Isotonic regression in general dimensions
Abstract
We study the least squares regression function estimator over the class of realvalued 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 polylogarithmic 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 polylogarithmic 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 polylogarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shapeconstrained estimators can be strictly worse than the parametric rate.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 arXiv:
 arXiv:1708.09468
 Bibcode:
 2017arXiv170809468H
 Keywords:

 Mathematics  Statistics Theory;
 62G08;
 62G05
 EPrint:
 36 pages