Isotonic regression in general dimensions

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.