Reviews: Geometrically Coupled Monte Carlo Sampling

Summary: -------------- This paper proposes a number of expectation estimation strategies which strive to attain lower error than naive methods relying on iid sampling. This is especially important in settings where evaluating the objective function at the sampled variates is expensive (e.g. The advantage of Monte Carlo over deterministic methods is that theoretical assurances are more readily obtained. The approach is based on (numerically) finding optimal couplings, i.e. joint distributions over an augmented state space marginalizing to the expectation distribution of interest. While a uniformly optimal coupling does not exist over generic function classes, as is appreciated in statistical decision theory, the problem is well-defined in both expected and minimax variants.