Bayesian nonparametric multivariate convex regression

X, where f(x) is the gradient of f at x. This is called the convex regression problem. Convex regression can easily be modified to allow concave regression by multiplying all of the values by negative one. Convex regression problems are common in economics, operations research and reinforcement learning. In economics, production functions (Skiba 1978) and consumer preferences (Meyer & Pratt 1968) are often convex, while in operations research and reinforcement learning, value functions for stochastic optimization problems can be convex (Shapiro et al. 2009). If a problem is known to be convex, a convex regression estimate provides advantages over an unrestricted estimate. First, convexity is a powerful regularizer: it places strong conditions on the derivatives--and hence smoothness--of a function. Convexity constraints can substantially reduce overfitting and lead to more accurate predictions. Second, maintaining convexity allows the use of convex optimization solvers when the regression estimate is used in an objective function of an optimization problem. 1 Multivariate convex regression has received relatively little attention in the literature.