  There is a long history in machine learning of model ensembling, beginning
with boosting and bagging and continuing to the present day. Much of this
history has focused on combining models for classification and regression, but
recently there is interest in more complex settings such as ensembling policies
in reinforcement learning. Strong connections have also emerged between
ensembling and multicalibration techniques. In this work, we further
investigate these themes by considering a setting in which we wish to ensemble
models for multidimensional output predictions that are in turn used for
downstream optimization. More precisely, we imagine we are given a number of
models mapping a state space to multidimensional real-valued predictions. These
predictions form the coefficients of a linear objective that we would like to
optimize under specified constraints. The fundamental question we address is
how to improve and combine such models in a way that outperforms the best of
them in the downstream optimization problem. We apply multicalibration
techniques that lead to two provably efficient and convergent algorithms. The
first of these (the white box approach) requires being given models that map
states to output predictions, while the second (the \emph{black box} approach)
requires only policies (mappings from states to solutions to the optimization
problem). For both, we provide convergence and utility guarantees. We conclude
by investigating the performance and behavior of the two algorithms in a
controlled experimental setting.
