Exponential convergence of general iterative proportional fitting procedures
Stephan Eckstein, Aziz Lakhal
TL;DR
The paper analyzes IPFP for information projection problems where the constraint set is defined by linear function spaces, establishing exponential convergence under strong duality and a bounded-iterate regime. The key methodological advance is a dual formulation paired with embedding into a quotient space to recover strong convexity, enabling a Polyak-Lojasiewicz-based linear convergence analysis. A central contribution is linking the IPFP rate to geometry among constraint subspaces via Friedrichs angles, yielding explicit contraction coefficients that depend on the subspaces’ sum-closedness and condition numbers. The results unify and extend convergence insights across multi-marginal, adapted, and martingale OT, with concrete bounds and an application to Martingale OT illustrating practical implications for the rate via subspace angles.
Abstract
Motivated by the success of Sinkhorn's algorithm for entropic optimal transport, we study convergence properties of iterative proportional fitting procedures (IPFP) used to solve more general information projection problems. We establish exponential convergence guarantees for the IPFP whenever the set of probability measures which is projected onto is defined through constraints arising from linear function spaces. This unifies and extends recent results from multi-marginal, adapted and martingale optimal transport. The proofs are based on strong convexity arguments for the dual problem, and the key contribution is to illuminate the role of the geometric interplay between the subspaces defining the constraints. In this regard, we show that the larger the angle (in the sense of Friedrichs) between the linear function spaces, the better the rate of contraction of the IPFP.