We consider the dynamic of gradient descent for learning a two-layer neural network. We assume the input $x\in\mathbb{R}^d$ is drawn from a Gaussian distribution and the label of $x$ satisfies $f^{\star}(x) = a^{\top}|W^{\star}x|$, where $a\in\mathbb{R}^d$ is a nonnegative vector and $W^{\star} \in\mathbb{R}^{d\times d}$ is an orthonormal matrix. We show that an over-parametrized two-layer neural network with ReLU activation, trained by gradient descent from random initialization, can provably learn the ground truth network with population loss at most $o(1/d)$ in polynomial time with polynomial samples. On the other hand, we prove that any kernel method, including Neural Tangent Kernel, with a polynomial number of samples in $d$, has population loss at least $Ω(1 / d)$.
We give a simple proof for the global convergence of gradient descent in training deep ReLU networks with the standard square loss, and show some of its improvements over the state-of-the-art. In particular, while prior works require all the hidden layers to be wide with width at least $Ω(N^8)$ ($N$ being the number of training samples), we require a single wide layer of linear, quadratic or cubic width depending on the type of initialization. Unlike many recent proofs based on the Neural Tangent Kernel (NTK), our proof need not track the evolution of the entire NTK matrix, or more generally, …
MPI-MIS（马克斯·普朗克数学研究所）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；UCLA（加州大学洛杉矶分校）
A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global convergence of gradient descent algorithms and the generalization of deep nets. However, existing results either provide bounds in the two-layer setting or assume that the spectrum of the NTK matrices is bounded away from 0 for multi-layer networks. In this paper, we provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limit…
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…
We focus on estimating \emph{a priori} generalization error of two-layer ReLU neural networks (NNs) trained by mean squared error, which only depends on initial parameters and the target function, through the following research line. We first estimate \emph{a priori} generalization error of finite-width two-layer ReLU NN with constraint of minimal norm solution, which is proved by \cite{zhang2019type} to be an equivalent solution of a linearized (w.r.t. parameter) finite-width two-layer NN. As the width goes to infinity, the linearized NN converges to the NN in Neural Tangent Kernel (NTK) regi…
Recent theoretical work has established connections between over-parametrized neural networks and linearized models governed by he Neural Tangent Kernels (NTKs). NTK theory leads to concrete convergence and generalization results, yet the empirical performance of neural networks are observed to exceed their linearized models, suggesting insufficiency of this theory. Towards closing this gap, we investigate the training of over-parametrized neural networks that are beyond the NTK regime yet still governed by the Taylor expansion of the network. We bring forward the idea of \emph{randomizing} th…
Texas A&M University（得克萨斯农工大学）；University of Toronto（多伦多大学）
We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not deterministic if depth and width simultaneously tend to infinity. Moreover, we prove that for such deep and wide networks, the NTK has a non-trivial evolution during training by showing that the mean of its first SGD update is also exponential in the ratio of network depth to width. This is…
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…
Peking University（北京大学）；Princeton University（普林斯顿大学）；The University of Washington（华盛顿大学）
Federated Learning (FL) is an emerging learning scheme that allows different distributed clients to train deep neural networks together without data sharing. Neural networks have become popular due to their unprecedented success. To the best of our knowledge, the theoretical guarantees of FL concerning neural networks with explicit forms and multi-step updates are unexplored. Nevertheless, training analysis of neural networks in FL is non-trivial for two reasons: first, the objective loss function we are optimizing is non-smooth and non-convex, and second, we are even not updating in the gradi…
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…
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…
Yale University（耶鲁大学）；Max-Planck-Institut für Informatik（马克斯·普朗克信息学研究所）；Google Research（谷歌研究院）
Infinite width limit has shed light on generalization and optimization aspects of deep learning by establishing connections between neural networks and kernel methods. Despite their importance, the utility of these kernel methods was limited in large-scale learning settings due to their (super-)quadratic runtime and memory complexities. Moreover, most prior works on neural kernels have focused on the ReLU activation, mainly due to its popularity but also due to the difficulty of computing such kernels for general activations. In this work, we overcome such difficulties by providing methods to …
This work establishes low test error of gradient flow (GF) and stochastic gradient descent (SGD) on two-layer ReLU networks with standard initialization, in three regimes where key sets of weights rotate little (either naturally due to GF and SGD, or due to an artificial constraint), and making use of margins as the core analytic technique. The first regime is near initialization, specifically until the weights have moved by $\mathcal{O}(\sqrt m)$, where $m$ denotes the network width, which is in sharp contrast to the $\mathcal{O}(1)$ weight motion allowed by the Neural Tangent Kernel (NTK); h…
The study of Neural Tangent Kernels (NTKs) has provided much needed insight into convergence and generalization properties of neural networks in the over-parametrized (wide) limit by approximating the network using a first-order Taylor expansion with respect to its weights in the neighborhood of their initialization values. This allows neural network training to be analyzed from the perspective of reproducing kernel Hilbert spaces (RKHS), which is informative in the over-parametrized regime, but a poor approximation for narrower networks as the weights change more during training. Our goal is …
Deep neural networks are often trained in the over-parametrized regime (i.e. with far more parameters than training examples), and understanding why the training converges to solutions that generalize remains an open problem. Several studies have highlighted the fact that the training procedure, i.e. mini-batch Stochastic Gradient Descent (SGD) leads to solutions that have specific properties in the loss landscape. However, even with plain Gradient Descent (GD) the solutions found in the over-parametrized regime are pretty good and this phenomenon is poorly understood. We propose an analysis o…
Large neural network models have been successful in learning functions of importance in many branches of science, including physics, chemistry and biology. Recent theoretical work has shown explicit learning bounds for wide networks and kernel methods on some simple classes of functions, but not on more complex functions which arise in practice. We extend these techniques to provide learning bounds for analytic functions on the sphere for any kernel method or equivalent infinitely-wide network with the corresponding activation function trained with SGD. We show that a wide, one-hidden layer Re…
Optimal Rates for Generalization of Gradient Descent for Deep ReLU Classification
School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）；Department of Mathematics, The University of Hong Kong（香港大学数学系）；School of mathematics and statistics, University of Sydney（悉尼大学数学与统计学学院）
Characterization of local minima draws much attention in theoretical studies of deep learning. In this study, we investigate the distribution of parameters in an over-parametrized finite neural network trained by ridge regularized empirical square risk minimization (RERM). We develop a new theory of ridgelet transform, a wavelet-like integral transform that provides a powerful and general framework for the theoretical study of neural networks involving not only the ReLU but general activation functions. We show that the distribution of the parameters converges to a spectrum of the ridgelet tra…
For small training set sizes $P$, the generalization error of wide neural networks is well-approximated by the error of an infinite width neural network (NN), either in the kernel or mean-field/feature-learning regime. However, after a critical sample size $P^*$, we empirically find the finite-width network generalization becomes worse than that of the infinite width network. In this work, we empirically study the transition from infinite-width behavior to this variance limited regime as a function of sample size $P$ and network width $N$. We find that finite-size effects can become relevant f…
Optimal Rates for Generalization of Gradient Descent Methods with Deep Neural Networks
Mathematical Institute for Machine Learning and Data Science, KU Eichstätt-Ingolstadt（机器学习与数据科学数学研究所，埃施特哈特-英戈尔施塔特大学）；Department of Computer Science, RPTU Kaiserslautern-Landau（计算机科学系，凯撒斯劳滕-兰道大学）；Department of Mathematics, University of Hong Kong（数学系，香港大学）；School of Mathematics and Statistics, University of Sydney（数学与统计学学院，悉尼大学）
Purdue University（普渡大学）；The Ohio State University（俄亥俄州立大学）
In this paper, we study the generalization performance of min $\ell_2$-norm overfitting solutions for the neural tangent kernel (NTK) model of a two-layer neural network with ReLU activation that has no bias term. We show that, depending on the ground-truth function, the test error of overfitted NTK models exhibits characteristics that are different from the "double-descent" of other overparameterized linear models with simple Fourier or Gaussian features. Specifically, for a class of learnable functions, we provide a new upper bound of the generalization error that approaches a small limiting…
Better NTK Conditioning: A Free Lunch from (ReLU) Nonlinear Activation in Wide Neural Networks
Elmore Family School of Electrical and Computer Engineering（埃尔莫尔电气与计算机工程学院）；Purdue University（普渡大学）；Department of Physics and Astronomy（物理与天文学系）；Halıcıoğlu Data Science Institute（哈利奇奥格卢数据科学研究所）；University of California San Diego（加州大学圣地亚哥分校）；Rosen Center For Advanced Computing（罗森先进计算中心）
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）
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We study the training and generalization of deep neural networks (DNNs) in the over-parameterized regime, where the network width (i.e., number of hidden nodes per layer) is much larger than the number of training data points. We show that, the expected $0$-$1$ loss of a wide enough ReLU network trained with stochastic gradient descent (SGD) and random initialization can be bounded by the training loss of a random feature model induced by the network gradient at initialization, which we call a neural tangent random feature (NTRF) model. For data distributions that can be classified by NTRF mod…
Stanford University（斯坦福大学）；Princeton University（普林斯顿大学）；University of Texas at Austin（得克萨斯大学奥斯汀分校）
Recent works have shown that on sufficiently over-parametrized neural nets, gradient descent with relatively large initialization optimizes a prediction function in the RKHS of the Neural Tangent Kernel (NTK). This analysis leads to global convergence results but does not work when there is a standard $\ell_2$ regularizer, which is useful to have in practice. We show that sample efficiency can indeed depend on the presence of the regularizer: we construct a simple distribution in d dimensions which the optimal regularized neural net learns with $O(d)$ samples but the NTK requires $Ω(d^2)$ samp…
Optimization dependent generalization bound for ReLU networks based on sensitivity in the tangent bundle
Institute for Computer Science and Control (SZTAKI)（计算机科学与控制研究所（SZTAKI））；Eötvös Loránd University（厄特沃什·罗兰大学）；CNRS CRIStAL（法国国家科学研究中心信号与自动化信息处理实验室）；Université de Lille（里尔大学）；Széchenyi István University（塞切尼·伊什特万大学）
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 Chicago（芝加哥大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Steklov Mathematical Institute（斯捷克洛夫数学研究所）
We continue a long line of research aimed at proving convergence of depth 2 neural networks, trained via gradient descent, to a global minimum. Like in many previous works, our model has the following features: regression with quadratic loss function, fully connected feedforward architecture, RelU activations, Gaussian data instances and network initialization, adversarial labels. It is more general in the sense that we allow both layers to be trained simultaneously and at {\em different} rates. Our results improve on state-of-the-art [Oymak Soltanolkotabi 20] (training the first layer only) a…
Duke University（杜克大学）；Princeton University（普林斯顿大学）；Stanford University（斯坦福大学）
Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an $l$-th order tensor in $(R^d)^{\otimes l}$ of rank $r$ (where $r\ll d$), can variants of gradient descent find a rank $m$ decomposition where $m > r$? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least $m = Ω(d^{l-1})$, while a variant of gradient descent can fi…
Microsoft Research AI（微软研究院AI）；Stanford University（斯坦福大学）；Princeton University（普林斯顿大学）；UT-Austin（德克萨斯大学奥斯汀分校）；University of Washington（华盛顿大学）；Harvard University（哈佛大学）
Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely unsettled. In this work, we prove why stochastic gradient descent (SGD) can find $\textit{global minima}$ on the training objective of DNNs in $\textit{polynomial time}$. We only make two assumptions: the inputs are non-degenerate and the network is over-parameterized. The latter…
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 …
When equipped with efficient optimization algorithms, the over-parameterized neural networks have demonstrated high level of performance even though the loss function is non-convex and non-smooth. While many works have been focusing on understanding the loss dynamics by training neural networks with the gradient descent (GD), in this work, we consider a broad class of optimization algorithms that are commonly used in practice. For example, we show from a dynamical system perspective that the Heavy Ball (HB) method can converge to global minimum on mean squared error (MSE) at a linear rate (sim…
School of Computer Science（计算机学院）；Tel Aviv University（特拉维夫大学）；Microsoft Research AI（微软研究院人工智能部门）
{\em Hypernetworks} are architectures that produce the weights of a task-specific {\em primary network}. A notable application of hypernetworks in the recent literature involves learning to output functional representations. In these scenarios, the hypernetwork learns a representation corresponding to the weights of a shallow MLP, which typically encodes shape or image information. While such representations have seen considerable success in practice, they remain lacking in the theoretical guarantees in the wide regime of the standard architectures. In this work, we study wide over-parameteriz…
University of Torino（都灵大学）；Collegio Carlo Alberto（卡洛·阿尔贝托学院）；Bocconi University（博科尼大学）；Cogent Labs（Cogent实验室）
There is a recent and growing literature on large-width asymptotic properties of Gaussian neural networks (NNs), namely NNs whose weights are initialized as Gaussian distributions. Two popular problems are: i) the study of the large-width distributions of NNs, which characterizes the infinitely wide limit of a rescaled NN in terms of a Gaussian stochastic process; ii) the study of the large-width training dynamics of NNs, which characterizes the infinitely wide dynamics in terms of a deterministic kernel, referred to as the neural tangent kernel (NTK), and shows that, for a sufficiently large …
Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
We analyze architectural features of Deep Neural Networks (DNNs) using the so-called Neural Tangent Kernel (NTK), which describes the training and generalization of DNNs in the infinite-width setting. In this setting, we show that for fully-connected DNNs, as the depth grows, two regimes appear: "order", where the (scaled) NTK converges to a constant, and "chaos", where it converges to a Kronecker delta. Extreme order slows down training while extreme chaos hinders generalization. Using the scaled ReLU as a nonlinearity, we end up in the ordered regime. In contrast, Layer Normalization brings …
Tel Aviv University（特拉维夫大学）；New York University（纽约大学）
A major factor in the success of deep neural networks is the use of sophisticated architectures rather than the classical multilayer perceptron (MLP). Residual networks (ResNets) stand out among these powerful modern architectures. Previous works focused on the optimization advantages of deep ResNets over deep MLPs. In this paper, we show another distinction between the two models, namely, a tendency of ResNets to promote smoother interpolations than MLPs. We analyze this phenomenon via the neural tangent kernel (NTK) approach. First, we compute the NTK for a considered ResNet model and prove …
Neural Tangent Kernel (NTK) is widely used to analyze overparametrized neural networks due to the famous result by Jacot et al. (2018): in the infinite-width limit, the NTK is deterministic and constant during training. However, this result cannot explain the behavior of deep networks, since it generally does not hold if depth and width tend to infinity simultaneously. In this paper, we study the NTK of fully-connected ReLU networks with depth comparable to width. We prove that the NTK properties depend significantly on the depth-to-width ratio and the distribution of parameters at initializat…
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…
Tsinghua University（清华大学）；Beijing Academy of Artificial Intelligence（北京人工智能研究院）
We perform a study on the generalization ability of the wide two-layer ReLU neural network on $\mathbb{R}$. We first establish some spectral properties of the neural tangent kernel (NTK): $a)$ $K_{d}$, the NTK defined on $\mathbb{R}^{d}$, is positive definite; $b)$ $λ_{i}(K_{1})$, the $i$-th largest eigenvalue of $K_{1}$, is proportional to $i^{-2}$. We then show that: $i)$ when the width $m\rightarrow\infty$, the neural network kernel (NNK) uniformly converges to the NTK; $ii)$ the minimax rate of regression over the RKHS associated to $K_{1}$ is $n^{-2/3}$; $iii)$ if one adopts the early sto…
The generalization mystery of overparametrized deep nets has motivated efforts to understand how gradient descent (GD) converges to low-loss solutions that generalize well. Real-life neural networks are initialized from small random values and trained with cross-entropy loss for classification (unlike the "lazy" or "NTK" regime of training where analysis was more successful), and a recent sequence of results (Lyu and Li, 2020; Chizat and Bach, 2020; Ji and Telgarsky, 2020) provide theoretical evidence that GD may converge to the "max-margin" solution with zero loss, which presumably generalize…
Deep residual network architectures have been shown to achieve superior accuracy over classical feed-forward networks, yet their success is still not fully understood. Focusing on massively over-parameterized, fully connected residual networks with ReLU activation through their respective neural tangent kernels (ResNTK), we provide here a spectral analysis of these kernels. Specifically, we show that, much like NTK for fully connected networks (FC-NTK), for input distributed uniformly on the hypersphere $\mathbb{S}^{d-1}$, the eigenfunctions of ResNTK are the spherical harmonics and the eigenv…
