In recent years we see a rapidly growing line of research which shows learnability of various models via common neural network algorithms. Yet, besides a very few outliers, these results show learnability of models that can be learned using linear methods. Namely, such results show that learning neural-networks with gradient-descent is competitive with learning a linear classifier on top of a data-independent representation of the examples. This leaves much to be desired, as neural networks are far more successful than linear methods. Furthermore, on the more conceptual level, linear models don't seem to capture the "deepness" of deep networks. In this paper we make a step towards showing leanability of models that are inherently non-linear. We show that under certain distributions, sparse parities are learnable via gradient decent on depth-two network. On the other hand, under the same distributions, these parities cannot be learned efficiently by linear methods.
Learning High-Degree Parities: The Crucial Role of the Initialization
Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Massachusetts Institute of Technology (MIT)（麻省理工学院）；Inria Paris（法国国家信息与自动化研究所巴黎分部）；PSL University（巴黎文理研究大学）；African Institute for Mathematical Sciences (AIMS)（非洲数学科学研究院）
Harvard University（哈佛大学）；Microsoft Research（微软研究院）；University of Pennsylvania（宾夕法尼亚大学）；Hebrew University of Jerusalem（希伯来大学）
There is mounting evidence of emergent phenomena in the capabilities of deep learning methods as we scale up datasets, model sizes, and training times. While there are some accounts of how these resources modulate statistical capacity, far less is known about their effect on the computational problem of model training. This work conducts such an exploration through the lens of learning a $k$-sparse parity of $n$ bits, a canonical discrete search problem which is statistically easy but computationally hard. Empirically, we find that a variety of neural networks successfully learn sparse paritie…
Apple Inc.（苹果公司）；Florida State University（佛罗里达州立大学）
Artificial Neural Networks form the basis of very powerful learning methods. It has been observed that a naive application of fully connected neural networks to data with many irrelevant variables often leads to overfitting. In an attempt to circumvent this issue, a prior knowledge pertaining to what features are relevant and their possible feature interactions can be encoded into these networks. In this work, we use decision trees to capture such relevant features and their interactions and define a mapping to encode extracted relationships into a neural network. This addresses the initializa…
The evolution of a deep neural network trained by the gradient descent can be described by its neural tangent kernel (NTK) as introduced in [20], where it was proven that in the infinite width limit the NTK converges to an explicit limiting kernel and it stays constant during training. The NTK was also implicit in some other recent papers [6,13,14]. In the overparametrization regime, a fully-trained deep neural network is indeed equivalent to the kernel regression predictor using the limiting NTK. And the gradient descent achieves zero training loss for a deep overparameterized neural network.…
The massive size of modern neural networks has motivated substantial recent interest in neural network quantization. We introduce Stochastic Markov Gradient Descent (SMGD), a discrete optimization method applicable to training quantized neural networks. The SMGD algorithm is designed for settings where memory is highly constrained during training. We provide theoretical guarantees of algorithm performance as well as encouraging numerical results.
Adversarial attacks during the testing phase of neural networks pose a challenge for the deployment of neural networks in security critical settings. These attacks can be performed by adding noise that is imperceptible to humans on top of the original data. By doing so, an attacker can create an adversarial sample, which will cause neural networks to misclassify. In this paper, we seek to understand the theoretical limits of what can be learned by neural networks in the presence of an adversary. We first defined the hypothesis space of a neural network, and showed the relationship between the …
University of Pennsylvania（宾夕法尼亚大学）；Harvard University（哈佛大学）；Microsoft Research（微软研究院）
Neural networks (NNs) struggle to efficiently solve certain problems, such as learning parities, even when there are simple learning algorithms for those problems. Can NNs discover learning algorithms on their own? We exhibit a NN architecture that, in polynomial time, learns as well as any efficient learning algorithm describable by a constant-sized program. For example, on parity problems, the NN learns as well as Gaussian elimination, an efficient algorithm that can be succinctly described. Our architecture combines both recurrent weight sharing between layers and convolutional weight shari…
Barcelona Supercomputing Center（巴塞罗那超级计算中心）；SEGITTUR（西班牙旅游技术与创新管理公司）
The training of neural networks is usually monitored with a validation (holdout) set to estimate the generalization of the model. This is done instead of measuring intrinsic properties of the model to determine whether it is learning appropriately. In this work, we suggest studying the training of neural networks with Algebraic Topology, specifically Persistent Homology (PH). Using simplicial complex representations of neural networks, we study the PH diagram distance evolution on the neural network learning process with different architectures and several datasets. Results show that the PH di…
Modern neural networks often contain significantly more parameters than the size of their training data. We show that this excess capacity provides an opportunity for embedding secret machine learning models within a trained neural network. Our novel framework hides the existence of a secret neural network with arbitrary desired functionality within a carrier network. We prove theoretically that the secret network's detection is computationally infeasible and demonstrate empirically that the carrier network does not compromise the secret network's disguise. Our paper introduces a previously un…
The transfer learning technique is widely used to learning in one context and applying it to another, i.e. the capacity to apply acquired knowledge and skills to new situations. But is it possible to transfer the learning from a deep neural network to a weaker neural network? Is it possible to improve the performance of a weak neural network using the knowledge acquired by a more powerful neural network? In this work, during the training process of a weak network, we add a loss function that minimizes the distance between the features previously learned from a strong neural network with the fe…
Hypernetworks are neural networks that generate weights for another neural network. We formulate the hypernetwork training objective as a compromise between accuracy and diversity, where the diversity takes into account trivial symmetry transformations of the target network. We explain how this simple formulation generalizes variational inference. We use multi-layered perceptrons to form the mapping from the low dimensional input random vector to the high dimensional weight space, and demonstrate how to reduce the number of parameters in this mapping by parameter sharing. We perform experiment…
School of Information Science and Engineering（信息科学与工程学院）；Lanzhou University（兰州大学）
We introduce the chi-square test neural network: a single hidden layer backpropagation neural network using chi-square test theorem to redefine the cost function and the error function. The weights and thresholds are modified using standard backpropagation algorithm. The proposed approach has the advantage of making consistent data distribution over training and testing sets. It can be used for binary classification. The experimental results on real world data sets indicate that the proposed algorithm can significantly improve the classification accuracy comparing to related approaches.
Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院）；National Research Data Infrastructure (NFDI)（国家研究数据基础设施）
Research on neural networks has gained significant momentum over the past few years. Because training is a resource-intensive process and training data cannot always be made available to everyone, there has been a trend to reuse pre-trained neural networks. As such, neural networks themselves have become research data. In this paper, we first present the neural network ontology FAIRnets Ontology, an ontology to make existing neural network models findable, accessible, interoperable, and reusable according to the FAIR principles. Our ontology allows us to model neural networks on a meta-level i…
Recent advances in neuroscience have revealed many principles about neural processing. In particular, many biological systems were found to reconfigure/recruit single neurons to generate multiple kinds of decisions. Such findings have the potential to advance our understanding of the design and optimization process of artificial neural networks. Previous work demonstrated that dense neural networks are needed to shape complex decision surfaces required for AI-level recognition tasks. We investigate the ability to model high dimensional recognition problems using single or several neurons netwo…
Formal verification of neural networks is essential for their deployment in safety-critical areas. Many available formal verification methods have been shown to be instances of a unified Branch and Bound (BaB) formulation. We propose a novel framework for designing an effective branching strategy for BaB. Specifically, we learn a graph neural network (GNN) to imitate the strong branching heuristic behaviour. Our framework differs from previous methods for learning to branch in two main aspects. Firstly, our framework directly treats the neural network we want to verify as a graph input for the…
Shenzhen Research Institute of Big Data（深圳大数据研究院）；Princeton University（普林斯顿大学）；HKUST（香港科技大学）
Analysis of over-parameterized neural networks has drawn significant attention in recentyears. It was shown that such systems behave like convex systems under various restrictedsettings, such as for two-level neural networks, and when learning is only restricted locally inthe so-called neural tangent kernel space around specialized initializations. However, there areno theoretical techniques that can analyze fully trained deep neural networks encountered inpractice. This paper solves this fundamental problem by investigating such overparameterizeddeep neural networks when fully trained. We gen…
Deep neural networks (DNN) are versatile parametric models utilised successfully in a diverse number of tasks and domains. However, they have limitations---particularly from their lack of robustness and over-sensitivity to out of distribution samples. Bayesian Neural Networks, due to their formulation under the Bayesian framework, provide a principled approach to building neural networks that address these limitations. This paper describes a study that empirically evaluates and compares Bayesian Neural Networks to their equivalent point estimate Deep Neural Networks to quantify the predictive …
National University of Singapore（新加坡国立大学）；Nanjing University（南京大学）
Most of the successful deep neural network architectures are structured, often consisting of elements like convolutional neural networks and gated recurrent neural networks. Recently, graph neural networks have been successfully applied to graph structured data such as point cloud and molecular data. These networks often only consider pairwise dependencies, as they operate on a graph structure. We generalize the graph neural network into a factor graph neural network (FGNN) in order to capture higher order dependencies. We show that FGNN is able to represent Max-Product Belief Propagation, an …
HRL Laboratories, LLC（HRL实验室有限责任公司）；University of Virginia（弗吉尼亚大学）
Connections between integration along hypersufaces, Radon transforms, and neural networks are exploited to highlight an integral geometric mathematical interpretation of neural networks. By analyzing the properties of neural networks as operators on probability distributions for observed data, we show that the distribution of outputs for any node in a neural network can be interpreted as a nonlinear projection along hypersurfaces defined by level surfaces over the input data space. We utilize these descriptions to provide new interpretation for phenomena such as nonlinearity, pooling, activati…
Neural Tangents is a library designed to enable research into infinite-width neural networks. It provides a high-level API for specifying complex and hierarchical neural network architectures. These networks can then be trained and evaluated either at finite-width as usual or in their infinite-width limit. Infinite-width networks can be trained analytically using exact Bayesian inference or using gradient descent via the Neural Tangent Kernel. Additionally, Neural Tangents provides tools to study gradient descent training dynamics of wide but finite networks in either function space or weight …
EDF R&D（法国电力公司研发部）；FiME, Laboratoire de Finance des Marchés de l’Energie（能源市场金融实验室（FiME））
We study news neural networks to approximate function of distributions in a probability space. Two classes of neural networks based on quantile and moment approximation are proposed to learn these functions and are theoretically supported by universal approximation theorems. By mixing the quantile and moment features in other new networks, we develop schemes that outperform existing networks on numerical test cases involving univariate distributions. For bivariate distributions, the moment neural network outperforms all other networks.
Different neural networks trained on the same dataset often learn similar input-output mappings with very different weights. Is there some correspondence between these neural network solutions? For linear networks, it has been shown that different instances of the same network architecture encode the same representational similarity matrix, and their neural activity patterns are connected by orthogonal transformations. However, it is unclear if this holds for non-linear networks. Using a shared response model, we show that different neural networks encode the same input examples as different o…
Ho Chi Minh City University of Technology (HCMUT), VNU-HCM（胡志明市理工大学（HCMUT），越南国家大学胡志明市分校）；EPHE-PSL University（巴黎高等研究实践学院-巴黎文理研究大学）
This paper presents a novel version of the hypergraph neural network method. This method is utilized to solve the noisy label learning problem. First, we apply the PCA dimensional reduction technique to the feature matrices of the image datasets in order to reduce the "noise" and the redundant features in the feature matrices of the image datasets and to reduce the runtime constructing the hypergraph of the hypergraph neural network method. Then, the classic graph-based semi-supervised learning method, the classic hypergraph based semi-supervised learning method, the graph neural network, the …
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Tensor decomposition is an effective approach to compress over-parameterized neural networks and to enable their deployment on resource-constrained hardware platforms. However, directly applying tensor compression in the training process is a challenging task due to the difficulty of choosing a proper tensor rank. In order to achieve this goal, this paper proposes a Bayesian tensorized neural network. Our Bayesian method performs automatic model compression via an adaptive tensor rank determination. We also present approaches for posterior density calculation and maximum a posteriori (MAP) est…
Institute for Mathematics, Carl von Ossietzky University Oldenburg（奥尔登堡卡尔·冯·奥西茨基大学数学研究所）
Neural networks are an indispensable model class for many complex learning tasks. Despite the popularity and importance of neural networks and many different established techniques from literature for stabilization and robustification of the training, the classical concepts from robust statistics have rarely been considered so far in the context of neural networks. Therefore, we adapt the notion of the regression breakdown point to regression neural networks and compute the breakdown point for different feed-forward network configurations and contamination settings. In an extensive simulation …
National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）
In this paper, we theoretically prove that the deep ReLU neural networks do not lie in spurious local minima in the loss landscape under the Neural Tangent Kernel (NTK) regime, that is, in the gradient descent training dynamics of the deep ReLU neural networks whose parameters are initialized by a normal distribution in the limit as the widths of the hidden layers tend to infinity.
Machine learning models are notoriously difficult to interpret and debug. This is particularly true of neural networks. In this work, we introduce automated software testing techniques for neural networks that are well-suited to discovering errors which occur only for rare inputs. Specifically, we develop coverage-guided fuzzing (CGF) methods for neural networks. In CGF, random mutations of inputs to a neural network are guided by a coverage metric toward the goal of satisfying user-specified constraints. We describe how fast approximate nearest neighbor algorithms can provide this coverage me…
Studies show that neural networks, not unlike traditional programs, are subject to bugs, e.g., adversarial samples that cause classification errors and discriminatory instances that demonstrate the lack of fairness. Given that neural networks are increasingly applied in critical applications (e.g., self-driving cars, face recognition systems and personal credit rating systems), it is desirable that systematic methods are developed to analyze (e.g., test or verify) neural networks against desirable properties. Recently, a number of approaches have been developed for analyzing neural networks. T…
Texas State University（德克萨斯州立大学）；Yale University（耶鲁大学）
Neural networks have attracted a lot of attention due to its success in applications such as natural language processing and computer vision. For large scale data, due to the tremendous number of parameters in neural networks, overfitting is an issue in training neural networks. To avoid overfitting, one common approach is to penalize the parameters especially the weights in neural networks. Although neural networks has demonstrated its advantages in many applications, the theoretical foundation of penalized neural networks has not been well-established. Our goal of this paper is to propose th…
Seoul National University（首尔国立大学）；Hongik University（弘益大学）
The tremendous recent progress in analyzing the training dynamics of overparameterized neural networks has primarily focused on wide networks and therefore does not sufficiently address the role of depth in deep learning. In this work, we present the first trainability guarantee of infinitely deep but narrow neural networks. We study the infinite-depth limit of a multilayer perceptron (MLP) with a specific initialization and establish a trainability guarantee using the NTK theory. We then extend the analysis to an infinitely deep convolutional neural network (CNN) and perform brief experiments…
We study the generalization capacity of group convolutional neural networks. We identify precise estimates for the VC dimensions of simple sets of group convolutional neural networks. In particular, we find that for infinite groups and appropriately chosen convolutional kernels, already two-parameter families of convolutional neural networks have an infinite VC dimension, despite being invariant to the action of an infinite group.
Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences（阿塞拜疆国家科学院数学与力学研究所）
In 1987, Hecht-Nielsen showed that any continuous multivariate function can be implemented by a certain type three-layer neural network. This result was very much discussed in neural network literature. In this paper we prove that not only continuous functions but also all discontinuous functions can be implemented by such neural networks.
As a generalization of the work in [Lee et al., 2017], this note briefly discusses when the prior of a neural network output follows a Gaussian process, and how a neural-network-induced Gaussian process is formulated. The posterior mean functions of such a Gaussian process regression lie in the reproducing kernel Hilbert space defined by the neural-network-induced kernel. In the case of two-layer neural networks, the induced Gaussian processes provide an interpretation of the reproducing kernel Hilbert spaces whose union forms a Barron space.
The Neural Tangent Kernel (NTK) is an important milestone in the ongoing effort to build a theory for deep learning. Its prediction that sufficiently wide neural networks behave as kernel methods, or equivalently as random feature models, has been confirmed empirically for certain wide architectures. It remains an open question how well NTK theory models standard neural network architectures of widths common in practice, trained on complex datasets such as ImageNet. We study this question empirically for two well-known convolutional neural network architectures, namely AlexNet and LeNet, and f…
We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks corresp…
We propose a new type of neural networks, Kronecker neural networks (KNNs), that form a general framework for neural networks with adaptive activation functions. KNNs employ the Kronecker product, which provides an efficient way of constructing a very wide network while keeping the number of parameters low. Our theoretical analysis reveals that under suitable conditions, KNNs induce a faster decay of the loss than that by the feed-forward networks. This is also empirically verified through a set of computational examples. Furthermore, under certain technical assumptions, we establish global co…
Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
The success of Neural networks in providing miraculous results when applied to a wide variety of tasks is astonishing. Insight in the working can be obtained by studying the universal approximation property of neural networks. It is proved extensively that neural networks are universal approximators. Further it is proved that deep Neural networks are better approximators. It is specifically proved that for a narrow neural network to approximate a function which is otherwise implemented by a deep Neural network, the network take exponentially large number of neurons. In this work, we have imple…
Neural Ordinary Differential Equation (Neural ODE) has been proposed as a continuous approximation to the ResNet architecture. Some commonly used regularization mechanisms in discrete neural networks (e.g. dropout, Gaussian noise) are missing in current Neural ODE networks. In this paper, we propose a new continuous neural network framework called Neural Stochastic Differential Equation (Neural SDE) network, which naturally incorporates various commonly used regularization mechanisms based on random noise injection. Our framework can model various types of noise injection frequently used in di…
As neural networks have become more powerful, there has been a rising desire to deploy them in the real world; however, the power and accuracy of neural networks is largely due to their depth and complexity, making them difficult to deploy, especially in resource-constrained devices. Neural network quantization has recently arisen to meet this demand of reducing the size and complexity of neural networks by reducing the precision of a network. With smaller and simpler networks, it becomes possible to run neural networks within the constraints of their target hardware. This paper surveys the ma…
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…
Stanford University（斯坦福大学）；Facebook AI Research（Facebook人工智能研究院）
Neural networks are often represented as graphs of connections between neurons. However, despite their wide use, there is currently little understanding of the relationship between the graph structure of the neural network and its predictive performance. Here we systematically investigate how does the graph structure of neural networks affect their predictive performance. To this end, we develop a novel graph-based representation of neural networks called relational graph, where layers of neural network computation correspond to rounds of message exchange along the graph structure. Using this …
We present a derivation of the gradients of feedforward neural networks using Fréchet calculus which is arguably more compact than the ones usually presented in the literature. We first derive the gradients for ordinary neural networks working on vectorial data and show how these derived formulas can be used to derive a simple and efficient algorithm for calculating a neural networks gradients. Subsequently we show how our analysis generalizes to more general neural network architectures including, but not limited to, convolutional networks.
Université Paris-Dauphine -- PSL Research University（巴黎第九大学（多菲纳大学）-- PSL研究大学）
Deep neural networks are state-of-the-art in a wide variety of tasks, however, they exhibit important limitations which hinder their use and deployment in real-world applications. When developing and training neural networks, the accuracy should not be the only concern, neural networks must also be cost-effective and reliable. Although accurate, large neural networks often lack these properties. This thesis focuses on the problem of training neural networks which are not only accurate but also compact, easy to train, reliable and robust to adversarial examples. To tackle these problems, we lev…
We propose firefly neural architecture descent, a general framework for progressively and dynamically growing neural networks to jointly optimize the networks' parameters and architectures. Our method works in a steepest descent fashion, which iteratively finds the best network within a functional neighborhood of the original network that includes a diverse set of candidate network structures. By using Taylor approximation, the optimal network structure in the neighborhood can be found with a greedy selection procedure. We show that firefly descent can flexibly grow networks both wider and dee…
Queen Mary University of London（伦敦玛丽女王大学）；American University of Beirut（贝鲁特美国大学）
Graph Neural Networks have gained huge interest in the past few years. These powerful algorithms expanded deep learning models to non-Euclidean space and were able to achieve state of art performance in various applications including recommender systems and social networks. However, this performance is based on static graph structures assumption which limits the Graph Neural Networks performance when the data varies with time. Spatiotemporal Graph Neural Networks are extension of Graph Neural Networks that takes the time factor into account. Recently, various Spatiotemporal Graph Neural Networ…
The use of artificial neural networks as models of chaotic dynamics has been rapidly expanding. Still, a theoretical understanding of how neural networks learn chaos is lacking. Here, we employ a geometric perspective to show that neural networks can efficiently model chaotic dynamics by becoming structurally chaotic themselves. We first confirm neural network's efficiency in emulating chaos by showing that a parsimonious neural network trained only on few data points can reconstruct strange attractors, extrapolate outside training data boundaries, and accurately predict local divergence rates…
