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On the Inductive Bias of Neural Tangent Kernels

Neural Information Processing Systems

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 properties of functions with finite norm, including stability to image deformations in the case of convolutional networks, and compare to other known kernels for similar architectures. Papers published at the Neural Information Processing Systems Conference.


Kernel and Deep Regimes in Overparametrized Models

arXiv.org Machine Learning

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 ``deep'' (aka active) regimes and affects generalization properties in multilayer homogeneous models. We provide a complete and detailed analysis for a simple two-layer model that already exhibits an interesting and meaningful transition between the kernel and deep regimes, and we demonstrate the transition for more complex matrix factorization models.


A Communication-Efficient Distributed Algorithm for Kernel Principal Component Analysis

arXiv.org Machine Learning

Principal Component Analysis (PCA) is a fundamental technology in machine learning. Nowadays many high-dimension large datasets are acquired in a distributed manner, which precludes the use of centralized PCA due to the high communication cost and privacy risk. Thus, many distributed PCA algorithms are proposed, most of which, however, focus on linear cases. To efficiently extract non-linear features, this brief proposes a communication-efficient distributed kernel PCA algorithm, where linear and RBF kernels are applied. The key is to estimate the global empirical kernel matrix from the eigenvectors of local kernel matrices. The approximate error of the estimators is theoretically analyzed for both linear and RBF kernels. The result suggests that when eigenvalues decay fast, which is common for RBF kernels, the proposed algorithm gives high quality results with low communication cost. Results of simulation experiments verify our theory analysis and experiments on GSE2187 dataset show the effectiveness of the proposed algorithm.


Kernel Methods and Multi-layer Perceptrons Learn Linear Models in High Dimensions

arXiv.org Machine Learning

Empirical observation of high dimensional phenomena, such as the double descent behaviour, has attracted a lot of interest in understanding classical techniques such as kernel methods, and their implications to explain generalization properties of neural networks. Many recent works analyze such models in a certain high-dimensional regime where the covariates are independent and the number of samples and the number of covariates grow at a fixed ratio (i.e. proportional asymptotics). In this work we show that for a large class of kernels, including the neural tangent kernel of fully connected networks, kernel methods can only perform as well as linear models in this regime. More surprisingly, when the data is generated by a kernel model where the relationship between input and the response could be very nonlinear, we show that linear models are in fact optimal, i.e. linear models achieve the minimum risk among all models, linear or nonlinear. These results suggest that more complex models for the data other than independent features are needed for high-dimensional analysis.


Classifying high-dimensional Gaussian mixtures: Where kernel methods fail and neural networks succeed

arXiv.org Machine Learning

Explaining the success of deep neural networks in many areas of machine learning remains a key challenge for learning theory. A series of recent theoretical works made progress towards this goal by proving trainability of two-layer neural networks (2LNN) with gradient-based methods [1-6]. These results are based on the observation that strongly over-parameterised 2LNN can achieve good performance even if their first-layer weights remain almost constant throughout training. This is the case if the initial weights are chosen with a particular scaling, which was dubbed the "lazy regime" by Chizat et al. [7]. This behaviour is to be contrasted with the "feature learning regime", where the weights of the first layer move significantly during training. Going a step further, simply fixing the first-layer weights of a 2LNN at their initial values yields the well-known random features model of Rahimi & Recht [8, 9], and can be seen as an approximation of kernel learning [10]. Recent empirical studies showed that on some benchmark data sets in computer vision, kernels derived from neural networks achieve comparable performance to neural networks [11-16]. These results raise the question of whether neural networks only learn successfully if random features can also learn successfully, and have led to a renewed interest in the exact conditions under which neural networks trained with gradient descent achieve a better performance than random features [17-20].