Robust Sublinear Convergence Rates for Iterative Bregman Projections

Entropic regularization provides a simple way to approximate linear programs whose constraints split into two (or more) tractable blocks. The resulting objectives are amenable to cyclic Kullback-Leibler (KL) Bregman projections, with the classical Sinkhorn algorithm for optimal transport (balanced, unbalanced, gradient flows, barycenters, \dots) as the canonical example. Assuming uniformly bounded primal mass and dual radius, we prove that the dual objective of these KL projections decreases at an $O(1/k)$ rate with a constant that scales only linearly in $1/γ$, where $γ$ is the entropic regularization parameter. This extends the guarantees known for entropic optimal transport to any such linearly constrained problem. Following the terminology introduced in [Chizat et al 2025], we call such rates "robust", because this mild dependence on $γ$ underpins favorable complexity bounds for approximating the unregularized problem via alternating KL projections. The crucial aspect of the analysis is that the dual radius should be measured according to a block-quotient dual seminorm, which depends on the structure of the split of the constraint into blocks. As an application, we derive the flow-Sinkhorn algorithm for the Wasserstein-1 distance on graphs. It achieves $ε$-additive accuracy on the transshipment cost in $O(p/ε^{4})$ arithmetic operations, where $p$ is the number of edges.