Conference Agenda

In this presentation we investigate the discrepancy of a sample. The discrepancy measures how well a distribution is represented by a sample (for example in Monte-Carlo simulation) and how evenly distributed the realizations are. We focus on calculating a-priori probabilistic bounds for the discrepancy which means that before sampling we want to know how probable it is that the discrepancy is below a specific value. We do this for pseudo-random draws but also for variance reduction techniques like systematic sampling or antithetics. We will see that the runtime to calculate probabilistic bounds decreases significantly if we apply variance reduction techniques.

The information projection is a frequently occurring optimization problem, wherein the goal is to compute the projection of a probability measure onto a set of measures with respect to the relative entropy. A famous instance of this problem arises from entropic regularization of optimal transport, which has recently seen a surge in applications because it allows for the use of the iterative proportional fitting procedure (IPFP, also called Sinkhorn's algorithm). In this work, we study convergence properties of generalized IPFPs for more general sets of probability measures than those arising from optimal transport. In particular, we establish exponential convergence guarantees for general information projection problems whenever the set which is projected onto is defined through constraints arising from linear function spaces. This unifies and generalizes recent results from multi-marginal, adapted and martingale optimal transport. The proofs are based on duality and establishing a Polyak-Lojasiewicz inequality. A key contribution is to illuminate the role of the geometric interplay between the linear function spaces determining the constraints.