A Computational Framework for Multivariate Convex Regression and Its Variants
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Author(s)
Choudhury, Arkopal
Iyengar, Garud
Sen, Bodhisattva
Mazumder, Rahul
Date Issued
August 2018
Journal
Journal of the American Statistical Association
Publisher
Taylor & Francis
Citation
Mazumder, Rahul et al. “A Computational Framework for Multivariate Convex Regression and Its Variants.” Journal of the American Statistical Association (January 2018): 1–14 © 2018 American Statistical Association
Version
Original manuscript
Abstract
We study the nonparametric least squares estimator (LSE) of a multivariate convex regression function. The LSE, given as the solution to a quadratic program with O(n²) linear constraints (n being the sample size), is difficult to compute for large problems. Exploiting problem specific structure, we propose a scalable algorithmic framework based on the augmented Lagrangian method to compute the LSE. We develop a novel approach to obtain smooth convex approximations to the fitted (piecewise affine) convex LSE and provide formal bounds on the quality of approximation. When the number of samples is not too large compared to the dimension of the predictor, we propose a regularization scheme—Lipschitz convex regression—where we constrain the norm of the subgradients, and study the rates of convergence of the obtained LSE. Our algorithmic framework is simple and flexible and can be easily adapted to handle variants: estimation of a nondecreasing/nonincreasing convex/concave (with or without a Lipschitz bound) function. We perform numerical studies illustrating the scalability of the proposed algorithm—on some instances our proposal leads to more than a 10,000-fold improvement in runtime when compared to off-the-shelf interior point solvers for problems with n = 500. Keywords: Augmented Lagrangian method; Lipschitz convex regression; Non parametric least squares estimator; Scalable quadratic programming; Smooth convex regression
MIT Department
Sloan School of Management
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1080/01621459.2017.1407771
