Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure

Abstract

We establish the uniform in time stability, w.r.t. the marginals, of the Iterative Proportional Fitting Procedure, also known as Sinkhorn algorithm, used to solve entropyregularised Optimal Transport problems. Our result is quantitative and stated in terms of the 1-Wasserstein metric. As a corollary we establish a quantitative stability result for Schrödinger bridges.

Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm

“…More importantly, the continuity result is purely qualitative, and that is the main difference with the present results. Most recently, and partly concurrently with the present study, a beautiful result of [22] establishes the uniform stability of Sinkhorn's algorithm with respect to the marginals, in a bounded setting. As a consequence, the authors deduce Lipschitzianity in W 1 of the optimal couplings with respect to the marginals; the assumptions include bounded Lipschitz costs and bounded spaces.…”

“…The argument is based on the Hilbert-Birkhoff projective metric which has also been used successfully to show linear convergence of Sinkhorn's algorithm [13,27]. A crucial additional step accomplished in [22] is to pass from this metric to a more standard norm on the potentials. The techniques involving the projective metric are less probabilistic in nature, which may be one reason why it is wide open how to relax the boundedness conditions.…”

Martingale Schrödinger Bridges and Optimal Semistatic Portfolios

“…A series of results revealed that the additive form f (x) + g(y) always holds, but also that the measurability of (f, g) fails without additional conditions; moreover, even when measurability holds, integrability fails without further conditions (see [8,10,16,35,36]). The study of Schrödinger potentials remains an area of active study (see for instance [2,14,20,30,31]) that we have benefited from, especially for our companion paper [32]. For the present work, we have not been able to transfer as many of those techniques.…”

Stability of Schrödinger Potentials and Convergence of Sinkhorn's Algorithm

“…The authors show by a differential approach that the potentials are continuous in L p relative to the marginal densities. Still with bounded cost (and some other conditions), [15] establishes uniform continuity of the potentials relative to the marginals in Wasserstein distance W 1 ; this result is based on the Hilbert-Birkhoff projective metric. Closer to the present unbounded setting, [20] obtains stability of the optimal couplings in weak convergence for general continuous costs.…”