"Non UBC"@en . "DSpace"@en . "Chatterjee, Sabyasachi"@en . "2018-07-29T05:00:43Z"@* . "2018-01-29T10:31"@en . "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."@en . "https://circle.library.ubc.ca/rest/handle/2429/66602?expand=metadata"@en . "41 minutes"@en . "video/mp4"@en . ""@en . "Author affiliation: University of Illinois at Urbana-Champaign"@en . "10.14288/1.0369235"@en . "eng"@en . "Unreviewed"@en . "Vancouver : University of British Columbia Library"@en . "Banff International Research Station for Mathematical Innovation and Discovery"@en . "Attribution-NonCommercial-NoDerivatives 4.0 International"@en . "http://creativecommons.org/licenses/by-nc-nd/4.0/"@en . "Researcher"@en . "BIRS Workshop Lecture Videos (Banff, Alta)"@en . "Mathematics"@en . "Statistics"@en . "Operations research, mathematical programming"@en . "Isotonic Regression in General Dimensions"@en . "Moving Image"@en . "http://hdl.handle.net/2429/66602"@en .