Isotonic regression in general dimensions

TL;DR

This paper analyzes isotonic regression estimators in multiple dimensions, establishing minimax rates and adaptive convergence properties, including in random design settings, revealing surprising aspects of shape-constrained estimation.

Contribution

It provides the first minimax rate results for multivariate isotonic regression with fixed design and extends bounds to random design, highlighting adaptive rates and surprising estimation behaviors.

Findings

Achieves minimax rate of $n^{- ext{min}\{2/(d+2),1/d ight"} in empirical $L_2$ loss.

Shows adaptive convergence rate of $(k/n)^{ ext{min}(1,2/d)}$ for piecewise constant functions.

Establishes new bounds in random design setting, even for $d=2$, demonstrating rate optimality and adaptation challenges.

Abstract

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.