To understand deep learning we need to understand kernel learning

TL;DR

This paper investigates why over-parameterized models like deep neural networks generalize well despite fitting noisy data, highlighting the role of kernel properties and challenging existing theoretical bounds.

Contribution

It demonstrates experimentally that kernel machines can generalize well in overfitted regimes and provides theoretical bounds showing the limitations of current understanding.

Findings

Kernel machines perform well on test data even with noisy labels.

Fitting noisy data with Laplacian kernels requires many epochs, unlike Gaussian kernels.

Generalization is more influenced by kernel properties than optimization processes.

Abstract

Generalization performance of classifiers in deep learning has recently become a subject of intense study. Deep models, typically over-parametrized, tend to fit the training data exactly. Despite this "overfitting", they perform well on test data, a phenomenon not yet fully understood. The first point of our paper is that strong performance of overfitted classifiers is not a unique feature of deep learning. Using six real-world and two synthetic datasets, we establish experimentally that kernel machines trained to have zero classification or near zero regression error perform very well on test data, even when the labels are corrupted with a high level of noise. We proceed to give a lower bound on the norm of zero loss solutions for smooth kernels, showing that they increase nearly exponentially with data size. We point out that this is difficult to reconcile with the existing generalization bounds. Moreover, none of the bounds produce non-trivial results for interpolating solutions. Second, we show experimentally that (non-smooth) Laplacian kernels easily fit random labels, a finding that parallels results for ReLU neural networks. In contrast, fitting noisy data requires many more epochs for smooth Gaussian kernels. Similar performance of overfitted Laplacian and Gaussian classifiers on test, suggests that generalization is tied to the properties of the kernel function rather than the optimization process. Certain key phenomena of deep learning are manifested similarly in kernel methods in the modern "overfitted" regime. The combination of the experimental and theoretical results presented in this paper indicates a need for new theoretical ideas for understanding properties of classical kernel methods. We argue that progress on understanding deep learning will be difficult until more tractable "shallow" kernel methods are better understood.