Attention based neural networks are state of the art in a large range of applications. However, their performance tends to degrade when the number of layers increases. In this work, we show that enforcing Lipschitz continuity by normalizing the attention scores can significantly improve the performance of deep attention models. First, we show that, for deep graph attention networks (GAT), gradient explosion appears during training, leading to poor performance of gradient-based training algorithms. To address this issue, we derive a theoretical analysis of the Lipschitz continuity of attention modules and introduce LipschitzNorm, a simple and parameter-free normalization for self-attention mechanisms that enforces the model to be Lipschitz continuous. We then apply LipschitzNorm to GAT and Graph Transformers and show that their performance is substantially improved in the deep setting (10 to 30 layers). More specifically, we show that a deep GAT model with LipschitzNorm achieves state of the art results for node label prediction tasks that exhibit long-range dependencies, while showing consistent improvements over their unnormalized counterparts in benchmark node classification tasks.
Lipschitz continuity recently becomes popular in generative adversarial networks (GANs). It was observed that the Lipschitz regularized discriminator leads to improved training stability and sample quality. The mainstream implementations of Lipschitz continuity include gradient penalty and spectral normalization. In this paper, we demonstrate that gradient penalty introduces undesired bias, while spectral normalization might be over restrictive. We accordingly propose a new method which is efficient and unbiased. Our experiments verify our analysis and show that the proposed method is able to …
University of Oxford（牛津大学）；The Alan Turing Institute（艾伦·图灵研究所）
Exciting new work on the generalization bounds for neural networks (NN) given by Neyshabur et al. , Bartlett et al. closely depend on two parameter-depenedent quantities: the Lipschitz constant upper-bound and the stable rank (a softer version of the rank operator). This leads to an interesting question of whether controlling these quantities might improve the generalization behaviour of NNs. To this end, we propose stable rank normalization (SRN), a novel, optimal, and computationally efficient weight-normalization scheme which minimizes the stable rank of a linear operator. Surprisingly we f…
Ping An Property & Casualty Insurance（平安财产保险股份有限公司）；Beijing University of Posts and Telecommunications（北京邮电大学）
Graph Neural Networks (GNNs) have attracted considerable attention and have emerged as a new promising paradigm to process graph-structured data. GNNs are usually stacked to multiple layers and the node representations in each layer are computed through propagating and aggregating the neighboring node features with respect to the graph. By stacking to multiple layers, GNNs are able to capture the long-range dependencies among the data on the graph and thus bring performance improvements. To train a GNN with multiple layers effectively, some normalization techniques (e.g., node-wise normalizati…
Lipschitz constants of neural networks have been explored in various contexts in deep learning, such as provable adversarial robustness, estimating Wasserstein distance, stabilising training of GANs, and formulating invertible neural networks. Such works have focused on bounding the Lipschitz constant of fully connected or convolutional networks, composed of linear maps and pointwise non-linearities. In this paper, we investigate the Lipschitz constant of self-attention, a non-linear neural network module widely used in sequence modelling. We prove that the standard dot-product self-attention …
Samsung AI Centre Toronto（多伦多三星人工智能中心）；University of Toronto（多伦多大学）；Vector Institute for AI（矢量人工智能研究所）
Modern generative adversarial networks (GANs) predominantly use piecewise linear activation functions in discriminators (or critics), including ReLU and LeakyReLU. Such models learn piecewise linear mappings, where each piece handles a subset of the input space, and the gradients per subset are piecewise constant. Under such a class of discriminator (or critic) functions, we present Gradient Normalization (GraN), a novel input-dependent normalization method, which guarantees a piecewise K-Lipschitz constraint in the input space. In contrast to spectral normalization, GraN does not constrain pr…
As the representations output by Graph Neural Networks (GNNs) are increasingly employed in real-world applications, it becomes important to ensure that these representations are fair and stable. In this work, we establish a key connection between counterfactual fairness and stability and leverage it to propose a novel framework, NIFTY (uNIfying Fairness and stabiliTY), which can be used with any GNN to learn fair and stable representations. We introduce a novel objective function that simultaneously accounts for fairness and stability and develop a layer-wise weight normalization using the Lip…
Spectral Normalization for Lipschitz-Constrained Policies on Learning Humanoid Locomotion
Department of Intelligence and Information, Graduate School of Convergence Science and Technology, Seoul National University（智能信息系，融合科学与技术研究院，首尔国立大学）；X Technologies（1X技术公司）；Department of Intelligence and Information, Graduate School of Convergence Science and Technology, ASRI, AIIS, Seoul National University, Republic of Korea, and Advanced Institute of Convergence Technology (AICT), Suwon, Republic of Korea（智能信息系，融合科学与技术研究院，ASRI、AIIS、首尔国立大学，韩国，以及融合技术研究所（AICT），Suwon，韩国）
In this paper, we propose a novel normalization method called gradient normalization (GN) to tackle the training instability of Generative Adversarial Networks (GANs) caused by the sharp gradient space. Unlike existing work such as gradient penalty and spectral normalization, the proposed GN only imposes a hard 1-Lipschitz constraint on the discriminator function, which increases the capacity of the discriminator. Moreover, the proposed gradient normalization can be applied to different GAN architectures with little modification. Extensive experiments on four datasets show that GANs trained wi…
In this paper, we determine analytical bounds on the local Lipschitz constants of of affine functions composed with rectified linear units (ReLUs). Affine-ReLU functions represent a widely used layer in deep neural networks, due to the fact that convolution, fully-connected, and normalization functions are all affine, and are often followed by a ReLU activation function. Using an analytical approach, we mathematically determine upper bounds on the local Lipschitz constant of an affine-ReLU function, show how these bounds can be combined to determine a bound on an entire network, and discuss ho…
Microsoft（微软）；Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；Microsoft Research Montréal（微软蒙特利尔研究院）
Attention is a powerful component of modern neural networks across a wide variety of domains. In this paper, we seek to quantify the regularity (i.e. the amount of smoothness) of the attention operation. To accomplish this goal, we propose a new mathematical framework that uses measure theory and integral operators to model attention. We show that this framework is consistent with the usual definition, and that it captures the essential properties of attention. Then we use this framework to prove that, on compact domains, the attention operation is Lipschitz continuous and provide an estimate …
Lipschitz constrained networks have gathered considerable attention in the deep learning community, with usages ranging from Wasserstein distance estimation to the training of certifiably robust classifiers. However they remain commonly considered as less accurate, and their properties in learning are still not fully understood. In this paper we clarify the matter: when it comes to classification 1-Lipschitz neural networks enjoy several advantages over their unconstrained counterpart. First, we show that these networks are as accurate as classical ones, and can fit arbitrarily difficult bound…
Hanoi University of Science and Technology（河内科技大学）
Generative adversarial networks (GANs) are so complex that the existing learning theories do not provide a satisfactory explanation for why GANs have great success in practice. The same situation also remains largely open for deep neural networks. To fill this gap, we introduce a Lipschitz theory to analyze generalization. We demonstrate its simplicity by analyzing generalization and consistency of overparameterized neural networks. We then use this theory to derive Lipschitz-based generalization bounds for GANs. Our bounds show that penalizing the Lipschitz constant of the GAN loss can improv…
University of Notre Dame（圣母大学）；Amazon AWS AI（亚马逊AWS人工智能）
We evaluate three simple, normalization-centric changes to improve Transformer training. First, we show that pre-norm residual connections (PreNorm) and smaller initializations enable warmup-free, validation-based training with large learning rates. Second, we propose $\ell_2$ normalization with a single scale parameter (ScaleNorm) for faster training and better performance. Finally, we reaffirm the effectiveness of normalizing word embeddings to a fixed length (FixNorm). On five low-resource translation pairs from TED Talks-based corpora, these changes always converge, giving an average +1.1 …
Columbia University Irving Medical Center（哥伦比亚大学欧文医学中心）
Inspired by the adaptation phenomenon of neuronal firing, we propose the regularity normalization (RN) as an unsupervised attention mechanism (UAM) which computes the statistical regularity in the implicit space of neural networks under the Minimum Description Length (MDL) principle. Treating the neural network optimization process as a partially observable model selection problem, the regularity normalization constrains the implicit space by a normalization factor, the universal code length. We compute this universal code incrementally across neural network layers and demonstrate the flexibil…
Mila（Mila（魁北克人工智能研究所））；Université de Montréal（蒙特利尔大学）；Microsoft Research Montreal（微软研究院蒙特利尔分院）；Microsoft（微软）
Attention is a powerful component of modern neural networks across a wide variety of domains. However, despite its ubiquity in machine learning, there is a gap in our understanding of attention from a theoretical point of view. We propose a framework to fill this gap by building a mathematically equivalent model of attention using measure theory. With this model, we are able to interpret self-attention as a system of self-interacting particles, we shed light on self-attention from a maximum entropy perspective, and we show that attention is actually Lipschitz-continuous (with an appropriate me…
Apple Inc.（苹果公司）；Carnegie Mellon University（卡内基梅隆大学）
We introduce a new routing algorithm for capsule networks, in which a child capsule is routed to a parent based only on agreement between the parent's state and the child's vote. The new mechanism 1) designs routing via inverted dot-product attention; 2) imposes Layer Normalization as normalization; and 3) replaces sequential iterative routing with concurrent iterative routing. When compared to previously proposed routing algorithms, our method improves performance on benchmark datasets such as CIFAR-10 and CIFAR-100, and it performs at-par with a powerful CNN (ResNet-18) with 4x fewer paramet…
Artistic style transfer aims to transfer the style characteristics of one image onto another image while retaining its content. Existing approaches commonly leverage various normalization techniques, although these face limitations in adequately transferring diverse textures to different spatial locations. Self-Attention-based approaches have tackled this issue with partial success but suffer from unwanted artifacts. Motivated by these observations, this paper aims to combine the best of both worlds: self-attention and normalization. That yields a new plug-and-play module that we name Self-Att…
National University of Singapore（新加坡国立大学）；Northwestern University（西北大学）；Purdue University（普渡大学）；New Oriental AI Research Academy（新东方人工智能研究院）
Batch Normalization (BN)(Ioffe and Szegedy 2015) normalizes the features of an input image via statistics of a batch of images and hence BN will bring the noise to the gradient of the training loss. Previous works indicate that the noise is important for the optimization and generalization of deep neural networks, but too much noise will harm the performance of networks. In our paper, we offer a new point of view that self-attention mechanism can help to regulate the noise by enhancing instance-specific information to obtain a better regularization effect. Therefore, we propose an attention-ba…
North Carolina State University（北卡罗来纳州立大学）；BITS Pilani K K Birla Goa Campus（比尔拉理工学院皮拉尼 K K 比尔拉果阿校区）；PES University（PES大学）
Optimizing deep neural networks is largely thought to be an empirical process, requiring manual tuning of several hyper-parameters, such as learning rate, weight decay, and dropout rate. Arguably, the learning rate is the most important of these to tune, and this has gained more attention in recent works. In this paper, we propose a novel method to compute the learning rate for training deep neural networks with stochastic gradient descent. We first derive a theoretical framework to compute learning rates dynamically based on the Lipschitz constant of the loss function. We then extend this fra…
On the Lipschitz Continuity of Set Aggregation Functions and Neural Networks for Sets
Department of Informatics and Telecommunications University of Peloponnese（信息与通信技术系希腊皮洛斯大学）；Department of Computer Science & Engineering University of Ioannina（计算机科学与工程系伊奥尼亚大学）
Phasor Attention: Mean Root Square Normalization for Phase Manifold Preservation
College of Pharmacy, Chungnam National University（忠南国立大学药学院）；Department of Computer Science & Engineering, Chungnam National University（忠南国立大学计算机科学与工程系）
Text normalization is an important enabling technology for several NLP tasks. Recently, neural-network-based approaches have outperformed well-established models in this task. However, in languages other than English, there has been little exploration in this direction. Both the scarcity of annotated data and the complexity of the language increase the difficulty of the problem. To address these challenges, we use a sequence-to-sequence model with character-based attention, which in addition to its self-learned character embeddings, uses word embeddings pre-trained with an approach that also m…
Beihang University（北京航空航天大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
Many real-world problems can be represented as graph-based learning problems. In this paper, we propose a novel framework for learning spatial and attentional convolution neural networks on arbitrary graphs. Different from previous convolutional neural networks on graphs, we first design a motif-matching guided subgraph normalization method to capture neighborhood information. Then we implement subgraph-level self-attentional layers to learn different importances from different subgraphs to solve graph classification problems. Analogous to image-based attentional convolution networks that oper…
NSF AI Institute for Artificial Intelligence and Fundamental Interactions（NSF人工智能与基本相互作用人工智能研究所）；Laboratory for Nuclear Science, MIT（麻省理工学院核科学实验室）；MIT（麻省理工学院）
An important part of problems in statistical physics and computer science can be expressed as the computation of marginal probabilities over a Markov Random Field. The belief propagation algorithm, which is an exact procedure to compute these marginals when the underlying graph is a tree, has gained its popularity as an efficient way to approximate them in the more general case. In this paper, we focus on an aspect of the algorithm that did not get that much attention in the literature, which is the effect of the normalization of the messages. We show in particular that, for a large class of n…
Text normalization is a ubiquitous process that appears as the first step of many Natural Language Processing problems. However, previous Deep Learning approaches have suffered from so-called silly errors, which are undetectable on unsupervised frameworks, making those models unsuitable for deployment. In this work, we make use of an attention-based encoder-decoder architecture that overcomes these undetectable errors by using a fine-grained character-level approach rather than a word-level one. Furthermore, our new general-purpose encoder based on causal convolutions, called Causal Feature Ex…
Recurrent neural networks (RNN) are used in many real-world text and speech applications. They include complex modules such as recurrence, exponential-based activation, gate interaction, unfoldable normalization, bi-directional dependence, and attention. The interaction between these elements prevents running them on integer-only operations without a significant performance drop. Deploying RNNs that include layer normalization and attention on integer-only arithmetic is still an open problem. We present a quantization-aware training method for obtaining a highly accurate integer-only recurrent…
An Architecture Built for Federated Learning: Addressing Data Heterogeneity through Adaptive Normalization-Free Feature Recalibration
Department of Computer Science, City St. George’s, University of London（伦敦大学城市圣乔治学院计算机科学系）；BioMedIA, Department of Computing, Imperial College London（伦敦帝国理工学院计算系BioMedIA）
ESPFormer: Doubly-Stochastic Attention with Expected Sliced Transport Plans
Department of Computer Science, Vanderbilt University, Nashville, TN, USA.（计算机科学系，范德比尔特大学）；Department of Computer Science, Duke University, Durham, NC, USA.（计算机科学系，杜克大学）；Department of Biostatistics and Bioinformatics, Duke University, Durham, NC, USA.（生物统计学与生物信息学系，杜克大学）
