Neural networks (NN)-based learning algorithms are strongly affected by the choices of initialization and data distribution. Different optimization strategies have been proposed for improving the learning trajectory and finding a better optima. However, designing improved optimization strategies is a difficult task under the conventional landscape view. Here, we propose persistent neurons, a trajectory-based strategy that optimizes the learning task using information from previous converged solutions. More precisely, we utilize the end of trajectories and let the parameters explore new landscapes by penalizing the model from converging to the previous solutions under the same initialization. Persistent neurons can be regarded as a stochastic gradient method with informed bias where individual updates are corrupted by deterministic error terms. Specifically, we show that persistent neurons, under certain data distribution, is able to converge to more optimal solutions while initializations under popular framework find bad local minima. We further demonstrate that persistent neurons helps improve the model's performance under both good and poor initializations. We evaluate the full and partial persistent model and show it can be used to boost the performance on a range of NN structures, such as AlexNet and residual neural network (ResNet).
University of São Paulo（圣保罗大学）；Ghent University（根特大学）
The deep learning literature is continuously updated with new architectures and training techniques. However, weight initialization is overlooked by most recent research, despite some intriguing findings regarding random weights. On the other hand, recent works have been approaching Network Science to understand the structure and dynamics of Artificial Neural Networks (ANNs) after training. Therefore, in this work, we analyze the centrality of neurons in randomly initialized networks. We show that a higher neuronal strength variance may decrease performance, while a lower neuronal strength var…
University of Electronic Science and Technology of China（电子科技大学）；Peking University（北京大学）；Microsoft Research Asia（微软亚洲研究院）
Initialization plays a critical role in the training of deep neural networks (DNN). Existing initialization strategies mainly focus on stabilizing the training process to mitigate gradient vanish/explosion problems. However, these initialization methods are lacking in consideration about how to enhance generalization ability. The Information Bottleneck (IB) theory is a well-known understanding framework to provide an explanation about the generalization of DNN. Guided by the insights provided by IB theory, we design two criteria for better initializing DNN. And we further design a neuron campa…
In this paper, we study the trainability of rectified linear unit (ReLU) networks. A ReLU neuron is said to be dead if it only outputs a constant for any input. Two death states of neurons are introduced; tentative and permanent death. A network is then said to be trainable if the number of permanently dead neurons is sufficiently small for a learning task. We refer to the probability of a network being trainable as trainability. We show that a network being trainable is a necessary condition for successful training and the trainability serves as an upper bound of successful training rates. In…
Australian National University（澳大利亚国立大学）；Australian Centre for Robotic Vision（澳大利亚机器人视觉中心）；Data61, CSIRO（澳大利亚联邦科学与工业研究组织Data61）；Microsoft Research（微软研究院）
Although 3D Convolutional Neural Networks (CNNs) are essential for most learning based applications involving dense 3D data, their applicability is limited due to excessive memory and computational requirements. Compressing such networks by pruning therefore becomes highly desirable. However, pruning 3D CNNs is largely unexplored possibly because of the complex nature of typical pruning algorithms that embeds pruning into an iterative optimization paradigm. In this work, we introduce a Resource Aware Neuron Pruning (RANP) algorithm that prunes 3D CNNs at initialization to high sparsity levels.…
Massachusetts Institute of Technology（麻省理工学院）；Brown University（布朗大学）；Fuzhou University（福州大学）
The dying ReLU refers to the problem when ReLU neurons become inactive and only output 0 for any input. There are many empirical and heuristic explanations of why ReLU neurons die. However, little is known about its theoretical analysis. In this paper, we rigorously prove that a deep ReLU network will eventually die in probability as the depth goes to infinite. Several methods have been proposed to alleviate the dying ReLU. Perhaps, one of the simplest treatments is to modify the initialization procedure. One common way of initializing weights and biases uses symmetric probability distribution…
IRIT（图卢兹信息学研究所）；University of Toulouse（图卢兹大学）；ISAE-SUPAERO（法国高等航空航天学院）
Neurogenesis in ANNs is an understudied and difficult problem, even compared to other forms of structural learning like pruning. By decomposing it into triggers and initializations, we introduce a framework for studying the various facets of neurogenesis: when, where, and how to add neurons during the learning process. We present the Neural Orthogonality (NORTH*) suite of neurogenesis strategies, combining layer-wise triggers and initializations based on the orthogonality of activations or weights to dynamically grow performant networks that converge to an efficient size. We evaluate our contr…
The over-parametrized nature of Deep Neural Networks leads to considerable hindrances during deployment on low-end devices with time and space constraints. Network pruning strategies that sparsify DNNs using iterative prune-train schemes are often computationally expensive. As a result, techniques that prune at initialization, prior to training, have become increasingly popular. In this work, we propose neuron-to-neuron skip connections, which act as sparse weighted skip connections, to enhance the overall connectivity of pruned DNNs. Following a preliminary pruning step, N2NSkip connections a…
Small neural networks with a constrained number of trainable parameters, can be suitable resource-efficient candidates for many simple tasks, where now excessively large models are used. However, such models face several problems during the learning process, mainly due to the redundancy of the individual neurons, which results in sub-optimal accuracy or the need for additional training steps. Here, we explore the diversity of the neurons within the hidden layer during the learning process, and analyze how the diversity of the neurons affects predictions of the model. As following, we introduce…
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…
Spiking Neural Networks (SNNs) have been attached great importance due to the distinctive properties of low power consumption, biological plausibility, and adversarial robustness. The most effective way to train deep SNNs is through ANN-to-SNN conversion, which have yielded the best performance in deep network structure and large-scale datasets. However, there is a trade-off between accuracy and latency. In order to achieve high precision as original ANNs, a long simulation time is needed to match the firing rate of a spiking neuron with the activation value of an analog neuron, which impedes …
State-of-the-art neural networks are getting deeper and wider. While their performance increases with the increasing number of layers and neurons, it is crucial to design an efficient deep architecture in order to reduce computational and memory costs. Designing an efficient neural network, however, is labor intensive requiring many experiments, and fine-tunings. In this paper, we introduce network trimming which iteratively optimizes the network by pruning unimportant neurons based on analysis of their outputs on a large dataset. Our algorithm is inspired by an observation that the outputs of…
Training feedforward neural networks with standard logistic activations is considered difficult because of the intrinsic properties of these sigmoidal functions. This work aims at showing that these networks can be trained to achieve generalization performance comparable to those based on hyperbolic tangent activations. The solution consists on applying a set of conditions in parameter initialization, which have been derived from the study of the properties of a single neuron from an information-theoretic perspective. The proposed initialization is validated through an extensive experimental a…
Neural network pruning typically removes connections or neurons from a pretrained converged model; while a new pruning paradigm, pruning at initialization (PaI), attempts to prune a randomly initialized network. This paper offers the first survey concentrated on this emerging pruning fashion. We first introduce a generic formulation of neural network pruning, followed by the major classic pruning topics. Then, as the main body of this paper, a thorough and structured literature review of PaI methods is presented, consisting of two major tracks (sparse training and sparse selection). Finally, w…
Technische Universität Dresden（德累斯顿工业大学）；Ghent University（根特大学）；Princeton University（普林斯顿大学）
Echo State Networks (ESNs) are a special type of recurrent neural networks (RNNs), in which the input and recurrent connections are traditionally generated randomly, and only the output weights are trained. Despite the recent success of ESNs in various tasks of audio, image and radar recognition, we postulate that a purely random initialization is not the ideal way of initializing ESNs. The aim of this work is to propose an unsupervised initialization of the input connections using the $K$-Means algorithm on the training data. We show that for a large variety of datasets this initialization pe…
Understanding implicit bias of gradient descent for generalization capability of ReLU networks has been an important research topic in machine learning research. Unfortunately, even for a single ReLU neuron trained with the square loss, it was recently shown impossible to characterize the implicit regularization in terms of a norm of model parameters (Vardi & Shamir, 2021). In order to close the gap toward understanding intriguing generalization behavior of ReLU networks, here we examine the gradient flow dynamics in the parameter space when training single-neuron ReLU networks. Specifically, …
The inner representation of deep neural networks (DNNs) is indecipherable, which makes it difficult to tune DNN models, control their training process, and interpret their outputs. In this paper, we propose a novel approach to investigate the inner representation of DNNs through topological data analysis (TDA). Persistent homology (PH), one of the outstanding methods in TDA, was employed for investigating the complexities of trained DNNs. We constructed clique complexes on trained DNNs and calculated the one-dimensional PH of DNNs. The PH reveals the combinational effects of multiple neurons i…
Neuron death is a complex phenomenon with implications for model trainability: the deeper the network, the lower the probability of finding a valid initialization. In this work, we derive both upper and lower bounds on the probability that a ReLU network is initialized to a trainable point, as a function of model hyperparameters. We show that it is possible to increase the depth of a network indefinitely, so long as the width increases as well. Furthermore, our bounds are asymptotically tight under reasonable assumptions: first, the upper bound coincides with the true probability for a single-…
Successfully training deep neural networks often requires either batch normalization, appropriate weight initialization, both of which come with their own challenges. We propose an alternative, geometrically motivated method for training. Using elementary results from linear programming, we introduce Farkas layers: a method that ensures at least one neuron is active at a given layer. Focusing on residual networks with ReLU activation, we empirically demonstrate a significant improvement in training capacity in the absence of batch normalization or methods of initialization across a broad range…
Peking University（北京大学）；Beijing Institute of Big Data Research（北京大数据研究院）；Stanford University（斯坦福大学）；Huazhong University of Science and Technology（华中科技大学）；Cornell University（康奈尔大学）
It is widely believed that learning good representations is one of the main reasons for the success of deep neural networks. Although highly intuitive, there is a lack of theory and systematic approach quantitatively characterizing what representations do deep neural networks learn. In this work, we move a tiny step towards a theory and better understanding of the representations. Specifically, we study a simpler problem: How similar are the representations learned by two networks with identical architecture but trained from different initializations. We develop a rigorous theory based on the …
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Solving for the frequency-domain scattered wavefield via physics-informed neural network (PINN) has great potential in seismic modeling and inversion. However, when dealing with high-frequency wavefields, its accuracy and training cost limits its applications. Thus, we propose a novel implementation of PINN using frequency upscaling and neuron splitting, which allows the neural network model to grow in size as we increase the frequency while leveraging the information from the pre-trained model for lower-frequency wavefields, resulting in fast convergence to high-accuracy solutions. Numerical …
Shanghai Jiao Tong University（上海交通大学）；CMA-Shanghai（上海数学中心）；Shanghai Center for Brain Science and Brain-Inspired Technology（上海脑科学与类脑研究中心）
Empirical works show that for ReLU neural networks (NNs) with small initialization, input weights of hidden neurons (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) condense on isolated orientations. The condensation dynamics implies that the training implicitly regularizes a NN towards one with a much smaller effective size. In this work, we illustrate the formation of the condensation in multi-layer fully connected NNs and show that the maximal number of condensed orientations in the initial training stage is twice the m…
We prove that a single step of gradient decent over depth two network, with $q$ hidden neurons, starting from orthogonal initialization, can memorize $Ω\left(\frac{dq}{\log^4(d)}\right)$ independent and randomly labeled Gaussians in $\mathbb{R}^d$. The result is valid for a large class of activation functions, which includes the absolute value.
The interpretability of neural networks (NNs) is a challenging but essential topic for transparency in the decision-making process using machine learning. One of the reasons for the lack of interpretability is random weight initialization, where the input is randomly embedded into a different feature space in each layer. In this paper, we propose an interpretation method for a deep multilayer perceptron, which is the most general architecture of NNs, based on identity initialization (namely, initialization using identity matrices). The proposed method allows us to analyze the contribution of e…
Faster Predictive Coding Networks via Better Initialization
Department of Computer Science, University of Oxford（牛津大学计算机科学系）；Institute of Logic and Computation, Vienna University of Technology（维也纳技术大学逻辑与计算研究所）；VERSES AI Research Lab（VERSES AI研究实验室）
Princeton University（普林斯顿大学）；Beijing Institute of Big Data Research（北京大数据研究院）
A numerical and phenomenological study of the gradient descent (GD) algorithm for training two-layer neural network models is carried out for different parameter regimes when the target function can be accurately approximated by a relatively small number of neurons. It is found that for Xavier-like initialization, there are two distinctive phases in the dynamic behavior of GD in the under-parametrized regime: An early phase in which the GD dynamics follows closely that of the corresponding random feature model and the neurons are effectively quenched, followed by a late phase in which the neur…
A common method in training neural networks is to initialize all the weights to be independent Gaussian vectors. We observe that by instead initializing the weights into independent pairs, where each pair consists of two identical Gaussian vectors, we can significantly improve the convergence analysis. While a similar technique has been studied for random inputs [Daniely, NeurIPS 2020], it has not been analyzed with arbitrary inputs. Using this technique, we show how to significantly reduce the number of neurons required for two-layer ReLU networks, both in the under-parameterized setting with…
Can persistent homology whiten Transformer-based black-box models? A case study on BERT compression
Department of Computer Science and Artificial Intelligence, DiCITS, iMUDS, DaSCI, E.T.S.I.I.T. University of Granada, Spain（计算机科学与人工智能系，DiCITS，iMUDS，DaSCI，E.T.S.I.I.T. 格拉纳达大学）
In a recent work, we introduced a rigorous framework to describe the mean field limit of the gradient-based learning dynamics of multilayer neural networks, based on the idea of a neuronal embedding. There we also proved a global convergence guarantee for three-layer (as well as two-layer) networks using this framework. In this companion note, we point out that the insights in our previous work can be readily extended to prove a global convergence guarantee for multilayer networks of any depths. Unlike our previous three-layer global convergence guarantee that assumes i.i.d. initializations, o…
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Academy for Advanced Interdisciplinary Studies, Peking University（北京大学前沿交叉学科研究院）
In this paper, we follow Eftekhari's work to give a non-local convergence analysis of deep linear networks. Specifically, we consider optimizing deep linear networks which have a layer with one neuron under quadratic loss. We describe the convergent point of trajectories with arbitrary starting point under gradient flow, including the paths which converge to one of the saddle points or the original point. We also show specific convergence rates of trajectories that converge to the global minimizer by stages. To achieve these results, this paper mainly extends the machinery in Eftekhari's work …
Shanghai Jiao Tong University（上海交通大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Deep neural networks have achieved state-of-the-art performance on various tasks. However, lack of interpretability and transparency makes it easier for malicious attackers to inject trojan backdoor into the neural networks, which will make the model behave abnormally when a backdoor sample with a specific trigger is input. In this paper, we propose NeuronInspect, a framework to detect trojan backdoors in deep neural networks via output explanation techniques. NeuronInspect first identifies the existence of backdoor attack targets by generating the explanation heatmap of the output layer. We o…
IDInit: A Universal and Stable Initialization Method for Neural Network Training
Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳））；The Chinese University of Hong Kong（香港中文大学）；The Hong Kong Polytechnic University（香港理工大学）；MetaAI（元宇宙人工智能实验室）；Fudan University（复旦大学）；Shanghai Academy of AI for Science（上海人工智能科学研究院）
Despite deep neural network (DNN)'s impressive prediction performance in various domains, it is well known now that a set of DNN models trained with the same model specification and the same data can produce very different prediction results. Ensemble method is one state-of-the-art benchmark for prediction uncertainty estimation. However, ensembles are expensive to train and serve for web-scale traffic. In this paper, we seek to advance the understanding of prediction variation estimated by the ensemble method. Through empirical experiments on two widely used benchmark datasets MovieLens and C…
Texas A&M University（得克萨斯农工大学）；Facebook AI Research（Facebook人工智能研究院）；University of Pennsylvania（宾夕法尼亚大学）
It is well-known that the expressivity of a neural network depends on its architecture, with deeper networks expressing more complex functions. In the case of networks that compute piecewise linear functions, such as those with ReLU activation, the number of distinct linear regions is a natural measure of expressivity. It is possible to construct networks with merely a single region, or for which the number of linear regions grows exponentially with depth; it is not clear where within this range most networks fall in practice, either before or after training. In this paper, we provide a mathem…
From Neurons to Neutrons: A Case Study in Interpretability
NSF Institute for Artificial Intelligence and Fundamental Interactions (IAIFI)（NSF人工智能与基础相互作用研究所（IAIFI））；Massachusetts Institute of Technology（麻省理工学院）；FAIR at Meta（元公司FAIR研究院）；Harvard John A. Paulson School of Engineering and Applied Sciences（哈佛大学约翰·A·保尔森工程与应用科学学院）；School of Engineering, Science and Technology, Universidad del Rosario（罗萨里奥大学工程、科学与技术学院）
The field of neuromorphic computing promises extremely low-power and low-latency sensing and processing. Challenges in transferring learning algorithms from traditional artificial neural networks (ANNs) to spiking neural networks (SNNs) have so far prevented their application to large-scale, complex regression tasks. Furthermore, realizing a truly asynchronous and fully neuromorphic pipeline that maximally attains the abovementioned benefits involves rethinking the way in which this pipeline takes in and accumulates information. In the case of perception, spikes would be passed as-is and one-b…
Shanghai Jiao Tong University（上海交通大学）；CMA-Shanghai（上海数学中心）；Shanghai Center for Brain Science and Brain-Inspired Technology（上海脑科学与类脑研究中心）
Substantial work indicates that the dynamics of neural networks (NNs) is closely related to their initialization of parameters. Inspired by the phase diagram for two-layer ReLU NNs with infinite width (Luo et al., 2021), we make a step towards drawing a phase diagram for three-layer ReLU NNs with infinite width. First, we derive a normalized gradient flow for three-layer ReLU NNs and obtain two key independent quantities to distinguish different dynamical regimes for common initialization methods. With carefully designed experiments and a large computation cost, for both synthetic datasets and…
