TL;DR
This paper develops efficient Sinkhorn-type algorithms for entropic optimal transport problems with martingale constraints, achieving fast convergence and robustness in approximate solutions.
Contribution
It introduces a novel entropic formulation and sparse Newton-based Sinkhorn algorithms tailored for martingale optimal transport problems, leveraging Hessian sparsity.
Findings
Algorithms exhibit super-exponential convergence.
Methods are robust with controllable constraint violations.
Approximations effectively satisfy martingale constraints.
Abstract
This work introduces novel computational methods for entropic optimal transport (OT) problems under martingale-type conditions. The considered problems include the discrete martingale optimal transport (MOT) problem. Moreover, as the (super-)martingale conditions are equivalent to row-wise (in-)equality constraints on the coupling matrix, our work applies to a prevalent class of OT problems with structural constraints. Inspired by the recent empirical success of Sinkhorn-type algorithms, we propose an entropic formulation for the MOT problem and introduce Sinkhorn-type algorithms with sparse Newton iterations that utilize the (approximate) sparsity of the Hessian matrix of the dual objective. As exact martingale conditions are typically infeasible, we adopt entropic regularization to find an approximate constraint-satisfied solution. We show that, in practice, the proposed algorithms…
Peer Reviews
Decision·Submitted to ICLR 2025
The paper provides and discusses multiple examples that well justifies the underlying problem in various fields. In support of the choice of entropic regularization, the paper proves that the effect of this term vanishes at an exponential speed w.r.t the growth of the regularization parameter. The resulting algorithm is compared to a state-of-the-art first-orde method, which shows remarkable improvement in the speed of convergence.
I have few concerns related to the algorithmic choices and the theory that will explain tn the next part (questions). The theoretical result is a streightforward generalization of the result in (Weed 2018), but still interesting. The exaperiments certainly support that the algorithm is applicable to relatively large problems, but the setup of the experiments is still considered small in certain fields such as machine learning. However, this is a general limitation of the Kantorovich formulati
Focusing on optimal transport with martingale conditions is interesting, given its applications, such as model-free optimal pricing, as highlighted by the authors. Additionally, it would be valuable to explore extensions of Sinkhorn-type and other OT solvers to address this novel class of optimal transport problems.
1. The paper lacks clarity in its structure. It is strongly recommended that the authors clearly state the main message at the beginning of each section and subsection. This would improve transitions and prevent readers from feeling confused. 2. Although the topic is interesting, the absence of a theoretical convergence analysis for the proposed algorithm raises concerns about its suitability for publication in high-tier machine learning conferences. 3. While the authors claim that the propose
The article is well organized, and the overall motivation and story is clear. The MOT problem considered is an extension to the well-known OT problem, and developing efficient algorithm for solving MOT is helpful.
1. One of my major concerns for this article is the necessity of using a Sinkhorn-type algorithm for solving the entropic MOT problem. The Sinkhorn algorithm is efficient partly because it has closed-form formulas for the two alternating minimization steps. However, in the entropic MOT problem, the author(s) show that the $g$ variables need to be updated using Newton's method. If this is the case, what is the benefit of using the alternating minimization method? We can simply use limited-memory
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