Isotonic regression in general dimensions
Isotonic regression in general dimensions
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DOI：
10.1214/18-aos1753
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发表时间：
2017-08
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影响因子：
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通讯作者：
Q. Han;Tengyao Wang;S. Chatterjee;R. Samworth
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作者：
Q. Han;Tengyao Wang;S. Chatterjee;R. Samworth
我们研究了在$[0,1]^d$上每一坐标递增的实值函数的最小二乘回归函数估计量。对于均匀有界信号和固定的三次晶格设计，我们建立了估计器在经验$L_2$损失中达到最小阶$n^{-\min\{2/(d+2),1/d\}}$的最大速率，最高可达多对数因子。进一步，我们证明了一个尖锐的oracle不等式，它特别揭示了当真实回归函数在$k$超矩形上是分段常数时，最小二乘估计量具有更快的自适应收敛速度$(k/n)^{\min(1,2/d)}$，再次达到多对数因子。先前的结果仅限于$d \leq 2$的情况。最后，我们在更具挑战性的随机设计设置中建立了相应的界限（即使在$d=2$的情况下也是新的）。这些结果有两个令人惊讶的特征：首先，它们证明，即使函数类的相应熵积分迅速发散，全局经验风险最小化程序也有可能达到多对数因子的最优率；其次，它们表明形状约束估计器的自适应率可能严格低于参数率。
We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^d$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that the estimator achieves the minimax rate of order $n^{-\min\{2/(d+2),1/d\}}$ in the empirical $L_2$ loss, up to poly-logarithmic factors. Further, we prove a sharp oracle inequality, which reveals in particular that when the true regression function is piecewise constant on $k$ hyperrectangles, the least squares estimator enjoys a faster, adaptive rate of convergence of $(k/n)^{\min(1,2/d)}$, again up to poly-logarithmic factors. Previous results are confined to the case $d \leq 2$. Finally, we establish corresponding bounds (which are new even in the case $d=2$) in the more challenging random design setting. There are two surprising features of these results: first, they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to poly-logarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shape-constrained estimators can be strictly worse than the parametric rate.