Shallow Neural Networks Learn Low-Degree Spherical Polynomials with Feature Learning by Learnable Channel Attention

We study the problem of learning a low-degree spherical polynomial of degree ${\mathrm{\ell }}_0=\mathrm{\Theta }(1)\ge 1$ defined on the unit sphere in ${\text{RR}}^d$ by training an over-parameterized two-layer neural network (NN) with channel attention in this paper. Our main result is the significantly improved sample complexity for learning such low-degree polynomials. We show that, for any regression risk $\text{eps}\in (0,1)$, a carefully designed two-layer NN with channel attention and finite width trained by the vanilla gradient descent (GD) requires the lowest sample complexity of $n\asymp \mathrm{\Theta }(d^{{\mathrm{\ell }}_0}\mathrm{/}\text{eps})$ with high probability, in contrast with the representative sample complexity Θ\pth{d^{\ell_0} \max\set{\eps^{-2},\log d}}, where $n$ is the training data size. Moreover, such sample complexity is not improvable since the trained network renders a sharp rate of the nonparametric regression risk of the order $\mathrm{\Theta }(d^{{\mathrm{\ell }}_0}\mathrm{/}n)$ with high probability. On the other hand, the minimax optimal rate for the regression risk with a kernel of rank $\mathrm{\Theta }(d^{{\mathrm{\ell }}_0})$ is $\mathrm{\Theta }(d^{{\mathrm{\ell }}_0}\mathrm{/}n)$, so that the rate of the nonparametric regression risk of the network trained by GD is minimax optimal. Training the two-layer NN with channel attention proceeds in two stages: (1) a provable learnable channel selection algorithm, as a learnable harmonic-degree selection process, identifies the ground truth channel number in the target function, ${\mathrm{\ell }}_0$, from $L\ge {\mathrm{\ell }}_0$ channels in the first-layer activation; (2) the second layer is trained by standard GD using the selected channels. To the best of our knowledge, this is the first time a minimax optimal risk bound is obtained by training an over-parameterized but finite-width neural network with feature learning capability to learn low-degree spherical polynomials.