and generalization error. It was empirically observed in He et al. (2015) that stacking more layers of residual blocks with shortcut 2 results in smaller training error, while it is not true for shortcut of length 1 or 3. We provide a theoretical explanation for the uniqueness of shortcut 2. We show that with or without nonlinearities, by adding shortcuts that have depth two, the condi…
Google Research（谷歌研究院）；ICSI and UC Berkeley（国际计算机科学研究所和加州大学伯克利分校）；University at Buffalo, SUNY（纽约州立大学布法罗分校）
Recent work has attempted to interpret residual networks (ResNets) as one step of a forward Euler discretization of an ordinary differential equation, focusing mainly on syntactic algebraic similarities between the two systems. Discrete dynamical integrators of continuous dynamical systems, however, have a much richer structure. We first show that ResNets fail to be meaningful dynamical integrators in this richer sense. We then demonstrate that neural network models can learn to represent continuous dynamical systems, with this richer structure and properties, by embedding them into higher-ord…
Microsoft Research Asia（微软亚洲研究院）；Sun Yat-sen University（中山大学）；University of Chinese Academy of Sciences（中国科学院大学）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）
We study the stability and convergence of training deep ResNets with gradient descent. Specifically, we show that the parametric branch in the residual block should be scaled down by a factor $τ=O(1/\sqrt{L})$ to guarantee stable forward/backward process, where $L$ is the number of residual blocks. Moreover, we establish a converse result that the forward process is unbounded when $τ>L^{-\frac{1}{2}+c}$, for any positive constant $c$. The above two results together establish a sharp value of the scaling factor in determining the stability of deep ResNet. Based on the stability result, we furth…
Princeton University（普林斯顿大学）；Northwestern University（西北大学）；University of Minnesota（明尼苏达大学）；Georgia Institute of Technology（佐治亚理工学院）
We establish a margin based data dependent generalization error bound for a general family of deep neural networks in terms of the depth and width, as well as the Jacobian of the networks. Through introducing a new characterization of the Lipschitz properties of neural network family, we achieve significantly tighter generalization bounds than existing results. Moreover, we show that the generalization bound can be further improved for bounded losses. Aside from the general feedforward deep neural networks, our results can be applied to derive new bounds for popular architectures, including co…
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 …
Peking University（北京大学）；Georgia Institute of Technology（佐治亚理工学院）
Deep residual networks (ResNets) have demonstrated better generalization performance than deep feedforward networks (FFNets). However, the theory behind such a phenomenon is still largely unknown. This paper studies this fundamental problem in deep learning from a so-called "neural tangent kernel" perspective. Specifically, we first show that under proper conditions, as the width goes to infinity, training deep ResNets can be viewed as learning reproducing kernel functions with some kernel function. We then compare the kernel of deep ResNets with that of deep FFNets and discover that the class…
The training of deep residual neural networks (ResNets) with backpropagation has a memory cost that increases linearly with respect to the depth of the network. A way to circumvent this issue is to use reversible architectures. In this paper, we propose to change the forward rule of a ResNet by adding a momentum term. The resulting networks, momentum residual neural networks (Momentum ResNets), are invertible. Unlike previous invertible architectures, they can be used as a drop-in replacement for any existing ResNet block. We show that Momentum ResNets can be interpreted in the infinitesimal s…
In this paper we introduce a novel method of gradient normalization and decay with respect to depth. Our method leverages the simple concept of normalizing all gradients in a deep neural network, and then decaying said gradients with respect to their depth in the network. Our proposed normalization and decay techniques can be used in conjunction with most current state of the art optimizers and are a very simple addition to any network. This method, although simple, showed improvements in convergence time on state of the art networks such as DenseNet and ResNet on image classification tasks, a…
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…
University of Oxford（牛津大学）；OATML, Department of Computer Science（OATML计算机科学系）
We challenge the longstanding assumption that the mean-field approximation for variational inference in Bayesian neural networks is severely restrictive, and show this is not the case in deep networks. We prove several results indicating that deep mean-field variational weight posteriors can induce similar distributions in function-space to those induced by shallower networks with complex weight posteriors. We validate our theoretical contributions empirically, both through examination of the weight posterior using Hamiltonian Monte Carlo in small models and by comparing diagonal- to structure…
We introduce a GP generalization of ResNets (including ResNets as a particular case). We show that ResNets (and their GP generalization) converge, in the infinite depth limit, to a generalization of image registration variational algorithms. Whereas computational anatomy aligns images via warping of the material space, this generalization aligns ideas (or abstract shapes as in Plato's theory of forms) via the warping of the RKHS of functions mapping the input space to the output space. While the Hamiltonian interpretation of ResNets is not new, it was based on an Ansatz. We do not rely on this…
Cogent Labs（Cogent实验室）；University of Torino（都灵大学）；Collegio Carlo Alberto（阿尔贝托国王学院）
Modern neural networks (NN) featuring a large number of layers (depth) and units per layer (width) have achieved a remarkable performance across many domains. While there exists a vast literature on the interplay between infinitely wide NNs and Gaussian processes, a little is known about analogous interplays with respect to infinitely deep NNs. NNs with independent and identically distributed (i.i.d.) initializations exhibit undesirable forward and backward propagation properties as the number of layers increases. To overcome these drawbacks, Peluchetti and Favaro (2020) considered fully-conne…
A long-standing obstacle to progress in deep learning is the problem of vanishing and exploding gradients. Although, the problem has largely been overcome via carefully constructed initializations and batch normalization, architectures incorporating skip-connections such as highway and resnets perform much better than standard feedforward architectures despite well-chosen initialization and batch normalization. In this paper, we identify the shattered gradients problem. Specifically, we show that the correlation between gradients in standard feedforward networks decays exponentially with depth…
This work gives an in-depth derivation of the trainable evidence lower bound obtained from the marginal joint log-Likelihood with the goal of training a Multi-Modal Variational Autoencoder (M$^2$VAE).
Recent works have demonstrated that increasing model capacity through width in over-parameterized neural networks leads to a decrease in test risk. For neural networks, however, model capacity can also be increased through depth, yet understanding the impact of increasing depth on test risk remains an open question. In this work, we demonstrate that the test risk of over-parameterized convolutional networks is a U-shaped curve (i.e. monotonically decreasing, then increasing) with increasing depth. We first provide empirical evidence for this phenomenon via image classification experiments usin…
Peking University（北京大学）；MGH/BWH Center for Clinical Data Science（麻省总医院/布列根和妇女医院临床数据科学中心）；Massachusetts General Hospital（麻省总医院）；Harvard Medical School（哈佛医学院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Center for Data Science, Peking University（北京大学数据科学中心）；Beijing Institute of Big Data Research（北京大数据研究院）
In our work, we bridge deep neural network design with numerical differential equations. We show that many effective networks, such as ResNet, PolyNet, FractalNet and RevNet, can be interpreted as different numerical discretizations of differential equations. This finding brings us a brand new perspective on the design of effective deep architectures. We can take advantage of the rich knowledge in numerical analysis to guide us in designing new and potentially more effective deep networks. As an example, we propose a linear multi-step architecture (LM-architecture) which is inspired by the lin…
Improving performance of deep learning models and reducing their training times are ongoing challenges in deep neural networks. There are several approaches proposed to address these challenges one of which is to increase the depth of the neural networks. Such deeper networks not only increase training times, but also suffer from vanishing gradients problem while training. In this work, we propose gradient amplification approach for training deep learning models to prevent vanishing gradients and also develop a training strategy to enable or disable gradient amplification method across several…
Field theory for optimal signal propagation in ResNets
Institute for Advanced Simulation (IAS-6), Jülich Research Centre（先进模拟研究所（IAS-6）、尤利希研究中心）；RWTH Aachen University（亚琛工业大学）；Department of Physics, Faculty 1, RWTH Aachen University（物理系，亚琛工业大学）
Duke University（杜克大学）；University of Nevada, Reno（内华达大学雷诺分校）
Depth is a key component of Deep Neural Networks (DNNs), however, designing depth is heuristic and requires many human efforts. We propose AutoGrow to automate depth discovery in DNNs: starting from a shallow seed architecture, AutoGrow grows new layers if the growth improves the accuracy; otherwise, stops growing and thus discovers the depth. We propose robust growing and stopping policies to generalize to different network architectures and datasets. Our experiments show that by applying the same policy to different network architectures, AutoGrow can always discover near-optimal depth on va…
Variational Learning is Effective for Large Deep Networks
Technical University of Munich（慕尼黑工业大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）；Technical University of Darmstadt（达姆施塔特工业大学）；Tokyo Institute of Technology（东京工业大学）；RIKEN Center for AI Project（理化学研究所人工智能项目中心）
