Beyond NTK: A Mean-Field Analysis of Neural Networks with Polynomial Width, Samples, and Time
summary
Claim: gradient-flow mean-field analysis proves unmodified SGD learns single-index targets with quartic activations using polynomial width, iterations, and d^{3.1} samples, improving on kernel’s d^4 barrier. Technique: reduce population dynamics on the sphere to a 1D moment system via spherical harmonics, analyze stationary points and power-method growth, then tightly couple finite-width/finite-sample dynamics (degree-8 concentration, discrete-time with small step). Result is a constructive polynomial-time guarantee for uniform-sphere inputs; restrictions and loosened exponents remain.
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