We quantify the minimax rate for a nonparametric regression model over a convex function class $\mathcal{F}$ with bounded diameter. We obtain a minimax rate of ${\varepsilon^{\ast}}^2\wedge\mathrm{diam}(\mathcal{F})^2$ where \[\varepsilon^{\ast} =\sup\{\varepsilon>0:n\varepsilon^2 \le \log M_{\mathcal{F}}^{\operatorname{loc}}(\varepsilon,c)\},\] where $M_{\mathcal{F}}^{\operatorname{loc}}(\cdot, c)$ is the local metric entropy of $\mathcal{F}$ 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 addition, we prove that our estimator is adaptive to the true point, 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.

翻译：我们针对直径有界的凸函数类 $\mathcal{F}$，量化了非参数回归模型的极小极大速率。我们得到的极小极大速率为 ${\varepsilon^{\ast}}^2\wedge\mathrm{diam}(\mathcal{F})^2$，其中 \[\varepsilon^{\ast} =\sup\{\varepsilon>0:n\varepsilon^2 \le \log M_{\mathcal{F}}^{\operatorname{loc}}(\varepsilon,c)\},\] 这里 $M_{\mathcal{F}}^{\operatorname{loc}}(\cdot, c)$ 是 $\mathcal{F}$ 的局部度量熵，而我们的损失函数是输入空间 $\mathcal{X}$ 上的总体 $L_2$ 距离的平方。与关于该主题的经典工作[参见 Yang 和 Barron, 1999]不同，我们的结果不要求 $\mathcal{F}$ 中的函数在 sup-范数下一致有界。此外，我们证明我们的估计量对真实点具有自适应性，据我们所知，这是该一般设定下首个具有此性质的估计量。本工作基于 Neykov [2022] 的高斯序列框架，采用类似的算法方案实现极小极大速率。我们的算法速率同样适用于次高斯噪声。我们通过多元单调函数、椭球上的线性泛函以及 Lipschitz 类等示例说明了该理论的实用性。