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Isotonic Regression in General Dimensions Chatterjee, Sabyasachi


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 $d = 2$. 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.

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