Characterizing the minimax rate of nonparametric regression under bounded star-shaped constraints

We quantify the minimax rate for a nonparametric regression model over a star-shaped function class $\mathcal{F}$ with bounded diameter. We obtain a minimax rate of ${{\epsilon }^{\ast }}^2\wedge \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\mathcal{F}{)}^2$ where $${\epsilon }^{\ast }=\sup \{\epsilon \ge 0:n{\epsilon }^2\le \log M_{\mathcal{F}}^{\mathrm{loc}}(\epsilon ,c)\},$$ where $\log M_{\mathcal{F}}^{\mathrm{loc}}(\cdot ,c)$ is the local metric entropy of $\mathcal{F}$, $c$ is some absolute constant scaling down the entropy radius, and our loss function is the squared population $L_2$ distance over our input space $\mathcal{X}$. In contrast to classical works on the topic [cf. Yang and Barron, 1999], our results do not require functions in $\mathcal{F}$ to be uniformly bounded in sup-norm. In fact, we propose a condition that simultaneously generalizes boundedness in sup-norm and the so-called $L$-sub-Gaussian assumption that appears in the prior literature. In addition, we prove that our estimator is adaptive to the true point in the convex-constrained case, and to the best of our knowledge this is the first such estimator in this general setting. This work builds on the Gaussian sequence framework of Neykov [2022] using a similar algorithmic scheme to achieve the minimax rate. Our algorithmic rate also applies with sub-Gaussian noise. We illustrate the utility of this theory with examples including multivariate monotone functions, linear functionals over ellipsoids, and Lipschitz classes.