One of the key problems of GNNs is how to describe the importance of neighbor nodes in the aggregation process for learning node representations. A class of GNNs solves this problem by learning implicit weights to represent the importance of neighbor nodes, which we call implicit GNNs such as Graph Attention Network. The basic idea of implicit GNNs is to introduce graph information with special properties followed by Learnable Transformation Structures (LTS) which encode the importance of neighbor nodes via a data-driven way. In this paper, we argue that LTS makes the special properties of graph information disappear during the learning process, resulting in graph information unhelpful for learning node representations. We call this phenomenon Graph Information Vanishing (GIV). Also, we find that LTS maps different graph information into highly similar results. To validate the above two points, we design two sets of 70 random experiments on five Implicit GNNs methods and seven benchmark datasets by using a random permutation operator to randomly disrupt the order of graph information and replacing graph information with random values. We find that randomization does not affect the model performance in 93\% of the cases, with about 7 percentage causing an average 0.5\% accuracy loss. And the cosine similarity of output results, generated by LTS mapping different graph information, over 99\% with an 81\% proportion. The experimental results provide evidence to support the existence of GIV in Implicit GNNs and imply that the existing methods of Implicit GNNs do not make good use of graph information. The relationship between graph information and LTS should be rethought to ensure that graph information is used in node representation.
Ruhr University Bochum（波鸿鲁尔大学）；University of Bonn（波恩大学）
Many popular variants of graph neural networks (GNNs) that are capable of handling multi-relational graphs may suffer from vanishing gradients. In this work, we propose a novel GNN architecture based on the Gated Graph Neural Network with an improved ability to handle long-range dependencies in multi-relational graphs. An experimental analysis on different synthetic tasks demonstrates that the proposed architecture outperforms several popular GNN models.
On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL condition
ESI Group-UZ Chair of the National Strategy on Artificial Intelligence（国家人工智能战略委员会ESI集团-乌达学院）；Aragon Institute of Engineering Research (I3A)（阿龙工程研究 institute）；Universidad de Zaragoza（萨拉戈塔大学）；Aragonese Foundation for Research and Development (ARAID)（阿拉贡研究与发展基金会）
It is well known that the problem of vanishing/exploding gradients is a challenge when training deep networks. In this paper, we describe another phenomenon, called vanishing nodes, that also increases the difficulty of training deep neural networks. As the depth of a neural network increases, the network's hidden nodes have more highly correlated behavior. This results in great similarities between these nodes. The redundancy of hidden nodes thus increases as the network becomes deeper. We call this problem vanishing nodes, and we propose the metric vanishing node indicator (VNI) for quantita…
University of Waterloo（滑铁卢大学）；Ruhr-University Bochum（波鸿鲁尔大学）
Deep learning requires several design choices, such as the nodes' activation functions and the widths, types, and arrangements of the layers. One consideration when making these choices is the vanishing-gradient problem, which is the phenomenon of algorithms getting stuck at suboptimal points due to small gradients. In this paper, we revisit the vanishing-gradient problem in the context of sigmoid-type activation. We use mathematical arguments to highlight two different sources of the phenomenon, namely large individual parameters and effects across layers, and to illustrate two simple remedie…
Rice University（莱斯大学）；Technical University of Munich（慕尼黑工业大学）
Normalization layers are widely used in deep neural networks to stabilize training. In this paper, we consider the training of convolutional neural networks with gradient descent on a single training example. This optimization problem arises in recent approaches for solving inverse problems such as the deep image prior or the deep decoder. We show that for this setup, channel normalization, which centers and normalizes each channel individually, avoids vanishing gradients, whereas, without normalization, gradients vanish which prevents efficient optimization. This effect prevails in deep singl…
The plateau phenomenon, wherein the loss value stops decreasing during the process of learning, has been reported by various researchers. The phenomenon is actively inspected in the 1990s and found to be due to the fundamental hierarchical structure of neural network models. Then the phenomenon has been thought as inevitable. However, the phenomenon seldom occurs in the context of recent deep learning. There is a gap between theory and reality. In this paper, using statistical mechanical formulation, we clarified the relationship between the plateau phenomenon and the statistical property of t…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Shanghai University（上海大学）
Deep Neural Networks (DNNs) training can be difficult due to vanishing and exploding gradients during weight optimization through backpropagation. To address this problem, we propose a general class of Hamiltonian DNNs (H-DNNs) that stem from the discretization of continuous-time Hamiltonian systems and include several existing DNN architectures based on ordinary differential equations. Our main result is that a broad set of H-DNNs ensures non-vanishing gradients by design for an arbitrary network depth. This is obtained by proving that, using a semi-implicit Euler discretization scheme, the b…
We give a rigorous analysis of the statistical behavior of gradients in a randomly initialized fully connected network N with ReLU activations. Our results show that the empirical variance of the squares of the entries in the input-output Jacobian of N is exponential in a simple architecture-dependent constant beta, given by the sum of the reciprocals of the hidden layer widths. When beta is large, the gradients computed by N at initialization vary wildly. Our approach complements the mean field theory analysis of random networks. From this point of view, we rigorously compute finite width cor…
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
In this paper, we propose a general scheme to analyze the gradient vanishing phenomenon, also known as the barren plateau phenomenon, in training quantum neural networks with the ZX-calculus. More precisely, we extend the barren plateaus theorem from unitary 2-design circuits to any parameterized quantum circuits under certain reasonable assumptions. The main technical contribution of this paper is representing certain integrations as ZX-diagrams and computing them with the ZX-calculus. The method is used to analyze four concrete quantum neural networks with different structures. It is shown t…
An overview of condensation phenomenon in deep learning
School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University（上海交通大学数学科学学院、自然科学研究院、教育部科学与工程计算重点实验室）；School of Artificial Intelligence, Shanghai Jiao Tong University（上海交通大学人工智能学院）
This paper revisits the so-called vanishing gradient phenomenon, which commonly occurs in deep randomly initialized neural networks. Leveraging an in-depth analysis of neural chains, we first show that vanishing gradients cannot be circumvented when the network width scales with less than O(depth), even when initialized with the popular Xavier and He initializations. Second, we extend the analysis to second-order derivatives and show that random i.i.d. initialization also gives rise to Hessian matrices with eigenspectra that vanish as networks grow in depth. Whenever this happens, optimizers a…
The Vanishing Gradient Problem for Stiff Neural Differential Equations
Department of Chemical Engineering（化学工程系）；University of California, Santa Barbara（加州大学圣芭芭拉分校）；Department of Mechanical Engineering（机械工程系）；Department of Computer Science（计算机科学系）
The reliability of a learning model is key to the successful deployment of machine learning in various industries. Creating a robust model, particularly one unaffected by adversarial attacks, requires a comprehensive understanding of the adversarial examples phenomenon. However, it is difficult to describe the phenomenon due to the complicated nature of the problems in machine learning. Consequently, many studies investigate the phenomenon by proposing a simplified model of how adversarial examples occur and validate it by predicting some aspect of the phenomenon. While these studies cover man…
This work develops rigorous theoretical basis for the fact that deep Bayesian neural network (BNN) is an effective tool for high-dimensional variable selection with rigorous uncertainty quantification. We develop new Bayesian non-parametric theorems to show that a properly configured deep BNN (1) learns the variable importance effectively in high dimensions, and its learning rate can sometimes "break" the curse of dimensionality. (2) BNN's uncertainty quantification for variable importance is rigorous, in the sense that its 95% credible intervals for variable importance indeed covers the truth…
In this paper, we study the adversarial examples existence and adversarial training from the standpoint of convergence and provide evidence that pointwise convergence in ANNs can explain these observations. The main contribution of our proposal is that it relates the objective of the evasion attacks and adversarial training with concepts already defined in learning theory. Also, we extend and unify some of the other proposals in the literature and provide alternative explanations on the observations made in those proposals. Through different experiments, we demonstrate that the framework is va…
Recent efforts to improve the interpretability of deep neural networks use saliency to characterize the importance of input features to predictions made by models. Work on interpretability using saliency-based methods on Recurrent Neural Networks (RNNs) has mostly targeted language tasks, and their applicability to time series data is less understood. In this work we analyze saliency-based methods for RNNs, both classical and gated cell architectures. We show that RNN saliency vanishes over time, biasing detection of salient features only to later time steps and are, therefore, incapable of re…
University of Washington（华盛顿大学）；Google AI Residency（谷歌人工智能 residency项目）；Google Brain（谷歌大脑）
Recent work has uncovered a striking phenomenon in large-capacity neural networks: they contain blocks of contiguous hidden layers with highly similar representations. This block structure has two seemingly contradictory properties: on the one hand, its constituent layers exhibit highly similar dominant first principal components (PCs), but on the other hand, their representations, and their common first PC, are highly dissimilar across different random seeds. Our work seeks to reconcile these discrepant properties by investigating the origin of the block structure in relation to the data and …
The inherent autocorrelation of time series data presents an ongoing challenge to multivariate time series prediction. Recently, a widely adopted approach has been the incorporation of frequency domain information to assist in long-term prediction tasks. Many researchers have independently observed the spectral bias phenomenon in neural networks, where models tend to fit low-frequency signals before high-frequency ones. However, these observations have often been attributed to the specific architectures designed by the researchers, rather than recognizing the phenomenon as a universal characte…
We propose a novel Shapley value approach to help address neural networks' interpretability and "vanishing gradient" problems. Our method is based on an accurate analytical approximation to the Shapley value of a neuron with ReLU activation. This analytical approximation admits a linear propagation of relevance across neural network layers, resulting in a simple, fast and sensible interpretation of neural networks' decision making process. We then derived a globally continuous and non-vanishing Shapley gradient, which can replace the conventional gradient in training neural network layers with…
The problem of vanishing and exploding gradients has been a long-standing obstacle that hinders the effective training of neural networks. Despite various tricks and techniques that have been employed to alleviate the problem in practice, there still lacks satisfactory theories or provable solutions. In this paper, we address the problem from the perspective of high-dimensional probability theory. We provide a rigorous result that shows, under mild conditions, how the vanishing/exploding gradients problem disappears with high probability if the neural networks have sufficient width. Our main i…
The recent work of Papyan, Han, & Donoho (2020) presented an intriguing "Neural Collapse" phenomenon, showing a structural property of interpolating classifiers in the late stage of training. This opened a rich area of exploration studying this phenomenon. Our motivation is to study the upper limits of this research program: How far will understanding Neural Collapse take us in understanding deep learning? First, we investigate its role in generalization. We refine the Neural Collapse conjecture into two separate conjectures: collapse on the train set (an optimization property) and collapse on…
Artificial neural networks (ANNs) suffer from catastrophic forgetting when trained on a sequence of tasks. While this phenomenon was studied in the past, there is only very limited recent research on this phenomenon. We propose a method for determining the contribution of individual parameters in an ANN to catastrophic forgetting. The method is used to analyze an ANNs response to three different continual learning scenarios.
Deep Learning models are vulnerable to adversarial examples, i.e.\ images obtained via deliberate imperceptible perturbations, such that the model misclassifies them with high confidence. However, class confidence by itself is an incomplete picture of uncertainty. We therefore use principled Bayesian methods to capture model uncertainty in prediction for observing adversarial misclassification. We provide an extensive study with different Bayesian neural networks attacked in both white-box and black-box setups. The behaviour of the networks for noise, attacks and clean test data is compared. W…
Incheon National University（仁川大学）；Dain Technology, Inc.
Modern convolutional neural networks (CNNs) have massive identical convolution blocks, and, hence, recursive sharing of parameters across these blocks has been proposed to reduce the amount of parameters. However, naive sharing of parameters poses many challenges such as limited representational power and the vanishing/exploding gradients problem of recursively shared parameters. In this paper, we present a recursive convolution block design and training method, in which a recursively shareable part, or a filter basis, is separated and learned while effectively avoiding the vanishing/exploding…
University of Pisa（比萨大学）；University of Exeter（埃克塞特大学）
Since the recognition in the early nineties of the vanishing/exploding (V/E) gradient issue plaguing the training of neural networks (NNs), significant efforts have been exerted to overcome this obstacle. However, a clear solution to the V/E issue remained elusive so far. In this manuscript a new architecture of NN is proposed, designed to mathematically prevent the V/E issue to occur. The pursuit of approximate dynamical isometry, i.e. parameter configurations where the singular values of the input-output Jacobian are tightly distributed around 1, leads to the derivation of a NN's architectur…
The vanishing gradient problem was a major obstacle for the success of deep learning. In recent years it was gradually alleviated through multiple different techniques. However the problem was not really overcome in a fundamental way, since it is inherent to neural networks with activation functions based on dot products. In a series of papers, we are going to analyze alternative neural network structures which are not based on dot products. In this first paper, we revisit neural networks built up of layers based on distance measures and Gaussian activation functions. These kinds of networks w…
Hyperbolic networks have shown prominent improvements over their Euclidean counterparts in several areas involving hierarchical datasets in various domains such as computer vision, graph analysis, and natural language processing. However, their adoption in practice remains restricted due to (i) non-scalability on accelerated deep learning hardware, (ii) vanishing gradients due to the closure of hyperbolic space, and (iii) information loss due to frequent mapping between local tangent space and fully hyperbolic space. To tackle these issues, we propose the approximation of hyperbolic operators …
$ε$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics
Department of Mathematics, SUSTech International Center for Mathematics & National Center for Applied Mathematics Shenzhen (NCAMS), Southern University of Science and Technology, Shenzhen, China（数学系、斯坦福大学国际数学中心及深圳应用数学国家中心、南方科技大学深圳中心，中国）；Department of Mathematics, Southern University of Science and Technology, Shenzhen, China（数学系、南方科技大学深圳中心，中国）
Recurrent neural networks (RNNs) are particularly well-suited for modeling long-term dependencies in sequential data, but are notoriously hard to train because the error backpropagated in time either vanishes or explodes at an exponential rate. While a number of works attempt to mitigate this effect through gated recurrent units, well-chosen parametric constraints, and skip-connections, we develop a novel perspective that seeks to evolve the hidden state on the equilibrium manifold of an ordinary differential equation (ODE). We propose a family of novel RNNs, namely {\em Equilibriated Recurren…
The Ohio State University（俄亥俄州立大学）；Western Washington University（西华盛顿大学）
Neural collapse is an emergent phenomenon in deep learning that was recently discovered by Papyan, Han and Donoho. We propose a simple "unconstrained features model" in which neural collapse also emerges empirically. By studying this model, we provide some explanation for the emergence of neural collapse in terms of the landscape of empirical risk.
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…
Convolutional Neural Networks spread through computer vision like a wildfire, impacting almost all visual tasks imaginable. Despite this, few researchers dare to train their models from scratch. Most work builds on one of a handful of ImageNet pre-trained models, and fine-tunes or adapts these for specific tasks. This is in large part due to the difficulty of properly initializing these networks from scratch. A small miscalibration of the initial weights leads to vanishing or exploding gradients, as well as poor convergence properties. In this work we present a fast and simple data-dependent i…
A numerical approach is developed for detecting the equivalence of deep learning architectures. The method is based on generating Mixed Matrix Ensembles (MMEs) out of deep neural network weight matrices and {\it conjugate circular ensemble} matching the neural architecture topology. Following this, the empirical evidence supports the {\it phenomenon} that difference between spectral densities of neural architectures and corresponding {\it conjugate circular ensemble} are vanishing with different decay rates at the long positive tail part of the spectrum i.e., cumulative Circular Spectral Diffe…
In this work, we study hallucinations in Neural Machine Translation (NMT), which lie at an extreme end on the spectrum of NMT pathologies. Firstly, we connect the phenomenon of hallucinations under source perturbation to the Long-Tail theory of Feldman (2020), and present an empirically validated hypothesis that explains hallucinations under source perturbation. Secondly, we consider hallucinations under corpus-level noise (without any source perturbation) and demonstrate that two prominent types of natural hallucinations (detached and oscillatory outputs) could be generated and explained thro…
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Large-scale cyber-physical systems require that control policies are distributed, that is, that they only rely on local real-time measurements and communication with neighboring agents. Optimal Distributed Control (ODC) problems are, however, highly intractable even in seemingly simple cases. Recent work has thus proposed training Neural Network (NN) distributed controllers. A main challenge of NN controllers is that they are not dependable during and after training, that is, the closed-loop system may be unstable, and the training may fail due to vanishing and exploding gradients. In this pap…
