We propose a novel regularization method, called \textit{volumization}, for neural networks. Inspired by physics, we define a physical volume for the weight parameters in neural networks, and we show that this method is an effective way of regularizing neural networks. Intuitively, this method interpolates between an $L_2$ and $L_\infty$ regularization. Therefore, weight decay and weight clipping become special cases of the proposed algorithm. We prove, on a toy example, that the essence of this method is a regularization technique to control bias-variance tradeoff. The method is shown to do well in the categories where the standard weight decay method is shown to work well, including improving the generalization of networks and preventing memorization. Moreover, we show that the volumization might lead to a simple method for training a neural network whose weight is binary or ternary.
University of Liverpool（利物浦大学）；Institute of Software Chinese Academy of Sciences（中国科学院软件研究所）
While dropout is known to be a successful regularization technique, insights into the mechanisms that lead to this success are still lacking. We introduce the concept of \emph{weight expansion}, an increase in the signed volume of a parallelotope spanned by the column or row vectors of the weight covariance matrix, and show that weight expansion is an effective means of increasing the generalization in a PAC-Bayesian setting. We provide a theoretical argument that dropout leads to weight expansion and extensive empirical support for the correlation between dropout and weight expansion. To supp…
RiskEcon® Lab（RiskEcon®实验室）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）；New York University（纽约大学）
Heavy-tailed distributions have been studied in statistics, random matrix theory, physics, and econometrics as models of correlated systems, among other domains. Further, heavy-tail distributed eigenvalues of the covariance matrix of the weight matrices in neural networks have been shown to empirically correlate with test set accuracy in several works (e.g. arXiv:1901.08276), but a formal relationship between heavy-tail distributed parameters and generalization bounds was yet to be demonstrated. In this work, the compression framework of arXiv:1802.05296 is utilized to show that matrices with …
Designing a deep neural network (DNN) with good generalization capability is a complex process especially when the weights are severely quantized. Model averaging is a promising approach for achieving the good generalization capability of DNNs, especially when the loss surface for training contains many sharp minima. We present a new quantized neural network optimization approach, stochastic quantized weight averaging (SQWA), to design low-precision DNNs with good generalization capability using model averaging. The proposed approach includes (1) floating-point model training, (2) direct quant…
We study generalization under labeled shift for categorical and general normed label spaces. We propose a series of methods to estimate the importance weights from labeled source to unlabeled target domain and provide confidence bounds for these estimators. We deploy these estimators and provide generalization bounds in the unlabeled target domain.
Using weight decay to penalize the L2 norms of weights in neural networks has been a standard training practice to regularize the complexity of networks. In this paper, we show that a family of regularizers, including weight decay, is ineffective at penalizing the intrinsic norms of weights for networks with positively homogeneous activation functions, such as linear, ReLU and max-pooling functions. As a result of homogeneity, functions specified by the networks are invariant to the shifting of weight scales between layers. The ineffective regularizers are sensitive to such shifting and thus p…
Information Sciences Institute, University of Southern California（南加州大学信息科学研究所）
In the presence of noisy or incorrect labels, neural networks have the undesirable tendency to memorize information about the noise. Standard regularization techniques such as dropout, weight decay or data augmentation sometimes help, but do not prevent this behavior. If one considers neural network weights as random variables that depend on the data and stochasticity of training, the amount of memorized information can be quantified with the Shannon mutual information between weights and the vector of all training labels given inputs, $I(w ; \mathbf{y} \mid \mathbf{x})$. We show that for any …
Koopman-based generalization bound: New aspect for full-rank weights
NTT Corporation（NTT公司）；Center for Advanced Intelligence Project, RIKEN（理化学研究所先进智能项目中心）；Center for Data Science, Ehime University（爱媛大学数据科学中心）；A*STAR Centre for Frontier AI and Research（新加坡科技研究局前沿人工智能与研究中心）；Graduate School of Information Science and Technology, The University of Tokyo（东京大学情报理工学研究科）
National Yang Ming Chiao Tung University（阳明交通大学）；IBM Research（IBM研究院）
Studying the sensitivity of weight perturbation in neural networks and its impacts on model performance, including generalization and robustness, is an active research topic due to its implications on a wide range of machine learning tasks such as model compression, generalization gap assessment, and adversarial attacks. In this paper, we provide the first integral study and analysis for feed-forward neural networks in terms of the robustness in pairwise class margin and its generalization behavior under weight perturbation. We further design a new theory-driven loss function for training gene…
Tsinghua University（清华大学）；PCL Research Center of Networks and Communications, Peng Cheng Laboratory（鹏城实验室网络通信PCL研究中心）；Key Lab. of Machine Perception (MoE), School of EECS, Peking University（北京大学信息科学技术学院机器感知教育部重点实验室）
The study on improving the robustness of deep neural networks against adversarial examples grows rapidly in recent years. Among them, adversarial training is the most promising one, which flattens the input loss landscape (loss change with respect to input) via training on adversarially perturbed examples. However, how the widely used weight loss landscape (loss change with respect to weight) performs in adversarial training is rarely explored. In this paper, we investigate the weight loss landscape from a new perspective, and identify a clear correlation between the flatness of weight loss la…
Generative Adversarial Networks (GANs), though powerful, is hard to train. Several recent works (brock2016neural,miyato2018spectral) suggest that controlling the spectra of weight matrices in the discriminator can significantly improve the training of GANs. Motivated by their discovery, we propose a new framework for training GANs, which allows more flexible spectrum control (e.g., making the weight matrices of the discriminator have slow singular value decays). Specifically, we propose a new reparameterization approach for the weight matrices of the discriminator in GANs, which allows us to d…
University of Pittsburgh（匹兹堡大学）；CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）
A lot of theoretical and empirical evidence shows that the flatter local minima tend to improve generalization. Adversarial Weight Perturbation (AWP) is an emerging technique to efficiently and effectively find such minima. In AWP we minimize the loss w.r.t. a bounded worst-case perturbation of the model parameters thereby favoring local minima with a small loss in a neighborhood around them. The benefits of AWP, and more generally the connections between flatness and generalization, have been extensively studied for i.i.d. data such as images. In this paper, we extensively study this phenomen…
Convolutional neural networks (CNNs) have achieved breakthrough performances in a wide range of applications including image classification, semantic segmentation, and object detection. Previous research on characterizing the generalization ability of neural networks mostly focuses on fully connected neural networks (FNNs), regarding CNNs as a special case of FNNs without taking into account the special structure of convolutional layers. In this work, we propose a tighter generalization bound for CNNs by exploiting the sparse and permutation structure of its weight matrices. As the generalizat…
Weight decay is often used to ensure good generalization in the training practice of deep neural networks with batch normalization (BN-DNNs), where some convolution layers are invariant to weight rescaling due to the normalization. In this paper, we demonstrate that the practical usage of weight decay still has some unsolved problems in spite of existing theoretical work on explaining the effect of weight decay in BN-DNNs. On the one hand, when the non-adaptive learning rate e.g. SGD with momentum is used, the effective learning rate continues to increase even after the initial training stage,…
Artificial neural networks (ANNs) may not be worth their computational/memory costs when used in mobile phones or embedded devices. Parameter-pruning algorithms combat these costs, with some algorithms capable of removing over 90% of an ANN's weights without harming the ANN's performance. Removing weights from an ANN is a form of regularization, but existing pruning algorithms do not significantly improve generalization error. We show that pruning ANNs can improve generalization if pruning targets large weights instead of small weights. Applying our pruning algorithm to an ANN leads to a highe…
Trainable Weight Averaging: Accelerating Training and Improving Generalization
Shanghai Jiao Tong University（上海交通大学）；KU Leuven（鲁汶大学）；National Taiwan University（台湾大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
Jilin University（吉林大学）；Rice University（莱斯大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
Despite the recent advances of graph neural networks (GNNs) in modeling graph data, the training of GNNs on large datasets is notoriously hard due to the overfitting. Adversarial training, which augments data with the worst-case adversarial examples, has been widely demonstrated to improve model's robustness against adversarial attacks and generalization ability. However, while the previous adversarial training generally focuses on protecting GNNs from spiteful attacks, it remains unclear how the adversarial training could improve the generalization abilities of GNNs in the graph analytics pro…
Quantized neural networks have drawn a lot of attention as they reduce the space and computational complexity during the inference. Moreover, there has been folklore that quantization acts as an implicit regularizer and thus can improve the generalizability of neural networks, yet no existing work formalizes this interesting folklore. In this paper, we take the binary weights in a neural network as random variables under stochastic rounding, and study the distribution propagation over different layers in the neural network. We propose a quasi neural network to approximate the distribution prop…
Our work presents extensive empirical evidence that layer rotation, i.e. the evolution across training of the cosine distance between each layer's weight vector and its initialization, constitutes an impressively consistent indicator of generalization performance. In particular, larger cosine distances between final and initial weights of each layer consistently translate into better generalization performance of the final model. Interestingly, this relation admits a network independent optimum: training procedures during which all layers' weights reach a cosine distance of 1 from their initia…
Cornell University（康奈尔大学）；Higher School of Economics（高等经济学院）；Samsung-HSE Laboratory（三星-高等经济学院实验室）；Samsung AI Center in Moscow（莫斯科三星人工智能中心）；Lomonosov Moscow State University（莫斯科国立大学（罗蒙诺索夫莫斯科国立大学））
Deep neural networks are typically trained by optimizing a loss function with an SGD variant, in conjunction with a decaying learning rate, until convergence. We show that simple averaging of multiple points along the trajectory of SGD, with a cyclical or constant learning rate, leads to better generalization than conventional training. We also show that this Stochastic Weight Averaging (SWA) procedure finds much flatter solutions than SGD, and approximates the recent Fast Geometric Ensembling (FGE) approach with a single model. Using SWA we achieve notable improvement in test accuracy over co…
Huawei Noah’s Ark Lab（华为诺亚方舟实验室）；Université de Montréal（蒙特利尔大学）；University of Waterloo（滑铁卢大学）
Knowledge Distillation (KD) is a commonly used technique for improving the generalization of compact Pre-trained Language Models (PLMs) on downstream tasks. However, such methods impose the additional burden of training a separate teacher model for every new dataset. Alternatively, one may directly work on the improvement of the optimization procedure of the compact model toward better generalization. Recent works observe that the flatness of the local minimum correlates well with better generalization. In this work, we adapt Stochastic Weight Averaging (SWA), a method encouraging convergence …
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Regularization is typically understood as improving generalization by altering the landscape of local extrema to which the model eventually converges. Deep neural networks (DNNs), however, challenge this view: We show that removing regularization after an initial transient period has little effect on generalization, even if the final loss landscape is the same as if there had been no regularization. In some cases, generalization even improves after interrupting regularization. Conversely, if regularization is applied only after the initial transient, it has no effect on the final solution, who…
Transformer models can use two fundamentally different kinds of information: information stored in weights during training, and information provided ``in-context'' at inference time. In this work, we show that transformers exhibit different inductive biases in how they represent and generalize from the information in these two sources. In particular, we characterize whether they generalize via parsimonious rules (rule-based generalization) or via direct comparison with observed examples (exemplar-based generalization). This is of important practical consequence, as it informs whether to encode…
Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；Criteo AI Lab（Criteo人工智能实验室）；Université de Rouen（鲁昂大学）
Standard neural networks struggle to generalize under distribution shifts in computer vision. Fortunately, combining multiple networks can consistently improve out-of-distribution generalization. In particular, weight averaging (WA) strategies were shown to perform best on the competitive DomainBed benchmark; they directly average the weights of multiple networks despite their nonlinearities. In this paper, we propose Diverse Weight Averaging (DiWA), a new WA strategy whose main motivation is to increase the functional diversity across averaged models. To this end, DiWA averages weights obtain…
We prove bounds on the generalization error of convolutional networks. The bounds are in terms of the training loss, the number of parameters, the Lipschitz constant of the loss and the distance from the weights to the initial weights. They are independent of the number of pixels in the input, and the height and width of hidden feature maps. We present experiments using CIFAR-10 with varying hyperparameters of a deep convolutional network, comparing our bounds with practical generalization gaps.
Batch normalization is currently the most widely used variant of internal normalization for deep neural networks. Additional work has shown that the normalization of weights and additional conditioning as well as the normalization of gradients further improve the generalization. In this work, we combine several of these methods and thereby increase the generalization of the networks. The advantage of the newer methods compared to the batch normalization is not only increased generalization, but also that these methods only have to be applied during training and, therefore, do not influence the…
MEAT: Median-Ensemble Adversarial Training for Improving Robustness and Generalization
Fujian Province Key Laboratory of Information Security（福建省信息安全重点实验室）；College of Computer Science and Big Data, Fuzhou University（福州大学计算机与大数据学院）；Yuan Ze University（元智大学）
To fully uncover the great potential of deep neural networks (DNNs), various learning algorithms have been developed to improve the model's generalization ability. Recently, sharpness-aware minimization (SAM) establishes a generic scheme for generalization improvements by minimizing the sharpness measure within a small neighborhood and achieves state-of-the-art performance. However, SAM requires two consecutive gradient evaluations for solving the min-max problem and inevitably doubles the training time. In this paper, we resort to filter-wise random weight perturbations (RWP) to decouple the …
Revisiting Weighted Aggregation in Federated Learning with Neural Networks
Zhejiang University（浙江大学）；Research Center for Industries of the Future, Westlake University（西湖大学未来产业研究中心）；Westlake University（西湖大学）；Xiamen University（厦门大学）
Improving Generalization of Adversarial Training via Robust Critical Fine-Tuning
University of Chinese Academy of Sciences（中国科学院大学）；Institute of Automation, CAS（中国科学院自动化研究所）；Microsoft Research（微软研究院）；City University of Hong Kong（香港城市大学）
Adversarial training has been proven to be a powerful regularization method to improve the generalization of models. However, current adversarial training methods only attack the original input sample or the embedding vectors, and their attacks lack coverage and diversity. To further enhance the breadth and depth of attack, we propose a novel masked weight adversarial training method called DropAttack, which enhances generalization of model by adding intentionally worst-case adversarial perturbations to both the input and hidden layers in different dimensions and minimize the adversarial risks…
School of Computer Science, Guangdong University of Technology（广东工业大学计算机科学学院）；Pazhou Laboratory (Huangpu)（黄埔实验室）；College of Science, Shantou University（汕头大学理学院）；Didi Chuxing（滴滴出行）
University of Southern California（南加州大学）；University of Cambridge（剑桥大学）；Carnegie Mellon University（卡内基梅隆大学）
Deep neural networks are often overparameterized and may not easily achieve model generalization. Adversarial training has shown effectiveness in improving generalization by regularizing the change of loss on top of adversarially chosen perturbations. The recently proposed sharpness-aware minimization (SAM) algorithm conducts adversarial weight perturbation, encouraging the model to converge to a flat minima. SAM finds a common adversarial weight perturbation per-batch. Although per-instance adversarial weight perturbations are stronger adversaries and can potentially lead to better generaliza…
A Spectral Theory of Grokking: Weight Decay induces Feature Learning
Arnold-Sommerfeld-Center for Theoretical Physics（阿诺德-索末菲理论物理中心）；Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）；University of Cambridge（剑桥大学）；Standford University（斯坦福大学）
