熵鞅最优传输中Sinkhorn算法的指数收敛性

中文摘要 AI 辅助

我们证明了熵鞅最优传输中Sinkhorn算法在相对熵意义下的指数收敛性。我们假设边际分布具有紧支撑且处于严格凸序，终端边际支撑为凸集，初始边际支撑位于终端边际支撑的相对内部；参考成本函数假定在每个变量上都是Lipschitz连续的。在这些假设下，我们建立了对偶变量在仿射规范下的一致界。这一结果使我们能够获得优化器的存在唯一性及其用对偶变量表示的指数形式。随后，我们建立了具有不同终端边际的鞅耦合的相对熵稳定性估计。证明稳定性估计中的常数在Sinkhorn迭代过程中保持一致，使我们能够确立Sinkhorn算法的指数收敛性。

英文摘要

We prove the exponential convergence in relative entropy of the Sinkhorn algorithm for the entropy martingale optimal transport. We assume that the marginals have compact supports and are in strict convex order, the terminal marginal support is convex, and the initial marginal support lies in the relative interior of the terminal marginal support; the reference cost is assumed to be Lipschitz in each variable. Under these assumptions we establish uniform bounds on the dual variables modulo affine gauges. This result allows us to obtain the existence and uniqueness of the optimizer, and its exponential representation in terms of dual variables. We then establish a relative-entropy stability estimate for martingale couplings with different terminal marginals. Proving that the constant in the stability estimate stays uniform throughout the Sinkhorn iteration allows us to establish the exponential convergence of the Sinkhorn algorithm.

发表机构

• LMO, Université Paris-Saclay（巴黎萨克雷大学）
• LaMME, Université Évry Paris-Saclay（巴黎萨克雷大学埃夫里校区）

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