What Can ResNet Learn Efficiently, Going Beyond Kernels?
This addresses a foundational gap in understanding the efficiency and superiority of neural networks over kernel methods for machine learning practitioners and theorists, offering a non-incremental theoretical breakthrough.
The paper tackles the problem of why neural networks like ResNet achieve high test accuracy on CIFAR-10 (over 96%) while kernel methods lag behind, and it provides theoretical justification by proving that neural networks can efficiently learn certain function classes with much smaller test error than any kernel method in a distribution-free setting.
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.