Bayesian nonparametric multivariate convex regression

TL;DR

This paper introduces a Bayesian nonparametric method for multivariate convex regression, modeling the function as a maximum of hyperplanes, with strong theoretical guarantees and practical application to value function approximation.

Contribution

It proposes a novel Bayesian approach for multivariate convex regression using hyperplanes, with proven posterior consistency and an efficient computational algorithm.

Findings

Achieves a convergence rate of log(n)^{-1} n^{-1/(d+2)} under certain conditions.

Demonstrates effective application to value function approximation.

Provides an MCMC algorithm for practical Bayesian inference.

Abstract

In many applications, such as economics, operations research and reinforcement learning, one often needs to estimate a multivariate regression function f subject to a convexity constraint. For example, in sequential decision processes the value of a state under optimal subsequent decisions may be known to be convex or concave. We propose a new Bayesian nonparametric multivariate approach based on characterizing the unknown regression function as the max of a random collection of unknown hyperplanes. This specification induces a prior with large support in a Kullback-Leibler sense on the space of convex functions, while also leading to strong posterior consistency. Although we assume that f is defined over R^p, we show that this model has a convergence rate of log(n)^{-1} n^{-1/(d+2)} under the empirical L2 norm when f actually maps a d dimensional linear subspace to R. We design an efficient reversible jump MCMC algorithm for posterior computation and demonstrate the methods through application to value function approximation.