A Function-Space Dichotomy for Compositional Learning: Exponential Sub-Optimality of the Neural Tangent Kernel

A persistent empirical observation is that trained neural networks outperform their neural tangent kernel (NTK) limit on tasks with compositional structure, yet a quantitative account of when and by how much has been lacking. Working on the unit circle, we give such an account through a dichotomy between two complexity measures of the target: its Fourier complexity, which controls NTK kernel regression, and its architectural complexity, which controls learning over depth-L, width-w ReLU networks with the variation norm of the weights bounded by R. We first characterize the minimax rate of the architecture class CL,w,R, pinning it down up to a single factor of L: between Omega(Lw2R2/n) and tildeO(L2w2R2/n). We then show the NTK estimator sits exponentially above this floor whenever the two complexities decouple: for the depth-L iterated sawtooth, NTK regression needs Omega(4L) samples while the minimax floor is polynomial in L. Numerical experiments confirm the theoretical claims: on bandlimited smooth targets, the NTK is competitive or better, while on the hypercube sparse-parity model, a standard two-layer network beats the NTK by four to six orders of magnitude in test error. The gap is thus a function-space property, a mismatch between the kernel's smoothness bias and the target's compositional structure, rather than a generic kernel-versus-network phenomenon.