Multivariate convex regression with adaptive partitioning

We propose a new, nonparametric method for multivariate regression subject to convexity or concavity constraints on the response function. Convexity constraints are common in economics, statistics, operations research and financial engineering, but there is currently no multivariate method that is computationally feasible for more than a few hundred observations. We introduce Convex Adaptive Partitioning (CAP), which creates a globally convex regression model from locally linear estimates fit on adaptively selected covariate partitions. Adaptive partitioning makes computation efficient even on large problems. Convexity itself acts as a regularizer, making CAP resistant to overfitting. We give consistency results for the univariate case. CAP is applied to value function approximation for pricing American basket options with a large number of underlying assets.