When Do Neural Networks Outperform Kernel Methods?

by   Behrooz Ghorbani, et al.

For a certain scaling of the initialization of stochastic gradient descent (SGD), wide neural networks (NN) have been shown to be well approximated by reproducing kernel Hilbert space (RKHS) methods. Recent empirical work showed that, for some classification tasks, RKHS methods can replace NNs without a large loss in performance. On the other hand, two-layers NNs are known to encode richer smoothness classes than RKHS and we know of special examples for which SGD-trained NN provably outperform RKHS. This is true even in the wide network limit, for a different scaling of the initialization. How can we reconcile the above claims? For which tasks do NNs outperform RKHS? If feature vectors are nearly isotropic, RKHS methods suffer from the curse of dimensionality, while NNs can overcome it by learning the best low-dimensional representation. Here we show that this curse of dimensionality becomes milder if the feature vectors display the same low-dimensional structure as the target function, and we precisely characterize this tradeoff. Building on these results, we present a model that can capture in a unified framework both behaviors observed in earlier work. We hypothesize that such a latent low-dimensional structure is present in image classification. We test numerically this hypothesis by showing that specific perturbations of the training distribution degrade the performances of RKHS methods much more significantly than NNs.


page 19

page 22

page 26


Neural Networks Efficiently Learn Low-Dimensional Representations with SGD

We study the problem of training a two-layer neural network (NN) of arbi...

Catapults in SGD: spikes in the training loss and their impact on generalization through feature learning

In this paper, we first present an explanation regarding the common occu...

Weighted Neural Tangent Kernel: A Generalized and Improved Network-Induced Kernel

The Neural Tangent Kernel (NTK) has recently attracted intense study, as...

Neural Networks Optimally Compress the Sawbridge

Neural-network-based compressors have proven to be remarkably effective ...

On One-Bit Quantization

We consider the one-bit quantizer that minimizes the mean squared error ...

Code Repositories


The code for the paper "When do neural networks outperform kernel methods"

view repo

Please sign up or login with your details

Forgot password? Click here to reset