The proper initialization of weights is crucial for the effective training and fast convergence of deep neural networks (DNNs). Prior work in this area has mostly focused on balancing the variance among weights per layer to maintain stability of (i) the input data propagated forwards through the network and (ii) the loss gradients propagated backwards, respectively. This prevalent heuristic is however agnostic of dependencies among gradients across the various layers and captures only firstorder effects. In this paper, we propose and discuss an initialization principle that is based on a rigorous estimation of the global curvature of weights across layers by approximating and controlling the norm of their Hessian matrix. The proposed approach is more systematic and recovers previous results for DNN activations such as smooth functions, dropouts, and ReLU. Our experiments on Word2Vec and the MNIST/CIFAR image classification tasks confirm that tracking the Hessian norm is a useful diagnostic tool which helps to more rigorously initialize weights
Sandia National Laboratories（桑迪亚国家实验室）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；University of New Mexico（新墨西哥大学）
This paper investigates multilevel initialization strategies for training very deep neural networks with a layer-parallel multigrid solver. The scheme is based on the continuous interpretation of the training problem as a problem of optimal control, in which neural networks are represented as discretizations of time-dependent ordinary differential equations. A key goal is to develop a method able to intelligently initialize the network parameters for the very deep networks enabled by scalable layer-parallel training. To do this, we apply a refinement strategy across the time domain, that is eq…
Institute for Stochastics and Applications（随机数学及其应用研究所）；Faculty 8: Mathematics and Physics（第8学院：数学与物理学院）；University of Stuttgart（斯图加特大学）
Initializing the weights and the biases is a key part of the training process of a neural network. Unlike the subsequent optimization phase, however, the initialization phase has gained only limited attention in the literature. In this paper we discuss some consequences of commonly used initialization strategies for vanilla DNNs with ReLU activations. Based on these insights we then develop an alternative initialization strategy. Finally, we present some large scale experiments assessing the quality of the new initialization strategy.
We propose a simple, data-driven approach to help guide hyperparameter selection for neural network initialization. We leverage the relationship between neural network and Gaussian process models having corresponding activation and covariance functions to infer the hyperparameter values desirable for model initialization. Our experiment shows that marginal likelihood maximization provides recommendations that yield near-optimal prediction performance on MNIST classification task under experiment constraints. Furthermore, our empirical results indicate consistency in the proposed technique, sug…
Convergence rate of training algorithms for neural networks is heavily affected by initialization of weights. In this paper, an original algorithm for initialization of weights in backpropagation neural net is presented with application to character recognition. The initialization method is mainly based on a customization of the Kalman filter, translating it into Bayesian statistics terms. A metrological approach is used in this context considering weights as measurements modeled by mutually dependent normal random variables. The algorithm performance is demonstrated by reporting and discussin…
From the statistical learning perspective, complexity control via explicit regularization is a necessity for improving the generalization of over-parameterized models. However, the impressive generalization performance of neural networks with only implicit regularization may be at odds with this conventional wisdom. In this work, we revisit the importance of explicit regularization for obtaining well-calibrated predictive uncertainty. Specifically, we introduce a probabilistic measure of calibration performance, which is lower bounded by the log-likelihood. We then explore explicit regularizat…
We revisit and extend the experiments of Goodfellow et al. (2014), who showed that - for then state-of-the-art networks - "the objective function has a simple, approximately convex shape" along the linear path between initialization and the trained weights. We do not find this to be the case for modern networks on CIFAR-10 and ImageNet. Instead, although loss is roughly monotonically non-increasing along this path, it remains high until close to the optimum. In addition, training quickly becomes linearly separated from the optimum by loss barriers. We conclude that, although Goodfellow et al.'…
University of Southern California（南加州大学）；Pinscreen, Inc.（Pinscreen公司）
In this paper, we provide some careful analysis of certain pathological behavior of Euler angles and unit quaternions encountered in previous works related to rotation representation in neural networks. In particular, we show that for certain problems, these two representations will provably produce completely wrong results for some inputs, and that this behavior is inherent in the topological property of the problem itself and is not caused by unsuitable network architectures or training procedures. We further show that previously proposed embeddings of $\mathrm{SO}(3)$ into higher dimensiona…
Deep neural networks are typically initialized with random weights, with variances chosen to facilitate signal propagation and stable gradients. It is also believed that diversity of features is an important property of these initializations. We construct a deep convolutional network with identical features by initializing almost all the weights to $0$. The architecture also enables perfect signal propagation and stable gradients, and achieves high accuracy on standard benchmarks. This indicates that random, diverse initializations are \textit{not} necessary for training neural networks. An es…
Standard practice in training neural networks involves initializing the weights in an independent fashion. The results of recent work suggest that feature "diversity" at initialization plays an important role in training the network. However, other initialization schemes with reduced feature diversity have also been shown to be viable. In this work, we conduct a series of experiments aimed at elucidating the importance of feature diversity at initialization. We show that a complete lack of diversity is harmful to training, but its effects can be counteracted by a relatively small addition of n…
A proper initialization of the weights in a neural network is critical to its convergence. Current insights into weight initialization come primarily from linear activation functions. In this paper, I develop a theory for weight initializations with non-linear activations. First, I derive a general weight initialization strategy for any neural network using activation functions differentiable at 0. Next, I derive the weight initialization strategy for the Rectified Linear Unit (RELU), and provide theoretical insights into why the Xavier initialization is a poor choice with RELU activations. My…
Department of EECS, Oregon State University（俄勒冈州立大学电子工程与计算机科学系）
Training a neural network (NN) depends on multiple factors, including but not limited to the initial weights. In this paper, we focus on initializing deep NN parameters such that it performs better, comparing to random or zero initialization. We do this by reducing the process of initialization into an SMT solver. Previous works consider certain activation functions on small NNs, however the studied NN is a deep network with different activation functions. Our experiments show that the proposed approach for parameter initialization achieves better performance comparing to randomly initialized …
Technion-IIT（以色列理工学院）；Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）
This work provides an additional step in the theoretical understanding of neural networks. We consider neural networks with one hidden layer and show that when learning symmetric functions, one can choose initial conditions so that standard SGD training efficiently produces generalization guarantees. We empirically verify this and show that this does not hold when the initial conditions are chosen at random. The proof of convergence investigates the interaction between the two layers of the network. Our results highlight the importance of using symmetry in the design of neural networks.
Accurate estimation of predictive uncertainty in modern neural networks is critical to achieve well calibrated predictions and detect out-of-distribution (OOD) inputs. The most promising approaches have been predominantly focused on improving model uncertainty (e.g. deep ensembles and Bayesian neural networks) and post-processing techniques for OOD detection (e.g. ODIN and Mahalanobis distance). However, there has been relatively little investigation into how the parametrization of the probabilities in discriminative classifiers affects the uncertainty estimates, and the dominant method, softm…
University of Fribourg（弗里堡大学）；University of Kaiserslautern（凯泽斯劳滕大学）
In this paper, we present a novel approach for initializing deep neural networks, i.e., by turning PCA into neural layers. Usually, the initialization of the weights of a deep neural network is done in one of the three following ways: 1) with random values, 2) layer-wise, usually as Deep Belief Network or as auto-encoder, and 3) re-use of layers from another network (transfer learning). Therefore, typically, many training epochs are needed before meaningful weights are learned, or a rather similar dataset is required for seeding a fine-tuning of transfer learning. In this paper, we describe ho…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Traditional landscape analysis of deep neural networks aims to show that no sub-optimal local minima exist in some appropriate sense. From this, one may be tempted to conclude that descent algorithms which escape saddle points will reach a good local minimum. However, basic optimization theory tell us that it is also possible for a descent algorithm to diverge to infinity if there are paths leading to infinity, along which the loss function decreases. It is not clear whether for non-linear neural networks there exists one setting that no bad local-min and no decreasing paths to infinity can be…
Informed by the basic geometry underlying feed forward neural networks, we initialize the weights of the first layer of a neural network using the linear discriminants which best distinguish individual classes. Networks initialized in this way take fewer training steps to reach the same level of training, and asymptotically have higher accuracy on training data.
University of Technology Sydney（悉尼科技大学）；Northwestern University（西北大学）
The prevailing thinking is that orthogonal weights are crucial to enforcing dynamical isometry and speeding up training. The increase in learning speed that results from orthogonal initialization in linear networks has been well-proven. However, while the same is believed to also hold for nonlinear networks when the dynamical isometry condition is satisfied, the training dynamics behind this contention have not been thoroughly explored. In this work, we study the dynamics of ultra-wide networks across a range of architectures, including Fully Connected Networks (FCNs) and Convolutional Neural …
University of Cambridge（剑桥大学）；Imperial College London（帝国理工学院）
Current graph representation learning techniques use Graph Neural Networks (GNNs) to extract features from dataset embeddings. In this work, we examine the quality of these embeddings and assess how changing them can affect the accuracy of GNNs. We explore different embedding extraction techniques for both images and texts; and find that the performance of different GNN architectures is dependent on the embedding style used. We see a prevalence of bag of words (BoW) embeddings and text classification tasks in available graph datasets. Given the impact embeddings has on GNN performance. this le…
Carnegie Mellon University（卡内基梅隆大学）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；University of Maryland（马里兰大学）；Cornell Tech（康奈尔科技学院）
Deep neural networks achieve state-of-the-art performance for a range of classification and inference tasks. However, the use of stochastic gradient descent combined with the nonconvexity of the underlying optimization problems renders parameter learning susceptible to initialization. To address this issue, a variety of methods that rely on random parameter initialization or knowledge distillation have been proposed in the past. In this paper, we propose FuseInit, a novel method to initialize shallower networks by fusing neighboring layers of deeper networks that are trained with random initia…
Zhejiang Lab（之江实验室）；School of Computer Science and Information Engineering, Hefei University of Technology（合肥工业大学计算机与信息工程学院）；School of Artificial Intelligence, Xidian University（西安电子科技大学人工智能学院）；School of Automation, Northwestern Polytechnical University（西北工业大学自动化学院）
The University of Hong Kong（香港大学）；Wells Fargo（富国银行）
Network initialization is the first and critical step for training neural networks. In this paper, we propose a novel network initialization scheme based on the celebrated Stein's identity. By viewing multi-layer feedforward neural networks as cascades of multi-index models, the projection weights to the first hidden layer are initialized using eigenvectors of the cross-moment matrix between the input's second-order score function and the response. The input data is then forward propagated to the next layer and such a procedure can be repeated until all the hidden layers are initialized. Final…
Huazhong University of Science and Technology（华中科技大学）
Recently neural networks and multiple instance learning are both attractive topics in Artificial Intelligence related research fields. Deep neural networks have achieved great success in supervised learning problems, and multiple instance learning as a typical weakly-supervised learning method is effective for many applications in computer vision, biometrics, nature language processing, etc. In this paper, we revisit the problem of solving multiple instance learning problems using neural networks. Neural networks are appealing for solving multiple instance learning problem. The multiple instan…
We propose a system for calculating a "scaling constant" for layers and weights of neural networks. We relate this scaling constant to two important quantities that relate to the optimizability of neural networks, and argue that a network that is "preconditioned" via scaling, in the sense that all weights have the same scaling constant, will be easier to train. This scaling calculus results in a number of consequences, among them the fact that the geometric mean of the fan-in and fan-out, rather than the fan-in, fan-out, or arithmetic mean, should be used for the initialization of the variance…
The weight initialization and the activation function of deep neural networks have a crucial impact on the performance of the training procedure. An inappropriate selection can lead to the loss of information of the input during forward propagation and the exponential vanishing/exploding of gradients during back-propagation. Understanding the theoretical properties of untrained random networks is key to identifying which deep networks may be trained successfully as recently demonstrated by Schoenholz et al. (2017) who showed that for deep feedforward neural networks only a specific choice of h…
Tokyo Institute of Technology（东京工业大学）；RIKEN（理化学研究所）
Graph neural networks have become one of the most important techniques to solve machine learning problems on graph-structured data. Recent work on vertex classification proposed deep and distributed learning models to achieve high performance and scalability. However, we find that the feature vectors of benchmark datasets are already quite informative for the classification task, and the graph structure only provides a means to denoise the data. In this paper, we develop a theoretical framework based on graph signal processing for analyzing graph neural networks. Our results indicate that grap…
R.V. College of Engineering（R.V. 工程学院）；Indian Institute of Science（印度科学理工学院）
We address the problem of learning vector representations for entities and relations in Knowledge Graphs (KGs) for Knowledge Base Completion (KBC). This problem has received significant attention in the past few years and multiple methods have been proposed. Most of the existing methods in the literature use a predefined characteristic scoring function for evaluating the correctness of KG triples. These scoring functions distinguish correct triples (high score) from incorrect ones (low score). However, their performance vary across different datasets. In this work, we demonstrate that a simple…
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.
A new initialization method for hidden parameters in a neural network is proposed. Derived from the integral representation of the neural network, a nonparametric probability distribution of hidden parameters is introduced. In this proposal, hidden parameters are initialized by samples drawn from this distribution, and output parameters are fitted by ordinary linear regression. Numerical experiments show that backpropagation with proposed initialization converges faster than uniformly random initialization. Also it is shown that the proposed method achieves enough accuracy by itself without ba…
Boston University（波士顿大学）；Sorbonne Université（索邦大学）；Ecole Polytechnique（巴黎综合理工学院）
Dedicated neural network (NN) architectures have been designed to handle specific data types (such as CNN for images or RNN for text), which ranks them among state-of-the-art methods for dealing with these data. Unfortunately, no architecture has been found for dealing with tabular data yet, for which tree ensemble methods (tree boosting, random forests) usually show the best predictive performances. In this work, we propose a new sparse initialization technique for (potentially deep) multilayer perceptrons (MLP): we first train a tree-based procedure to detect feature interactions and use the…
University of British Columbia（不列颠哥伦比亚大学）；University of California, Santa Barbara（加州大学圣巴巴拉分校）
Neural Collapse refers to the remarkable structural properties characterizing the geometry of class embeddings and classifier weights, found by deep nets when trained beyond zero training error. However, this characterization only holds for balanced data. Here we thus ask whether it can be made invariant to class imbalances. Towards this end, we adopt the unconstrained-features model (UFM), a recent theoretical model for studying neural collapse, and introduce Simplex-Encoded-Labels Interpolation (SELI) as an invariant characterization of the neural collapse phenomenon. Specifically, we prove …
Pennsylvania State University（宾夕法尼亚州立大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Weight initialization plays an important role in training neural networks and also affects tremendous deep learning applications. Various weight initialization strategies have already been developed for different activation functions with different neural networks. These initialization algorithms are based on minimizing the variance of the parameters between layers and might still fail when neural networks are deep, e.g., dying ReLU. To address this challenge, we study neural networks from a nonlinear computation point of view and propose a novel weight initialization strategy that is based on…
Department of Electrical Engineering Indian Institute of Technology Bombay（印度理工学院孟买分校电气工程系）
Deep neural networks (DNNs) form the backbone of almost every state-of-the-art technique in the fields such as computer vision, speech processing, and text analysis. The recent advances in computational technology have made the use of DNNs more practical. Despite the overwhelming performances by DNN and the advances in computational technology, it is seen that very few researchers try to train their models from the scratch. Training of DNNs still remains a difficult and tedious job. The main challenges that researchers face during training of DNNs are the vanishing/exploding gradient problem a…
School of Computer Science and Engineering, Nanyang Technological University（南洋理工大学计算机科学与工程学院）；Joint NTU-UBC Research Centre of Excellence in Active Living for the Elderly (LILY)（南洋理工大学-不列颠哥伦比亚大学老年活跃生活联合卓越研究中心）；Alibaba-NTU Singapore Joint Research Institute（阿里巴巴-南洋理工大学新加坡联合研究院）；Alibaba Group（阿里巴巴集团）
Proper initialization is crucial to the optimization and the generalization of neural networks. However, most existing neural recommendation systems initialize the user and item embeddings randomly. In this work, we propose a new initialization scheme for user and item embeddings called Laplacian Eigenmaps with Popularity-based Regularization for Isolated Data (LEPORID). LEPORID endows the embeddings with information regarding multi-scale neighborhood structures on the data manifold and performs adaptive regularization to compensate for high embedding variance on the tail of the data distribut…
Neural networks are the pinnacle of Artificial Intelligence, as in recent years we witnessed many novel architectures, learning and optimization techniques for deep learning. Capitalizing on the fact that neural networks inherently constitute multipartite graphs among neuron layers, we aim to analyze directly their structure to extract meaningful information that can improve the learning process. To our knowledge graph mining techniques for enhancing learning in neural networks have not been thoroughly investigated. In this paper we propose an adapted version of the k-core structure for the co…
We develop a general duality between neural networks and compositional kernels, striving towards a better understanding of deep learning. We show that initial representations generated by common random initializations are sufficiently rich to express all functions in the dual kernel space. Hence, though the training objective is hard to optimize in the worst case, the initial weights form a good starting point for optimization. Our dual view also reveals a pragmatic and aesthetic perspective of neural networks and underscores their expressive power.
Motivated by the gap between theoretical optimal approximation rates of deep neural networks (DNNs) and the accuracy realized in practice, we seek to improve the training of DNNs. The adoption of an adaptive basis viewpoint of DNNs leads to novel initializations and a hybrid least squares/gradient descent optimizer. We provide analysis of these techniques and illustrate via numerical examples dramatic increases in accuracy and convergence rate for benchmarks characterizing scientific applications where DNNs are currently used, including regression problems and physics-informed neural networks …
IRSTEA（法国国家环境与农业科技研究院）；UMR TETIS（TETIS联合研究单位）；LIRMM（蒙彼利埃计算机科学与机器人实验室）；Univ. of Montpellier（蒙彼利埃大学）；CIRAD（法国国际农业发展研究中心）
Recurrent Neural Networks (RNNs) can be seriously impacted by the initial parameters assignment, which may result in poor generalization performances on new unseen data. With the objective to tackle this crucial issue, in the context of RNN based classification, we propose a new supervised layer-wise pretraining strategy to initialize network parameters. The proposed approach leverages a data-aware strategy that sets up a taxonomy of classification problems automatically derived by the model behavior. To the best of our knowledge, despite the great interest in RNN-based classification, this is…
A traditional approach to initialization in deep neural networks (DNNs) is to sample the network weights randomly for preserving the variance of pre-activations. On the other hand, several studies show that during the training process, the distribution of stochastic gradients can be heavy-tailed especially for small batch sizes. In this case, weights and therefore pre-activations can be modeled with a heavy-tailed distribution that has an infinite variance but has a finite (non-integer) fractional moment of order $s$ with $s<2$. Motivated by this fact, we develop initialization schemes for ful…
We revisit and extend model stitching (Lenc & Vedaldi 2015) as a methodology to study the internal representations of neural networks. Given two trained and frozen models $A$ and $B$, we consider a "stitched model'' formed by connecting the bottom-layers of $A$ to the top-layers of $B$, with a simple trainable layer between them. We argue that model stitching is a powerful and perhaps under-appreciated tool, which reveals aspects of representations that measures such as centered kernel alignment (CKA) cannot. Through extensive experiments, we use model stitching to obtain quantitative verifica…
National Innovation Institute of Defense Technology, Chinese Academy of Military Science（中国人民解放军军事科学院国防科技创新研究院）
Physics-informed neural networks (PINNs) have been widely used to solve various scientific computing problems. However, large training costs limit PINNs for some real-time applications. Although some works have been proposed to improve the training efficiency of PINNs, few consider the influence of initialization. To this end, we propose a New Reptile initialization based Physics-Informed Neural Network (NRPINN). The original Reptile algorithm is a meta-learning initialization method based on labeled data. PINNs can be trained with less labeled data or even without any labeled data by adding p…
University of Oxford（牛津大学）；Australian National University（澳大利亚国立大学）
Network pruning is a promising avenue for compressing deep neural networks. A typical approach to pruning starts by training a model and then removing redundant parameters while minimizing the impact on what is learned. Alternatively, a recent approach shows that pruning can be done at initialization prior to training, based on a saliency criterion called connection sensitivity. However, it remains unclear exactly why pruning an untrained, randomly initialized neural network is effective. In this work, by noting connection sensitivity as a form of gradient, we formally characterize initializat…
National University of Singapore（新加坡国立大学）；University of Wisconsin（威斯康星大学）；N.1 Institute for Health（N.1健康研究所）
This paper revisits spectral graph convolutional neural networks (graph-CNNs) given in Defferrard (2016) and develops the Laplace-Beltrami CNN (LB-CNN) by replacing the graph Laplacian with the LB operator. We then define spectral filters via the LB operator on a graph. We explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters and define an update of the LB operator for pooling in the LBCNN. We employ the brain image data from Alzheimer's Disease Neuroimaging Initiative (ADNI) and demonstrate the use of the proposed LB-CNN. Based on the…
Ulm University（乌尔姆大学）；University College London（伦敦大学学院）
Appropriate weight initialization has been of key importance to successfully train neural networks. Recently, batch normalization has diminished the role of weight initialization by simply normalizing each layer based on batch statistics. Unfortunately, batch normalization has several drawbacks when applied to small batch sizes, as they are required to cope with memory limitations when learning on point clouds. While well-founded weight initialization strategies can render batch normalization unnecessary and thus avoid these drawbacks, no such approaches have been proposed for point convolutio…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Boston University（波士顿大学）
We analyze single-layer neural networks with the Xavier initialization in the asymptotic regime of large numbers of hidden units and large numbers of stochastic gradient descent training steps. The evolution of the neural network during training can be viewed as a stochastic system and, using techniques from stochastic analysis, we prove the neural network converges in distribution to a random ODE with a Gaussian distribution. The limit is completely different than in the typical mean-field results for neural networks due to the $\frac{1}{\sqrt{N}}$ normalization factor in the Xavier initializ…
