A Computational Framework for Multivariate Convex Regression

• Activity Number: 448 - The Contribution of Convex Optimization to New Statistical Concepts
• Type: Topic Contributed
• Date/Time: Thursday, August 6, 2020 : 10:00 AM to 11:50 AM
• Sponsor: Section on Statistical Computing
• Author(s): Rahul Mazumder*
• Companies: Massachusetts Institute of Technology
• Keywords: convex optimization; nonparametric regression; shape constrained regression

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(n^2) 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.