Characterizing the minimax rate of nonparametric regression under bounded star-shaped constraints
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 ${\varepsilon^{\ast}}^2\wedge\mathrm{diam}(\mathcal{F})^2$ where \[\varepsilon^{\ast} =\sup\{\varepsilon\ge 0:n\varepsilon^2 \le \log M_{\mathcal{F}}^{\operatorname{loc}}(\varepsilon,c)\},\] where $\log M_{\mathcal{F}}^{\operatorname{loc}}(\cdot, c)$ 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.
Akshay Prasadan、Matey Neykov
数学
Akshay Prasadan,Matey Neykov.Characterizing the minimax rate of nonparametric regression under bounded star-shaped constraints[EB/OL].(2025-06-27)[2025-07-16].https://arxiv.org/abs/2401.07968.点此复制
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