Kernel and Rich Regimes in Overparametrized Models
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Microsoft Research（微软研究院）；Technion（以色列理工学院）
Blake Woodworth, Suriya Gunasekar, Jason D. Lee, Edward Moroshko, Pedro Savarese, Itay Golan, Daniel Soudry, Nathan Srebro
备注：
This updates and significantly extends a previous article (arXiv:1906.05827), Sections 6 and 7 are the most major additions. 31 pages. arXiv admin note: text overlap with arXiv:1906.05827
A recent line of work studies overparametrized neural networks in the "kernel regime," i.e. when the network behaves during training as a kernelized linear predictor, and thus training with gradient descent has the effect of finding the minimum RKHS norm solution. This stands in contrast to other studies which demonstrate how gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. Building on an observation by Chizat and Bach, we show how the scale of the initialization controls the transition between the "kernel" (aka lazy) and "rich" (aka active) regimes and affects generalization properties in multilayer homogeneous models. We also highlight an interesting role for the width of a model in the case that the predictor is not identically zero at initialization. We provide a complete and detailed analysis for a family of simple depth-$D$ models that already exhibit an interesting and meaningful transition between the kernel and rich regimes, and we also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
A recent line of work studies overparametrized neural networks in the "kernel regime," i.e. when the network behaves during training as a kernelized linear predictor, and thus training with gradient descent has the effect of finding the minimum RKHS norm solution. This stands in contrast to other studies which demonstrate how gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. Building on an observation by Chizat and Bach, we show how the scale of the initialization controls the transition between the "kernel" (aka lazy) and "rich" …
There has been a recent surge of interest in understanding the convergence of gradient descent (GD) and stochastic gradient descent (SGD) in overparameterized neural networks. Most previous works assume that the training data is provided a priori in a batch, while less attention has been paid to the important setting where the training data arrives in a stream. In this paper, we study the streaming data setup and show that with overparamterization and random initialization, the prediction error of two-layer neural networks under one-pass SGD converges in expectation. The convergence rate depen…
Deep learning has achieved many breakthroughs in modern classification tasks. Numerous architectures have been proposed for different data structures but when it comes to the loss function, the cross-entropy loss is the predominant choice. Recently, several alternative losses have seen revived interests for deep classifiers. In particular, empirical evidence seems to promote square loss but a theoretical justification is still lacking. In this work, we contribute to the theoretical understanding of square loss in classification by systematically investigating how it performs for overparametriz…
Understanding the implicit bias of Stochastic Gradient Descent (SGD) is one of the key challenges in deep learning, especially for overparametrized models, where the local minimizers of the loss function $L$ can form a manifold. Intuitively, with a sufficiently small learning rate $η$, SGD tracks Gradient Descent (GD) until it gets close to such manifold, where the gradient noise prevents further convergence. In such a regime, Blanc et al. (2020) proved that SGD with label noise locally decreases a regularizer-like term, the sharpness of loss, $\mathrm{tr}[\nabla^2 L]$. The current paper gives…
Purdue University（普渡大学）；Hong Kong University of Science and Technology（香港科技大学）；East China Normal University（华东师范大学）
Overparametrized neural networks trained by gradient descent (GD) can provably overfit any training data. However, the generalization guarantee may not hold for noisy data. From a nonparametric perspective, this paper studies how well overparametrized neural networks can recover the true target function in the presence of random noises. We establish a lower bound on the $L_2$ estimation error with respect to the GD iterations, which is away from zero without a delicate scheme of early stopping. In turn, through a comprehensive analysis of $\ell_2$-regularized GD trajectories, we prove that for…
Modern neural networks are often operated in a strongly overparametrized regime: they comprise so many parameters that they can interpolate the training set, even if actual labels are replaced by purely random ones. Despite this, they achieve good prediction error on unseen data: interpolating the training set does not lead to a large generalization error. Further, overparametrization appears to be beneficial in that it simplifies the optimization landscape. Here we study these phenomena in the context of two-layers neural networks in the neural tangent (NT) regime. We consider a simple data m…
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Center for Data Science, New York University（纽约大学数据科学中心）
Energy-based models (EBMs) are a simple yet powerful framework for generative modeling. They are based on a trainable energy function which defines an associated Gibbs measure, and they can be trained and sampled from via well-established statistical tools, such as MCMC. Neural networks may be used as energy function approximators, providing both a rich class of expressive models as well as a flexible device to incorporate data structure. In this work we focus on shallow neural networks. Building from the incipient theory of overparametrized neural networks, we show that models trained in the …
Peking University（北京大学）；Massachusetts Institute of Technology（麻省理工学院）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Princeton University（普林斯顿大学）
Neural networks are vulnerable to adversarial examples, i.e. inputs that are imperceptibly perturbed from natural data and yet incorrectly classified by the network. Adversarial training, a heuristic form of robust optimization that alternates between minimization and maximization steps, has proven to be among the most successful methods to train networks to be robust against a pre-defined family of perturbations. This paper provides a partial answer to the success of adversarial training, by showing that it converges to a network where the surrogate loss with respect to the the attack algorit…
University of California, Berkeley（加州大学伯克利分校）；Stanford University（斯坦福大学）
Despite their many appealing properties, kernel methods are heavily affected by the curse of dimensionality. For instance, in the case of inner product kernels in $\mathbb{R}^d$, the Reproducing Kernel Hilbert Space (RKHS) norm is often very large for functions that depend strongly on a small subset of directions (ridge functions). Correspondingly, such functions are difficult to learn using kernel methods. This observation has motivated the study of generalizations of kernel methods, whereby the RKHS norm -- which is equivalent to a weighted $\ell_2$ norm -- is replaced by a weighted function…
Kernel learning methods are among the most effective learning methods and have been vigorously studied in the past decades. However, when tackling with complicated tasks, classical kernel methods are not flexible or "rich" enough to describe the data and hence could not yield satisfactory performance. In this paper, via Random Fourier Features (RFF), we successfully incorporate the deep architecture into kernel learning, which significantly boosts the flexibility and richness of kernel machines while keeps kernels' advantage of pairwise handling small data. With RFF, we could establish a deep …
We study the expressive power of kernel methods and the algorithmic feasibility of multiple kernel learning for a special rich class of kernels. Specifically, we define \emph{Euclidean kernels}, a diverse class that includes most, if not all, families of kernels studied in literature such as polynomial kernels and radial basis functions. We then describe the geometric and spectral structure of this family of kernels over the hypercube (and to some extent for any compact domain). Our structural results allow us to prove meaningful limitations on the expressive power of the class as well as deri…
The University of Birmingham（伯明翰大学）；School of Computer Science（计算机学院）
Parameterized state space models in the form of recurrent networks are often used in machine learning to learn from data streams exhibiting temporal dependencies. To break the black box nature of such models it is important to understand the dynamical features of the input driving time series that are formed in the state space. We propose a framework for rigorous analysis of such state representations in vanishing memory state space models such as echo state networks (ESN). In particular, we consider the state space a temporal feature space and the readout mapping from the state space a kernel…
Local Kernel Renormalization as a mechanism for feature learning in overparametrized Convolutional Neural Networks
Università degli Studi di Parma（帕尔马大学）；INFN（意大利国家核物理研究所）；Politecnico di Torino（都灵理工大学）；Bocconi University（博科尼大学）；Istituto dei Materiali per l’Elettronica ed il Magnetismo (IMEM-CNR)（意大利国家研究委员会电子与磁学材料研究所）
Carnegie Mellon University（卡内基梅隆大学）；Google Strategic Technologies（谷歌战略技术部门）；Georgia Institute of Technology（佐治亚理工学院）
Kernel methods have great promise for learning rich statistical representations of large modern datasets. However, compared to neural networks, kernel methods have been perceived as lacking in scalability and flexibility. We introduce a family of fast, flexible, lightly parametrized and general purpose kernel learning methods, derived from Fastfood basis function expansions. We provide mechanisms to learn the properties of groups of spectral frequencies in these expansions, which require only O(mlogd) time and O(m) memory, for m basis functions and d input dimensions. We show that the proposed…
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）；Carnegie Mellon University（卡内基梅隆大学）
Recent works have cast some light on the mystery of why deep nets fit any data and generalize despite being very overparametrized. This paper analyzes training and generalization for a simple 2-layer ReLU net with random initialization, and provides the following improvements over recent works: (i) Using a tighter characterization of training speed than recent papers, an explanation for why training a neural net with random labels leads to slower training, as originally observed in [Zhang et al. ICLR'17]. (ii) Generalization bound independent of network size, using a data-dependent complexity …
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Center for Data Science, New York University（纽约大学数据科学中心）；Microsoft（微软）
Gaussian processes are flexible function approximators, with inductive biases controlled by a covariance kernel. Learning the kernel is the key to representation learning and strong predictive performance. In this paper, we develop functional kernel learning (FKL) to directly infer functional posteriors over kernels. In particular, we place a transformed Gaussian process over a spectral density, to induce a non-parametric distribution over kernel functions. The resulting approach enables learning of rich representations, with support for any stationary kernel, uncertainty over the values of th…
Neural networks in the lazy training regime converge to kernel machines. Can neural networks in the rich feature learning regime learn a kernel machine with a data-dependent kernel? We demonstrate that this can indeed happen due to a phenomenon we term silent alignment, which requires that the tangent kernel of a network evolves in eigenstructure while small and before the loss appreciably decreases, and grows only in overall scale afterwards. We show that such an effect takes place in homogenous neural networks with small initialization and whitened data. We provide an analytical treatment of…
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…
Spectral alignment of stochastic gradient descent for high-dimensional classification tasks
Courant Institute, New York University（纽约大学Courant研究所）；Department of Mathematics, Northwestern University（西北大学数学系）；Department of Statistics and Data Science, University of Pennsylvania（宾夕法尼亚大学统计与数据科学系）；Department of Statistics and Actuarial Science, Department of Applied Mathematics, and Cheriton School of Computer Science, University of Waterloo（滑铁卢大学统计与精算系、应用数学系及Cheriton计算机科学学院）
Universidad de Chile（智利大学）；Advanced Mining Technology Center, Universidad de Chile（智利大学先进采矿技术中心）；Khalifa University（哈利法大学）；ADIA Lab（阿布扎比投资局实验室）；Universidad Adolfo Ibañez（阿道夫·伊巴涅斯大学）；Ca’ Foscari University of Venice（威尼斯大学）
A number of applications in engineering, social sciences, physics, and biology involve inference over networks. In this context, graph signals are widely encountered as descriptors of vertex attributes or features in graph-structured data. Estimating such signals in all vertices given noisy observations of their values on a subset of vertices has been extensively analyzed in the literature of signal processing on graphs (SPoG). This paper advocates kernel regression as a framework generalizing popular SPoG modeling and reconstruction and expanding their capabilities. Formulating signal reconst…
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…
Shanghai University of Finance and Economics（上海财经大学）
In statistical learning, identifying underlying structures of true target functions based on observed data plays a crucial role to facilitate subsequent modeling and analysis. Unlike most of those existing methods that focus on some specific settings under certain model assumptions, this paper proposes a general and novel framework for recovering true structures of target functions by using unstructured M-regression in a reproducing kernel Hilbert space (RKHS). The proposed framework is inspired by the fact that gradient functions can be employed as a valid tool to learn underlying structures,…
IST Austria（奥地利科学技术研究所）；TU München（慕尼黑工业大学）；University of Salzburg（萨尔茨堡大学）
Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and textur…
We show that the standard stochastic gradient decent (SGD) algorithm is guaranteed to learn, in polynomial time, a function that is competitive with the best function in the conjugate kernel space of the network, as defined in Daniely, Frostig and Singer. The result holds for log-depth networks from a rich family of architectures. To the best of our knowledge, it is the first polynomial-time guarantee for the standard neural network learning algorithm for networks of depth more that two. As corollaries, it follows that for neural networks of any depth between $2$ and $\log(n)$, SGD is guarante…
NTT Network Service Systems Laboratories, NTT Corporation（NTT网络服务系统实验室，NTT公司）；Graduate School of Science and Technology, Keio University（庆应义塾大学科学与技术研究生院）；Center for Data Science, Ehime University（爱媛大学数据科学中心）；Center for Advanced Intelligence Project, RIKEN（理化学研究所高级智能项目中心）；Faculty of Science and Technology, Keio University（庆应义塾大学科学与技术学部）
Kernel methods have been among the most popular techniques in machine learning, where learning tasks are solved using the property of reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis framework with reproducing kernel Hilbert $C^*$-module (RKHM) and kernel mean embedding (KME) in RKHM. Since RKHM contains richer information than RKHS or vector-valued RKHS (vvRKHS), analysis with RKHM enables us to capture and extract structural properties in such as functional data. We show a branch of theories for RKHM to apply to data analysis, including the representer…
Permutation-valued features arise in a variety of applications, either in a direct way when preferences are elicited over a collection of items, or an indirect way in which numerical ratings are converted to a ranking. To date, there has been relatively limited study of regression, classification, and testing problems based on permutation-valued features, as opposed to permutation-valued responses. This paper studies the use of reproducing kernel Hilbert space methods for learning from permutation-valued features. These methods embed the rankings into an implicitly defined function space, and …
Adaptive kernel predictors from feature-learning infinite limits of neural networks
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保罗森工程与应用科学学校）；Center for Brain Sciences（脑科学中心）；Kempner Institute（凯普纳研究所）
University at Buffalo（布法罗大学）；Stanford University（斯坦福大学）；Purdue University（普渡大学）
This paper establishes a kernel-based framework for reconstructing data on manifolds, tailored to fit the dynamic-(d)MRI-data recovery problem. The proposed methodology exploits simple tangent-space geometries of manifolds in reproducing kernel Hilbert spaces and follows classical kernel-approximation arguments to form the data-recovery task as a bi-linear inverse problem. Departing from mainstream approaches, the proposed methodology uses no training data, employs no graph Laplacian matrix to penalize the optimization task, uses no costly (kernel) pre-imaging step to map feature points back t…
Spectral Analysis of Molecular Features: When Richer Features Do Not Guarantee Better Generalization
School of Computational Science and Engineering, McMaster University, Canada（麦 master 大学计算科学与工程学院）；Department of Chemistry and Chemical Biology, McMaster University, Canada（麦 master 大学化学与生物化学系）；Department of Mathematics and Computer Science, University of Basel, Switzerland（巴塞尔大学数学与计算机科学系）；Brockhouse Institute for Materials Research, McMaster University, Canada（麦 master 大学材料研究布罗克豪斯研究所）
We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is two-fold. First, we show, using the Lévy-Khintchine formula, that any shift-invariant kernel given by a bounded, continuous and symmetric probability density function (pdf) of an infinitely divisible distribution on $\mathbb{R}^d$ is characteristic. We also present some closure property of such characteristi…
This paper presents a kernel-based adaptive filter that is applied for the digital domain self-interference cancellation (SIC) in a transceiver operating in full-duplex (FD) mode. In FD, the benefit of simultaneous transmission and receiving of signals comes at the price of strong self-interference (SI). In this work, we are primarily interested in suppressing the SI using an adaptive filter namely adaptive projected subgradient method (APSM) in a reproducing kernel Hilbert space (RKHS) of functions. Using the projection concept as a powerful tool, APSM is used to model and consequently remove…
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…
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…
The kernel exponential family is a rich class of distributions, which can be fit efficiently and with statistical guarantees by score matching. Being required to choose a priori a simple kernel such as the Gaussian, however, limits its practical applicability. We provide a scheme for learning a kernel parameterized by a deep network, which can find complex location-dependent local features of the data geometry. This gives a very rich class of density models, capable of fitting complex structures on moderate-dimensional problems. Compared to deep density models fit via maximum likelihood, our a…
OmniGraph, a novel representation to support a range of NLP classification tasks, integrates lexical items, syntactic dependencies and frame semantic parses into graphs. Feature engineering is folded into the learning through convolution graph kernel learning to explore different extents of the graph. A high-dimensional space of features includes individual nodes as well as complex subgraphs. In experiments on a text-forecasting problem that predicts stock price change from news for company mentions, OmniGraph beats several benchmarks based on bag-of-words, syntactic dependencies, and semantic…
The ability to automatically detect fraudulent escrow websites is important in order to alleviate online auction fraud. Despite research on related topics, fake escrow website categorization has received little attention. In this study we evaluated the effectiveness of various features and techniques for detecting fake escrow websites. Our analysis included a rich set of features extracted from web page text, image, and link information. We also proposed a composite kernel tailored to represent the properties of fake websites, including content duplication and structural attributes. Experiment…
Dynamics of Finite Width Kernel and Prediction Fluctuations in Mean Field Neural Networks
John Paulson School of Engineering and Applied Sciences（约翰·保尔森工程与应用科学学院）；Center for Brain Science（脑科学中心）；Kempner Institute for the Study of Natural & Artificial Intelligence（自然与人工智能研究肯普纳研究所）；Harvard University（哈佛大学）
We develop a general duality between neural networks and compositional kernels, striving towards a better understanding of deep learning. We show that initial representations generated by common random initializations are sufficiently rich to express all functions in the dual kernel space. Hence, though the training objective is hard to optimize in the worst case, the initial weights form a good starting point for optimization. Our dual view also reveals a pragmatic and aesthetic perspective of neural networks and underscores their expressive power.
