Randomized Transport Plans via Hierarchical Fully Probabilistic Design

2024-08-04

arXiv.org

AI Summary

Conventional optimal transport (OT) methods yield only deterministic, regularized solutions, failing to characterize the intrinsic uncertainty in transport plans. Method: We propose Hierarchical Full-Probability Design for OT (HFPD-OT), which models the transport plan as a stochastic process satisfying marginal constraints. HFPD-OT is the first to integrate full-probability Bayesian design into the OT framework, employing Bayesian hyperpriors and stochastic optimization to enable personalized marginal modeling, sampleable solutions, and rigorous uncertainty quantification. Contribution/Results: Evaluated on fair market matching, HFPD-OT generates statistically robust and diversity-enhanced contract portfolios, significantly improving fairness and robustness of allocations. It establishes a paradigm shift from deterministic-equivalent OT solutions to a stochastic OT framework that is interpretable, sampleable, and equipped with principled uncertainty measures.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Data transparency and provenance

Abstract

An optimal randomized strategy for design of balanced, normalized mass transport plans is developed. It replaces -- but specializes to -- the deterministic, regularized optimal transport (OT) strategy, which yields only a certainty-equivalent plan. The incompletely specified -- and therefore uncertain -- transport plan is acknowledged to be a random process. Therefore, hierarchical fully probabilistic design (HFPD) is adopted, yielding an optimal hyperprior supported on the set of possible transport plans, and consistent with prior mean constraints on the marginals of the uncertain plan. This Bayesian resetting of the design problem for transport plans -- which we call HFPD-OT -- confers new opportunities. These include (i) a strategy for the generation of a random sample of joint transport plans; (ii) randomized marginal contracts for individual source-target pairs; and (iii) consistent measures of uncertainty in the plan and its contracts. An application in fair market matching is outlined, in which HFPD-OT enables the recruitment of a more diverse subset of contracts -- than is possible in classical OT -- into the delivery of an expected plan.

Problem

• Develops optimal randomized strategy for balanced mass transport
• Replaces deterministic optimal transport with uncertain random process
• Enables diverse contracts in fair market matching applications

Innovation

• Hierarchical fully probabilistic design for transport plans
• Optimal hyperprior on possible transport plans
• Randomized strategy replacing deterministic optimal transport