On the Pinsker bound of inner product kernel regression in large dimensions

Building on recent studies of large-dimensional kernel regression, particularly those involving inner product kernels on the sphere ${\mathbb{S}}^d$, we investigate the Pinsker bound for inner product kernel regression in such settings. Specifically, we address the scenario where the sample size $n$ is given by $\alpha d^{\gamma }(1+o_d(1))$ for some $\alpha ,\gamma >0$. We have determined the exact minimax risk for kernel regression in this setting, not only identifying the minimax rate but also the exact constant, known as the Pinsker constant, associated with the excess risk.