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Fast Neural Kernel Embeddings for General Activations
Insu Han · Amir Zandieh · Jaehoon Lee · Roman Novak · Lechao Xiao · Amin Karbasi

Tue Nov 29 09:00 AM -- 11:00 AM (PST) @ Hall J #806
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 work with general activations. First, we compile and expand the list of activation functions admitting exact dual activation expressions to compute neural kernels. When the exact computation is unknown, we present methods to effectively approximate them. We propose a fast sketching method that approximates any multi-layered Neural Network Gaussian Process (NNGP) kernel and Neural Tangent Kernel (NTK) matrices for a wide range of activation functions, going beyond the commonly analyzed ReLU activation. This is done by showing how to approximate the neural kernels using the truncated Hermite expansion of any desired activation functions. While most prior works require data points on the unit sphere, our methods do not suffer from such limitations and are applicable to any dataset of points in $\mathbb{R}^d$. Furthermore, we provide a subspace embedding for NNGP and NTK matrices with near input-sparsity runtime and near-optimal target dimension which applies to any \emph{homogeneous} dual activation functions with rapidly convergent Taylor expansion. Empirically, with respect to exact convolutional NTK (CNTK) computation, our method achieves $106\times$ speedup for approximate CNTK of a 5-layer Myrtle network on CIFAR-10 dataset.

Author Information

Insu Han (Yale University)
Amir Zandieh (Max-Planck-Institut für Informatik)
Jaehoon Lee (Google Brain)
Roman Novak (Google Brain)
Lechao Xiao (Google Research)

Lechao is a research scientist in the Brain team in Google Research, where he is working on machine learning and deep learning. Prior to Google Brain, he was a Hans Rademacher Instructor of Mathematics at the University of Pennsylvania, where he was working on harmonic analysis. He earned his PhD in mathematics from the University of Illinois at Urbana-Champaign and his BA in pure and applied math from Zhejiang University, Hangzhou, China. Lechao research interests include theory of machine learning and deep learning, optimization, Gaussian process, generalization, etc.

Amin Karbasi (Yale University)

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