Kernels derived from deep neural networks (DNNs) in the infinite-width regime provide not only high performance in a range of machine learning tasks but also new theoretical insights into DNN training dynamics and generalization. In this paper, we extend the family of kernels associated with recurrent neural networks (RNNs), which were previously derived only for simple RNNs, to more complex architectures including bidirectional RNNs and RNNs with average pooling. We also develop a fast GPU implementation to exploit the full practical potential of the kernels. Though RNNs are typically only applied to time-series data, we demonstrate that classifiers using RNN-based kernels outperform a range of baseline methods on 90 non-time-series datasets from the UCI data repository.
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.
Center for Brain Science, Harvard University（哈佛大学脑科学中心）；School of Engineering and Applied Sciences, Harvard University（哈佛大学工程与应用科学学院）
The training dynamics and generalization properties of neural networks (NN) can be precisely characterized in function space via the neural tangent kernel (NTK). Structural changes to the NTK during training reflect feature learning and underlie the superior performance of networks outside of the static kernel regime. In this work, we seek to theoretically understand kernel alignment, a prominent and ubiquitous structural change that aligns the NTK with the target function. We first study a toy model of kernel evolution in which the NTK evolves to accelerate training and show that alignment na…
Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔理工学院）；Université Lyon 1（里昂第一大学）；Laboratoire de Biométrie et Biologie Evolutive UMR 5558（生物统计与进化生物学实验室 UMR 5558）
Substring kernels are classical tools for representing biological sequences or text. However, when large amounts of annotated data are available, models that allow end-to-end training such as neural networks are often preferred. Links between recurrent neural networks (RNNs) and substring kernels have recently been drawn, by formally showing that RNNs with specific activation functions were points in a reproducing kernel Hilbert space (RKHS). In this paper, we revisit this link by generalizing convolutional kernel networks---originally related to a relaxation of the mismatch kernel---to model …
There are currently two parameterizations used to derive fixed kernels corresponding to infinite width neural networks, the NTK (Neural Tangent Kernel) parameterization and the naive standard parameterization. However, the extrapolation of both of these parameterizations to infinite width is problematic. The standard parameterization leads to a divergent neural tangent kernel while the NTK parameterization fails to capture crucial aspects of finite width networks such as: the dependence of training dynamics on relative layer widths, the relative training dynamics of weights and biases, and ove…
The Neural Tangent Kernel (NTK) has recently attracted intense study, as it describes the evolution of an over-parameterized Neural Network (NN) trained by gradient descent. However, it is now well-known that gradient descent is not always a good optimizer for NNs, which can partially explain the unsatisfactory practical performance of the NTK regression estimator. In this paper, we introduce the Weighted Neural Tangent Kernel (WNTK), a generalized and improved tool, which can capture an over-parameterized NN's training dynamics under different optimizers. Theoretically, in the infinite-width …
Ecole Normale Supérieure（巴黎高等师范学院）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université de Paris（巴黎大学）；Collège de France（法兰西公学院）；LightOn（LightOn公司）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Reservoir Computing is a class of simple yet efficient Recurrent Neural Networks where internal weights are fixed at random and only a linear output layer is trained. In the large size limit, such random neural networks have a deep connection with kernel methods. Our contributions are threefold: a) We rigorously establish the recurrent kernel limit of Reservoir Computing and prove its convergence. b) We test our models on chaotic time series prediction, a classic but challenging benchmark in Reservoir Computing, and show how the Recurrent Kernel is competitive and computationally efficient whe…
University of California, Berkeley（加州大学伯克利分校）；Helen Wills Neuroscience Institute（海伦·威尔斯神经科学研究所）
The development of methods to guide the design of neural networks is an important open challenge for deep learning theory. As a paradigm for principled neural architecture design, we propose the translation of high-performing kernels, which are better-understood and amenable to first-principles design, into equivalent network architectures, which have superior efficiency, flexibility, and feature learning. To this end, we constructively prove that, with just an appropriate choice of activation function, any positive-semidefinite dot-product kernel can be realized as either the NNGP or neural t…
Stanford University（斯坦福大学）；Element AI（元素人工智能公司）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）
In suitably initialized wide networks, small learning rates transform deep neural networks (DNNs) into neural tangent kernel (NTK) machines, whose training dynamics is well-approximated by a linear weight expansion of the network at initialization. Standard training, however, diverges from its linearization in ways that are poorly understood. We study the relationship between the training dynamics of nonlinear deep networks, the geometry of the loss landscape, and the time evolution of a data-dependent NTK. We do so through a large-scale phenomenological analysis of training, synthesizing dive…
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…
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 …
Pacific Northwest National Laboratory（太平洋西北国家实验室）；University of California San Diego（加州大学圣迭戈分校）；Rutgers University - New Brunswick（罗格斯大学新布朗斯维克分校）
We introduce torchNTK, a python library to calculate the empirical neural tangent kernel (NTK) of neural network models in the PyTorch framework. We provide an efficient method to calculate the NTK of multilayer perceptrons. We compare the explicit differentiation implementation against autodifferentiation implementations, which have the benefit of extending the utility of the library to any architecture supported by PyTorch, such as convolutional networks. A feature of the library is that we expose the user to layerwise NTK components, and show that in some regimes a layerwise calculation is …
Yang (2020a) recently showed that the Neural Tangent Kernel (NTK) at initialization has an infinite-width limit for a large class of architectures including modern staples such as ResNet and Transformers. However, their analysis does not apply to training. Here, we show the same neural networks (in the so-called NTK parametrization) during training follow a kernel gradient descent dynamics in function space, where the kernel is the infinite-width NTK. This completes the proof of the *architectural universality* of NTK behavior. To achieve this result, we apply the Tensor Programs technique: Wr…
In a backdoor attack, an attacker injects corrupted examples into the training set. The goal of the attacker is to cause the final trained model to predict the attacker's desired target label when a predefined trigger is added to test inputs. Central to these attacks is the trade-off between the success rate of the attack and the number of corrupted training examples injected. We pose this attack as a novel bilevel optimization problem: construct strong poison examples that maximize the attack success rate of the trained model. We use neural tangent kernels to approximate the training dynamics…
Sapienza University of Rome（罗马第一大学）；University of Cantabria（坎塔布里亚大学）
Gated recurrent neural networks have achieved remarkable results in the analysis of sequential data. Inside these networks, gates are used to control the flow of information, allowing to model even very long-term dependencies in the data. In this paper, we investigate whether the original gate equation (a linear projection followed by an element-wise sigmoid) can be improved. In particular, we design a more flexible architecture, with a small number of adaptable parameters, which is able to model a wider range of gating functions than the classical one. To this end, we replace the sigmoid func…
Understanding the Evolution of the Neural Tangent Kernel at the Edge of Stability
Department of Electrical and Computer Engineering, Princeton University（普林斯顿大学电气工程与计算机科学系）；Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Oxford-Man Institute of Quantitative Finance, University of Oxford（牛津大学量化金融研究所）；OATML, University of Oxford（牛津大学OATML研究中心）；Department of Computer Science, University of Oxford（牛津大学计算机科学系）；Spotify
Instituto Universitário de Lisboa (ISCTE-IUL)（里斯本大学学院（ISCTE-IUL））；Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高级技术学院数学分析、几何与动力系统中心，里斯本大学）
Seoul National University（首尔国立大学）；NAVER AI Lab（NAVER人工智能实验室）；NAVER CLOVA
In Neural Architecture Search (NAS), reducing the cost of architecture evaluation remains one of the most crucial challenges. Among a plethora of efforts to bypass training of each candidate architecture to convergence for evaluation, the Neural Tangent Kernel (NTK) is emerging as a promising theoretical framework that can be utilized to estimate the performance of a neural architecture at initialization. In this work, we revisit several at-initialization metrics that can be derived from the NTK and reveal their key shortcomings. Then, through the empirical analysis of the time evolution of NT…
A generalized neural tangent kernel for surrogate gradient learning
Department of Physiology, University of Bern（伯恩大学生理学系）；Institute for Applied Mathematics, University of Bonn（波恩大学应用数学研究所）；Institute of Genetics, University of Bonn（波恩大学遗传学研究所）；Bernstein Center Freiburg, University of Freiburg（弗赖堡大学伯恩斯坦中心）
University of Minnesota（明尼苏达大学）；Sumo Logic（Sumo Logic公司）
A number of recent approaches have been proposed for pruning neural network parameters at initialization with the goal of reducing the size and computational burden of models while minimally affecting their training dynamics and generalization performance. While each of these approaches have some amount of well-founded motivation, a rigorous analysis of the effect of these pruning methods on network training dynamics and their formal relationship to each other has thus far received little attention. Leveraging recent theoretical approximations provided by the Neural Tangent Kernel, we unify a …
The study of deep neural networks (DNNs) in the infinite-width limit, via the so-called neural tangent kernel (NTK) approach, has provided new insights into the dynamics of learning, generalization, and the impact of initialization. One key DNN architecture remains to be kernelized, namely, the recurrent neural network (RNN). In this paper we introduce and study the Recurrent Neural Tangent Kernel (RNTK), which provides new insights into the behavior of overparametrized RNNs. A key property of the RNTK should greatly benefit practitioners is its ability to compare inputs of different length. T…
MIT Computer Science & Artificial Intelligence Laboratory（麻省理工学院计算机科学与人工智能实验室）
The design of neural architectures for structured objects is typically guided by experimental insights rather than a formal process. In this work, we appeal to kernels over combinatorial structures, such as sequences and graphs, to derive appropriate neural operations. We introduce a class of deep recurrent neural operations and formally characterize their associated kernel spaces. Our recurrent modules compare the input to virtual reference objects (cf. filters in CNN) via the kernels. Similar to traditional neural operations, these reference objects are parameterized and directly optimized i…
Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；EPFL（洛桑联邦理工学院）
We study the relative power of learning with gradient descent on differentiable models, such as neural networks, versus using the corresponding tangent kernels. We show that under certain conditions, gradient descent achieves small error only if a related tangent kernel method achieves a non-trivial advantage over random guessing (a.k.a. weak learning), though this advantage might be very small even when gradient descent can achieve arbitrarily high accuracy. Complementing this, we show that without these conditions, gradient descent can in fact learn with small error even when no kernel metho…
State-of-the-art neural networks are heavily over-parameterized, making the optimization algorithm a crucial ingredient for learning predictive models with good generalization properties. A recent line of work has shown that in a certain over-parameterized regime, the learning dynamics of gradient descent are governed by a certain kernel obtained at initialization, called the neural tangent kernel. We study the inductive bias of learning in such a regime by analyzing this kernel and the corresponding function space (RKHS). In particular, we study smoothness, approximation, and stability proper…
University of California, Berkeley（加利福尼亚大学伯克利分校）；MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）
We investigate the connections between neural networks and simple building blocks in kernel space. In particular, using well established feature space tools such as direct sum, averaging, and moment lifting, we present an algebra for creating "compositional" kernels from bags of features. We show that these operations correspond to many of the building blocks of "neural tangent kernels (NTK)". Experimentally, we show that there is a correlation in test error between neural network architectures and the associated kernels. We construct a simple neural network architecture using only 3x3 convolu…
Peking University（北京大学）；Georgia Institute of Technology（佐治亚理工学院）
Deep residual networks (ResNets) have demonstrated better generalization performance than deep feedforward networks (FFNets). However, the theory behind such a phenomenon is still largely unknown. This paper studies this fundamental problem in deep learning from a so-called "neural tangent kernel" perspective. Specifically, we first show that under proper conditions, as the width goes to infinity, training deep ResNets can be viewed as learning reproducing kernel functions with some kernel function. We then compare the kernel of deep ResNets with that of deep FFNets and discover that the class…
Weizmann Institute of Science（魏茨曼科学研究所）；University of Maryland, College Park（马里兰大学帕克分校）
Recent theoretical work has shown that massively overparameterized neural networks are equivalent to kernel regressors that use Neural Tangent Kernels(NTK). Experiments show that these kernel methods perform similarly to real neural networks. Here we show that NTK for fully connected networks is closely related to the standard Laplace kernel. We show theoretically that for normalized data on the hypersphere both kernels have the same eigenfunctions and their eigenvalues decay polynomially at the same rate, implying that their Reproducing Kernel Hilbert Spaces (RKHS) include the same sets of fu…
Department of Computer Science, The Ohio State University（俄亥俄州立大学计算机科学系）；Halicioğlu Data Science Institute, University of California, San Diego（加利福尼亚大学圣迭戈分校哈利西奥卢数据科学学院）
The goal of this work is to shed light on the remarkable phenomenon of transition to linearity of certain neural networks as their width approaches infinity. We show that the transition to linearity of the model and, equivalently, constancy of the (neural) tangent kernel (NTK) result from the scaling properties of the norm of the Hessian matrix of the network as a function of the network width. We present a general framework for understanding the constancy of the tangent kernel via Hessian scaling applicable to the standard classes of neural networks. Our analysis provides a new perspective on…
As a popular approach to modeling the dynamics of training overparametrized neural networks (NNs), the neural tangent kernels (NTK) are known to fall behind real-world NNs in generalization ability. This performance gap is in part due to the \textit{label agnostic} nature of the NTK, which renders the resulting kernel not as \textit{locally elastic} as NNs~\citep{he2019local}. In this paper, we introduce a novel approach from the perspective of \emph{label-awareness} to reduce this gap for the NTK. Specifically, we propose two label-aware kernels that are each a superimposition of a label-agno…
We analyze the convergence of the averaged stochastic gradient descent for overparameterized two-layer neural networks for regression problems. It was recently found that a neural tangent kernel (NTK) plays an important role in showing the global convergence of gradient-based methods under the NTK regime, where the learning dynamics for overparameterized neural networks can be almost characterized by that for the associated reproducing kernel Hilbert space (RKHS). However, there is still room for a convergence rate analysis in the NTK regime. In this study, we show that the averaged stochastic…
Neural Tangent Kernels and Fisher Information Matrices for Simple ReLU Networks with Random Hidden Weights
Faculty of Information Science and Electrical Engineering, Kyushu University（九州大学信息科学与电气工程学部）；Faculty of Science and Engineering, Waseda University（早稻田大学理工学部）；Faculty of Information Sciences, Hiroshima City University（广岛城大学信息科学学部）；Joint Graduate School of Math for Innovation, Kyushu University（九州大学创新数学研究生院）；The University of Electro-Communications（电子通信大学）
Simons Institute for the Theory of Computing, University of California, Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）
We prove that the reproducing kernel Hilbert spaces (RKHS) of a deep neural tangent kernel and the Laplace kernel include the same set of functions, when both kernels are restricted to the sphere $\mathbb{S}^{d-1}$. Additionally, we prove that the exponential power kernel with a smaller power (making the kernel less smooth) leads to a larger RKHS, when it is restricted to the sphere $\mathbb{S}^{d-1}$ and when it is defined on the entire $\mathbb{R}^d$.
National University of Singapore（新加坡国立大学）；University of Oxford（牛津大学）
Recent work by Jacot et al. (2018) has shown that training a neural network using gradient descent in parameter space is related to kernel gradient descent in function space with respect to the Neural Tangent Kernel (NTK). Lee et al. (2019) built on this result by establishing that the output of a neural network trained using gradient descent can be approximated by a linear model when the network width is large. Indeed, under regularity conditions, the NTK converges to a time-independent kernel in the infinite-width limit. This regime is often called the NTK regime. In parallel, recent works o…
The Graduate University for Advanced Studies, SOKENDAI（综合研究大学院大学（SOKENDAI））；National Institute of Informatics（国立信息学研究所）
A soft tree is an actively studied variant of a decision tree that updates splitting rules using the gradient method. Although soft trees can take various architectures, their impact is not theoretically well known. In this paper, we formulate and analyze the Neural Tangent Kernel (NTK) induced by soft tree ensembles for arbitrary tree architectures. This kernel leads to the remarkable finding that only the number of leaves at each depth is relevant for the tree architecture in ensemble learning with an infinite number of trees. In other words, if the number of leaves at each depth is fixed, t…
University of British Columbia（不列颠哥伦比亚大学）；UBC（不列颠哥伦比亚大学）；Amii（阿尔伯塔机器智能研究所）
We propose a new method for approximating active learning acquisition strategies that are based on retraining with hypothetically-labeled candidate data points. Although this is usually infeasible with deep networks, we use the neural tangent kernel to approximate the result of retraining, and prove that this approximation works asymptotically even in an active learning setup -- approximating "look-ahead" selection criteria with far less computation required. This also enables us to conduct sequential active learning, i.e. updating the model in a streaming regime, without needing to retrain th…
The Graduate University for Advanced Studies, SOKENDAI（综合研究大学院大学）；National Institute of Informatics（国立信息学研究所）
In practical situations, the tree ensemble is one of the most popular models along with neural networks. A soft tree is a variant of a decision tree. Instead of using a greedy method for searching splitting rules, the soft tree is trained using a gradient method in which the entire splitting operation is formulated in a differentiable form. Although ensembles of such soft trees have been used increasingly in recent years, little theoretical work has been done to understand their behavior. By considering an ensemble of infinite soft trees, this paper introduces and studies the Tree Neural Tange…
Korea Advanced Institute of Science and Technology（韩国科学技术院）；Tel Aviv University（特拉维夫大学）
The Neural Tangent Kernel (NTK) has discovered connections between deep neural networks and kernel methods with insights of optimization and generalization. Motivated by this, recent works report that NTK can achieve better performances compared to training neural networks on small-scale datasets. However, results under large-scale settings are hardly studied due to the computational limitation of kernel methods. In this work, we propose an efficient feature map construction of the NTK of fully-connected ReLU network which enables us to apply it to large-scale datasets. We combine random featu…
University of California, Los Angeles（加州大学洛杉矶分校）；The Hong Kong University of Science and Technology（香港科技大学）
A recent breakthrough in deep learning theory shows that the training of over-parameterized deep neural networks can be characterized by a kernel function called \textit{neural tangent kernel} (NTK). However, it is known that this type of results does not perfectly match the practice, as NTK-based analysis requires the network weights to stay very close to their initialization throughout training, and cannot handle regularizers or gradient noises. In this paper, we provide a generalized neural tangent kernel analysis and show that noisy gradient descent with weight decay can still exhibit a "k…
High dimensionality poses many challenges to the use of data, from visualization and interpretation, to prediction and storage for historical preservation. Techniques abound to reduce the dimensionality of fixed-length sequences, yet these methods rarely generalize to variable-length sequences. To address this gap, we extend existing methods that rely on the use of kernels to variable-length sequences via use of the Recurrent Neural Tangent Kernel (RNTK). Since a deep neural network with ReLu activation is a Max-Affine Spline Operator (MASO), we dub our approach Max-Affine Spline Kernel (MASK)…
Modern deep learning models employ considerably more parameters than required to fit the training data. Whereas conventional statistical wisdom suggests such models should drastically overfit, in practice these models generalize remarkably well. An emerging paradigm for describing this unexpected behavior is in terms of a \emph{double descent} curve, in which increasing a model's capacity causes its test error to first decrease, then increase to a maximum near the interpolation threshold, and then decrease again in the overparameterized regime. Recent efforts to explain this phenomenon theoret…
Deep neural networks have become essential for numerous applications due to their strong empirical performance such as vision, RL, and classification. Unfortunately, these networks are quite difficult to interpret, and this limits their applicability in settings where interpretability is important for safety, such as medical imaging. One type of deep neural network is neural tangent kernel that is similar to a kernel machine that provides some aspect of interpretability. To further contribute interpretability with respect to classification and the layers, we develop a new network as a combinat…
We explore the link between deep ensembles and Gaussian processes (GPs) through the lens of the Neural Tangent Kernel (NTK): a recent development in understanding the training dynamics of wide neural networks (NNs). Previous work has shown that even in the infinite width limit, when NNs become GPs, there is no GP posterior interpretation to a deep ensemble trained with squared error loss. We introduce a simple modification to standard deep ensembles training, through addition of a computationally-tractable, randomised and untrainable function to each ensemble member, that enables a posterior i…
Neural Tangent Kernel (NTK) theory is widely used to study the dynamics of infinitely-wide deep neural networks (DNNs) under gradient descent. But do the results for infinitely-wide networks give us hints about the behavior of real finite-width ones? In this paper, we study empirically when NTK theory is valid in practice for fully-connected ReLU and sigmoid DNNs. We find out that whether a network is in the NTK regime depends on the hyperparameters of random initialization and the network's depth. In particular, NTK theory does not explain the behavior of sufficiently deep networks initialize…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper establishes rates of universal approximation for the shallow neural tangent kernel (NTK): network weights are only allowed microscopic changes from random initialization, which entails that activations are mostly unchanged, and the network is nearly equivalent to its linearization. Concretely, the paper has two main contributions: a generic scheme to approximate functions with the NTK by sampling from transport mappings between the initial weights and their desired values, and the construction of transport mappings via Fourier transforms. Regarding the first contribution, the proof …
