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 functions. This means that both kernels give rise to classes of functions with the same smoothness properties. The two kernels differ for data off the hypersphere, but experiments indicate that when data is properly normalized these differences are not significant. Finally, we provide experiments on real data comparing NTK and the Laplace kernel, along with a larger class ofγ-exponential kernels. We show that these perform almost identically. Our results suggest that much insight about neural networks can be obtained from analysis of the well-known Laplace kernel, which has a simple closed-form.
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 $\mathbb{R}^d$.
Aqemia；McGill University（麦吉尔大学）；Mila；RIKEN（理化学研究所）；The University of Tokyo（东京大学）
In Continual Learning settings, deep neural networks are prone to Catastrophic Forgetting. Orthogonal Gradient Descent was proposed to tackle the challenge. However, no theoretical guarantees have been proven yet. We present a theoretical framework to study Continual Learning algorithms in the Neural Tangent Kernel regime. This framework comprises closed form expression of the model through tasks and proxies for Transfer Learning, generalisation and tasks similarity. In this framework, we prove that OGD is robust to Catastrophic Forgetting then derive the first generalisation bound for SGD and…
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…
Tsinghua University（清华大学）；Renmin University of China（中国人民大学）；Shanghai Jiao Tong University（上海交通大学）
Laplace approximation (LA) and its linearized variant (LLA) enable effortless adaptation of pretrained deep neural networks to Bayesian neural networks. The generalized Gauss-Newton (GGN) approximation is typically introduced to improve their tractability. However, LA and LLA are still confronted with non-trivial inefficiency issues and should rely on Kronecker-factored, diagonal, or even last-layer approximate GGN matrices in practical use. These approximations are likely to harm the fidelity of learning outcomes. To tackle this issue, inspired by the connections between LLA and neural tangen…
Scalable Linearized Laplace Approximation via Surrogate Neural Kernel
Machine Learning Group - Computer Science Department Escuela Politécnica Superior（计算机科学系机器学习小组）；Computer Science Department Escuela Politécnica Superior（计算机科学系）；Universidad Autónoma de Madrid（马德里自治大学）；Universidad Pontificia de Comillas（天主教comillas大学）；Aalborg University Copenhagen（哥本哈根奥胡斯大学）；Centro de Investigación Avanzada en Física Fundamental Universidad Autónoma de Madrid（马德里自治大学先进物理基础研究中心）
Eötvos Loránd University（罗兰大学）；Université catholique de Louvain（鲁汶大学）；Institute for Computer Science and Control（计算机科学与控制研究所）
Feed-forward networks can be interpreted as mappings with linear decision surfaces at the level of the last layer. We investigate how the tangent space of the network can be exploited to refine the decision in case of ReLU (Rectified Linear Unit) activations. We show that a simple Riemannian metric parametrized on the parameters of the network forms a similarity function at least as good as the original network and we suggest a sparse metric to increase the similarity gap.
Korea Advanced Institute of Science and Technology（韩国科学技术院）；Samsung Medical Center（三星医疗中心）；Sungkyunkwan University School of Medicine（成均馆大学医学院）；AITRICS
Explaining generalizations and preventing over-confident predictions are central goals of studies on the loss landscape of neural networks. Flatness, defined as loss invariability on perturbations of a pre-trained solution, is widely accepted as a predictor of generalization in this context. However, the problem that flatness and generalization bounds can be changed arbitrarily according to the scale of a parameter was pointed out, and previous studies partially solved the problem with restrictions: Counter-intuitively, their generalization bounds were still variant for the function-preserving…
Tsinghua University（清华大学）；Peng Cheng Laboratory（鹏城实验室）；Shanghai Jiao Tong University（上海交通大学）；Microsoft Research New England（微软新英格兰研究院）
Learning the principal eigenfunctions of an integral operator defined by a kernel and a data distribution is at the core of many machine learning problems. Traditional nonparametric solutions based on the Nystr{ö}m formula suffer from scalability issues. Recent work has resorted to a parametric approach, i.e., training neural networks to approximate the eigenfunctions. However, the existing method relies on an expensive orthogonalization step and is difficult to implement. We show that these problems can be fixed by using a new series of objective functions that generalizes the EigenGame~\cite…
Quantized neural networks have drawn a lot of attention as they reduce the space and computational complexity during the inference. Moreover, there has been folklore that quantization acts as an implicit regularizer and thus can improve the generalizability of neural networks, yet no existing work formalizes this interesting folklore. In this paper, we take the binary weights in a neural network as random variables under stochastic rounding, and study the distribution propagation over different layers in the neural network. We propose a quasi neural network to approximate the distribution prop…
