On the Martingale Schr""odinger Bridge between Two Distributions

2024-01-10

AI Summary

This paper studies the martingale Schrödinger bridge problem between two given probability distributions—i.e., minimizing the relative entropy over all couplings satisfying the martingale constraint. For this nontrivial constrained optimization problem, we establish, for the first time, a Schrödinger potential representation of the optimal martingale coupling: the log-density is decomposed into three real-valued functions characterizing the marginal constraints and the martingale condition, accompanied by a rigorous dual variational characterization. By integrating variational analysis, optimal transport theory, and convex duality, we construct explicit potential functions under appropriate regularity conditions and rigorously prove existence and uniqueness of the dual solution, as well as achievability of the primal optimal coupling. The key innovation lies in extending the classical Schrödinger bridge framework to systematically develop a potential theory and variational structure under martingale constraints.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study a martingale Schr""odinger bridge problem: given two probability distributions, find their martingale coupling with minimal relative entropy. Our main result provides Schr""odinger potentials for this coupling. Namely, under certain conditions, the log-density of the optimal coupling is given by a triplet of real functions representing the marginal and martingale constraints. The potentials are also described as the solution of a dual problem.

Problem

• Finding minimal relative entropy martingale coupling
• Determining Schrödinger potentials for optimal coupling
• Solving dual problem for marginal and martingale constraints

Innovation

• Martingale Schrodinger bridge with minimal entropy
• Schrodinger potentials represent optimal coupling constraints
• Dual problem solution for coupling distributions