Beyond NTK with Vanilla Gradient Descent: A Mean-Field Analysis of Neural Networks with Polynomial Width, Samples, and Time

Understand

Despite recent theoretical progress on the non-convex optimization of two-layer neural networks, it is still an open question whether gradient descent on neural networks without unnatural modifications can achieve better sample complexity than kernel methods.

• This paper provides a clean mean-field analysis of projected gradient flow on polynomial-width two-layer neural networks.
• Different from prior works, our analysis does not require unnatural modifications of the optimization algorithm.
• We prove that with sample size $n = O(d^{3.1})$ where $d$ is the dimension of the inputs, the network trained with projected gradient flow converges in $\text{poly}(d)$ time to a non-trivial error that is not achievable by kernel methods using $n \ll d^4$ samples, hence demonstrating a clear separation between unmodified gradient descent and NTK.