  A recent goal in the theory of deep learning is to identify how neural
networks can escape the "lazy training," or Neural Tangent Kernel (NTK) regime,
where the network is coupled with its first order Taylor expansion at
initialization. While the NTK is minimax optimal for learning dense polynomials
(Ghorbani et al, 2021), it cannot learn features, and hence has poor sample
complexity for learning many classes of functions including sparse polynomials.
Recent works have thus aimed to identify settings where gradient based
algorithms provably generalize better than the NTK. One such example is the
"QuadNTK" approach of Bai and Lee (2020), which analyzes the second-order term
in the Taylor expansion. Bai and Lee (2020) show that the second-order term can
learn sparse polynomials efficiently; however, it sacrifices the ability to
learn general dense polynomials.
  In this paper, we analyze how gradient descent on a two-layer neural network
can escape the NTK regime by utilizing a spectral characterization of the NTK
(Montanari and Zhong, 2020) and building on the QuadNTK approach. We first
expand upon the spectral analysis to identify "good" directions in parameter
space in which we can move without harming generalization. Next, we show that a
wide two-layer neural network can jointly use the NTK and QuadNTK to fit target
functions consisting of a dense low-degree term and a sparse high-degree term
-- something neither the NTK nor the QuadNTK can do on their own. Finally, we
construct a regularizer which encourages our parameter vector to move in the
"good" directions, and show that gradient descent on the regularized loss will
converge to a global minimizer, which also has low test error. This yields an
end to end convergence and generalization guarantee with provable sample
complexity improvement over both the NTK and QuadNTK on their own.
