In this paper we study minimax and adaptation rates in general isotonic regression. For uniform deterministic and random designs in $[0,1]^d$ with $d\ge 2$ and $N(0,1)$ noise, the minimax rate for the $\ell_2$ risk is known to be bounded from below by $n^{-1/d}$ when the unknown mean function $f$ is nondecreasing and its range is bounded by a constant, while the least squares estimator (LSE) is known to nearly achieve the minimax rate up to a factor $(\log n)^\gamma$ where $n$ is sample size, $\gamma = 4$ in the lattice design and $\gamma = \max\{9/2, (d^2+d+1)/2 \}$ in the random design. Moreover, the LSE is known to achieve the adaptation rate $(K/n)^{-2/d}\{1\vee \log(n/K)\}^{2\gamma}$ when $f$ is piecewise constant on $K$ hyperrectangles in a partition of $[0,1]^d$. Due to the minimax theorem, the LSE is identical on every design point to both the max-min and min-max estimators over all upper and lower sets containing the design point. This motivates our consideration of estimators which lie in-between the max-min and min-max estimators over possibly smaller classes of upper and lower sets, including a subclass of block estimators. Under a $q$-th moment condition on the noise, we develop $\ell_q$ risk bounds for such general estimators for isotonic regression on graphs. For uniform deterministic and random designs in $[0,1]^d$ with $d\ge 3$, our $\ell_2$ risk bound for the block estimator matches the minimax rate $n^{-1/d}$ when the range of $f$ is bounded and achieves the near parametric adaptation rate $(K/n)\{1\vee\log(n/K)\}^{d}$ when $f$ is $K$-piecewise constant. Furthermore, the block estimator possesses the following oracle property in variable selection: When $f$ depends on only a subset $S$ of variables, the $\ell_2$ risk of the block estimator automatically achieves up to a poly-logarithmic factor the minimax rate based on the oracular knowledge of $S$.

中文翻译：

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多维空间和图中的等渗回归

在本文中，我们研究了一般等渗回归中的极小极大和适应率。对于 $[0,1]^d$ 中具有 $d\ge 2$ 和 $N(0,1)$ 噪声的均匀确定性和随机设计，已知 $\ell_2$ 风险的最小最大率有界于低于 $n^{-1/d}$ 当未知均值函数 $f$ 为非递减且其范围受常数限制时，而已知最小二乘估计器 (LSE) 几乎可以达到最小最大率因子 $(\log n)^\gamma$ 其中 $n$ 是样本大小，格子设计中的 $\gamma = 4$ 和 $\gamma = \max\{9/2, (d^2+d+1 )/2 \}$ 在随机设计中。此外，当 $f$ 为 LSE 时，已知 LSE 可实现自适应率 $(K/n)^{-2/d}\{1\vee \log(n/K)\}^{2\gamma}$ $[0,1]^d$ 分区中 $K$ 超矩形上的分段常数。由于极大极小定理，LSE 在每个设计点上都与包含设计点的所有上和下集的 max-min 和 min-max 估计量相同。这促使我们考虑位于 max-min 和 min-max 估计量之间的估计量，这些估计量可能位于更小的上下集合类别，包括块估计量的子类。在噪声的 $q$-th 矩条件下，我们为图上的等渗回归的这种一般估计量开发了 $\ell_q$ 风险界限。对于 $[0,1]^d$ 和 $d\ge 3$ 中的统一确定性和随机设计，我们的 $\ell_2$ 块估计器的风险界限与最小最大率 $n^{-1/d}$ 匹配，当$f$ 的范围是有界的，当 $f$ 为 $K$-piecewise 时，达到接近参数化的适应率 $(K/n)\{1\vee\log(n/K)\}^{d}$不变。此外，