Sharp regression risk for learning low-degree polynomials

Determine sharp convergence rates for the population regression risk when learning low-degree polynomials on high-dimensional spheres, specifically degree-ℓ0 spherical polynomials on the unit sphere S^{d−1}, so that the rates match the minimax order associated with the effective rank of the target class.

Background

The paper reviews prior work on learning low-degree polynomials on the sphere using over-parameterized neural networks and kernel methods, noting that many results either do not provide sharp (minimax) rates or require restrictive conditions such as infinite network width. For example, some methods yield rates on the order of sqrt(d^{ℓ0}/n) or only show vanishing risk without a convergence rate.

The authors position this as a core unresolved issue motivating their study: establishing sharp risk rates for learning degree-ℓ0 polynomials. They later present a two-stage approach with learnable channel selection and gradient descent that attains the minimax-optimal order Θ(d^{ℓ0}/n) under their setting.