Journal Article

ISOTONIC REGRESSION IN GENERAL DIMENSIONS

Wang, Tengyao, Han, Qiyang, Chatterjee, Sabyasachi, Samworth, Richard J.

2019

Overview

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 L₂ 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 ≤ 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 shape-constrained estimators can be strictly worse than the parametric rate.

• Publisher: Institute of Mathematical Statistics
• Subject: Asymptotic methods / Convergence / Cubic lattice / Entropy / Estimating techniques / Lattice design / Least squares method / Mathematical functions / Minimax technique / Optimization / Regression / Regression analysis