Uncertainty Quantification with the Empirical Neural Tangent Kernel
School of Mathematics And Physics University of Queensland（数学与物理学院，昆士兰大学）；Dept. of Electrical and Electronic Engineering University of Melbourne（电子与电气工程系，墨尔本大学）；School of Mathematics and Statistics University of Melbourne（数学与统计学学院，墨尔本大学）；CIRES and School of Mathematics And Physics University of Queensland（CIRES和数学与物理学院，昆士兰大学）
Joseph Wilson, Chris van der Heide, Liam Hodgkinson, Fred Roosta
While neural networks have demonstrated impressive performance across various tasks, accurately quantifying uncertainty in their predictions is essential to ensure their trustworthiness and enable widespread adoption in critical systems. Several Bayesian uncertainty quantification (UQ) methods exist that are either cheap or reliable, but not both. We propose a post-hoc, sampling-based UQ method for over-parameterized networks at the end of training. Our approach constructs efficient and meaningful deep ensembles by employing a (stochastic) gradient-descent sampling process on appropriately linearized networks. We demonstrate that our method effectively approximates the posterior of a Gaussian process using the empirical Neural Tangent Kernel. Through a series of numerical experiments, we show that our method not only outperforms competing approaches in computational efficiency-often reducing costs by multiple factors-but also maintains state-of-the-art performance across a variety of UQ metrics for both regression and classification tasks.
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…
Distribution-Free Uncertainty Quantification for Kernel Methods by Gradient Perturbations
Hungarian Academy of Sciences（匈牙利科学院）；MTA SZTAKI: Institute for Computer Science and Control（MTA SZTAKI：计算机科学与控制研究所）；EPIC Centre of Excellence（EPIC卓越中心）
The ever-growing size of the datasets renders well-studied learning techniques, such as Kernel Ridge Regression, inapplicable, posing a serious computational challenge. Divide-and-conquer is a common remedy, suggesting to split the dataset into disjoint partitions, obtain the local estimates and average them, it allows to scale-up an otherwise ineffective base approach. In the current study we suggest a fully data-driven approach to quantify uncertainty of the averaged estimator. Namely, we construct simultaneous element-wise confidence bands for the predictions yielded by the averaged estimat…
University of Oxford（牛津大学）；Microsoft Research Cambridge（微软研究院剑桥分院）
While causal models are becoming one of the mainstays of machine learning, the problem of uncertainty quantification in causal inference remains challenging. In this paper, we study the causal data fusion problem, where datasets pertaining to multiple causal graphs are combined to estimate the average treatment effect of a target variable. As data arises from multiple sources and can vary in quality and quantity, principled uncertainty quantification becomes essential. To that end, we introduce Bayesian Interventional Mean Processes, a framework which combines ideas from probabilistic integrat…
Rutgers University Newark（罗格斯大学纽瓦克分校）；Institute for Advanced Study（高等研究院）
Deep Gaussian Process (DGP) as a model prior in Bayesian learning intuitively exploits the expressive power in function composition. DGPs also offer diverse modeling capabilities, but inference is challenging because marginalization in latent function space is not tractable. With Bochner's theorem, DGP with squared exponential kernel can be viewed as a deep trigonometric network consisting of the random feature layers, sine and cosine activation units, and random weight layers. In the wide limit with a bottleneck, we show that the weight space view yields the same effective covariance function…
Uncertainty Quantification for Regression: A Unified Framework based on kernel scores
LMU Munich（慕尼黑莱茵恩大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；DLR-German Aerospace Center University of Tromsø（德国航空航天中心-特罗姆瑟大学）；German Research Center for Artificial Intelligence (DFKI, DSA)（德国人工智能研究中心）
Computer Science and Artificial Intelligence Lab (CSAIL)（计算机科学与人工智能实验室）；Massachusetts Institute of Technology (MIT)（麻省理工学院）
Two key challenges facing modern deep learning are mitigating deep networks' vulnerability to adversarial attacks and understanding deep learning's generalization capabilities. Towards the first issue, many defense strategies have been developed, with the most common being Adversarial Training (AT). Towards the second challenge, one of the dominant theories that has emerged is the Neural Tangent Kernel (NTK) -- a characterization of neural network behavior in the infinite-width limit. In this limit, the kernel is frozen, and the underlying feature map is fixed. In finite widths, however, there…
Identifying unfamiliar inputs, also known as out-of-distribution (OOD) detection, is a crucial property of any decision making process. A simple and empirically validated technique is based on deep ensembles where the variance of predictions over different neural networks acts as a substitute for input uncertainty. Nevertheless, a theoretical understanding of the inductive biases leading to the performance of deep ensemble's uncertainty estimation is missing. To improve our description of their behavior, we study deep ensembles with large layer widths operating in simplified linear training re…
Princeton University（普林斯顿大学）；Carnegie Mellon University（卡内基梅隆大学）；Institute for Advanced Study（高等研究院）
Recent research shows that for training with $\ell_2$ loss, convolutional neural networks (CNNs) whose width (number of channels in convolutional layers) goes to infinity correspond to regression with respect to the CNN Gaussian Process kernel (CNN-GP) if only the last layer is trained, and correspond to regression with respect to the Convolutional Neural Tangent Kernel (CNTK) if all layers are trained. An exact algorithm to compute CNTK (Arora et al., 2019) yielded the finding that classification accuracy of CNTK on CIFAR-10 is within 6-7% of that of that of the corresponding CNN architecture…
In this manuscript, we propose a multiclass data description model based on kernel Mahalanobis distance (MDD-KM) with self-adapting hyperparameter setting. MDD-KM provides uncertainty quantification and can be deployed to build classification systems for the realistic scenario where out-of-distribution (OOD) samples are present among the test data. Given a test signal, a quantity related to empirical kernel Mahalanobis distance between the signal and each of the training classes is computed. Since these quantities correspond to the same reproducing kernel Hilbert space, they are commensurable …
Model-Free Kernel Conformal Depth Measures Algorithm for Uncertainty Quantification in Regression Models in Separable Hilbert Spaces
Department of Biostatistics（生物统计学系）；Harvard University（哈佛大学）；Department of Epidemiology and Biostatistics（流行病学与生物统计学系）；University of South Carolina（南卡罗来纳大学）；LTCI Telecom Paris Institut Polytechnique de Paris（巴黎电信理工学院LTCI）；UCLA（加州大学洛杉矶分校）
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）
To better understand the theoretical behavior of large neural networks, several works have analyzed the case where a network's width tends to infinity. In this regime, the effect of random initialization and the process of training a neural network can be formally expressed with analytical tools like Gaussian processes and neural tangent kernels. In this paper, we review methods for quantifying uncertainty in such infinite-width neural networks and compare their relationship to Gaussian processes in the Bayesian inference framework. We make use of several equivalence results along the way to o…
We investigate the frequentist coverage properties of credible sets resulting in from Gaussian process priors with squared exponential covariance kernel. First we show that by selecting the scaling hyper-parameter using the maximum marginal likelihood estimator in the (slightly modified) squared exponential covariance kernel the corresponding credible sets will provide overconfident, misleading uncertainty statements for a large, representative subclass of the functional parameters in context of the Gaussian white noise model. Then we show that by either blowing up the credible sets with a log…
Center for Data Science（数据科学中心）；New York University（纽约大学）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）
The adversarial vulnerability of neural nets, and subsequent techniques to create robust models have attracted significant attention; yet we still lack a full understanding of this phenomenon. Here, we study adversarial examples of trained neural networks through analytical tools afforded by recent theory advances connecting neural networks and kernel methods, namely the Neural Tangent Kernel (NTK), following a growing body of work that leverages the NTK approximation to successfully analyze important deep learning phenomena and design algorithms for new applications. We show how NTKs allow to…
Several recent trends in machine learning theory and practice, from the design of state-of-the-art Gaussian Process to the convergence analysis of deep neural nets (DNNs) under stochastic gradient descent (SGD), have found it fruitful to study wide random neural networks. Central to these approaches are certain scaling limits of such networks. We unify these results by introducing a notion of a straightline \emph{tensor program} that can express most neural network computations, and we characterize its scaling limit when its tensors are large and randomized. From our framework follows (1) the …
Traditional Bayesian approaches for model uncertainty quantification rely on notoriously difficult processes of marginalization over each network parameter to estimate its probability density function (PDF). Our hypothesis is that internal layer outputs of a trained neural network contain all of the information related to both its mapping function (quantified by its weights) as well as the input data distribution. We therefore propose a framework for predictive uncertainty quantification of a trained neural network that explicitly estimates the PDF of its raw prediction space (before activatio…
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…
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…
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…
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.
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…
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…
Rice University（莱斯大学）；Facebook AI Research（Facebook人工智能研究院）
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 ap…
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 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…
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…
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…
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…
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 …
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 a scale separation of Kolmogorov width type between subspaces of a given Banach space under the condition that a sequence of linear maps converges much faster on one of the subspaces. The general technique is then applied to show that reproducing kernel Hilbert spaces are poor $L^2$-approximators for the class of two-layer neural networks in high dimension, and that multi-layer networks with small path norm are poor approximators for certain Lipschitz functions, also in the $L^2$-topology.
Duke University（杜克大学）；Georgia Institute of Technology（佐治亚理工学院）
We present a novel neural network Maximum Mean Discrepancy (MMD) statistic by identifying a new connection between neural tangent kernel (NTK) and MMD. This connection enables us to develop a computationally efficient and memory-efficient approach to compute the MMD statistic and perform NTK based two-sample tests towards addressing the long-standing challenge of memory and computational complexity of the MMD statistic, which is essential for online implementation to assimilating new samples. Theoretically, such a connection allows us to understand the NTK test statistic properties, such as th…
