What Can ResNet Learn Efficiently, Going Beyond Kernels?

TL;DR

This paper demonstrates that neural networks, specifically ResNet, can efficiently learn certain functions with better test error than kernel methods, due to their hierarchical learning capabilities, without distributional assumptions.

Contribution

It proves neural networks can learn a class of functions more effectively than kernels in a distribution-free setting, highlighting hierarchical learning as a key advantage.

Findings

Neural networks can learn certain functions with much smaller test error than kernels.

Hierarchical learning reduces sample complexity compared to kernel methods.

ResNet has a computational complexity advantage over other learning methods.

Abstract

How can neural networks such as ResNet efficiently learn CIFAR-10 with test accuracy more than 96%, while other methods, especially kernel methods, fall relatively behind? Can we more provide theoretical justifications for this gap? Recently, there is an influential line of work relating neural networks to kernels in the over-parameterized regime, proving they can learn certain concept class that is also learnable by kernels with similar test error. Yet, can neural networks provably learn some concept class BETTER than kernels? We answer this positively in the distribution-free setting. We prove neural networks can efficiently learn a notable class of functions, including those defined by three-layer residual networks with smooth activations, without any distributional assumption. At the same time, we prove there are simple functions in this class such that with the same number of training examples, the test error obtained by neural networks can be MUCH SMALLER than ANY kernel method, including neural tangent kernels (NTK). The main intuition is that multi-layer neural networks can implicitly perform hierarchical learning using different layers, which reduces the sample complexity comparing to "one-shot" learning algorithms such as kernel methods. In a follow-up work [2], this theory of hierarchical learning is further strengthened to incorporate the "backward feature correction" process when training deep networks. In the end, we also prove a computation complexity advantage of ResNet with respect to other learning methods including linear regression over arbitrary feature mappings.