Randomized Transport Plans via Hierarchical Fully Probabilistic Design

TL;DR

The paper reframes optimal transport as a Bayesian hierarchical design by treating transport plans as random objects encoded with a hyperprior S^o(π|K). It derives a Gibbs-form optimal hyperprior, S^o(π|K) ∝ exp(-λ_1^o D_KL(μ||μ_0)) S_I(π|K) exp(-D_KL(π||π_I)) exp(-λ_2^o D_KL(ν||ν_0)), and proves strong duality between the primal and dual formulations, yielding Kantorovich potentials that govern uncertainty in the marginals. In the parametric finite setting, it provides a detailed form of S^o(π|K), describes a stochastic approximation scheme for the Kantorovich potentials via second-order updates and HMC, and demonstrates how sampling from the hyperprior enables an expected transport plan plus uncertainty measures. The paper further shows how HFPD-OT can enhance algorithmic fairness in market matching by promoting diversity and long-run fairness across agents and contracts, with simulations illustrating higher diversity and activation of more contracts than conventional EOT. Overall, HFPD-OT extends entropy-regularized OT by introducing a generative model over plans, enabling uncertainty-aware decision making and robust, fair transport applications.

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.

Figures (10)

• Figure 1: Schematics which distinguish conventional base-level (i.e. deterministic) OT, in (a), from HFPD-OT, in (b) and (c). For ease of illustration, we consider the finite dimensional specialization in Section \ref{['parametric_hyperprior']}, but the ideas extend to the continuous setting. In HFPD-OT, uncertainty in the marginals, $\mu_0$ and $\nu_0$, induce uncertainty in the joint ($\pi$) and conditional ($\pi_{x|y_{0}}$) transport plans, as well as in the individual contracts ($\pi_{i,j}$). All are optimally modeled in probability (i.e. they are random processes or variables, per the setting). Here, a contract, $\pi_{i,j}\in [0, 1]$---see (a) and (b)---refers to the normalized quantity of resource (information, assets, stock, etc.) transported from agent $x_{i} \in \mathbb{\Omega}_{X}$ to agent $y_{j} \in \mathbb{\Omega}_{Y}$, in delivering the (global) transport plan, $\pi$.
• Figure 2: The conditional independence graph associated with HFPD-OT. Shaded nodes are observed. The arrows indicate the causal structure, where an arrow from one variable to a second indicates that the first variable causes the second.
• Figure 3: A sequential information-processing view of the optimal hierarchical model, $\mathsf{M}^o \equiv\hat{\pi}_{{\mathsf S}^o} \mathsf{S}^o$, used in the proof method (\ref{['eq:Mo']}). First, the inductive biases expressed via the hierarchical ideal model, $\mathsf{M}_\mathsf{I} \in {\mathbb M}_{\mathsf{H}}^c$ (\ref{['eq:joint_ideal_design']}), are processed to yield a new optimization problem over a constrained set $\mathbb{M}_{\mathsf{H}}$, whose solution, $\tilde{\mathsf M}$ (\ref{['eq:Mtilde']}), is on the boundary of $\mathbb{M}_{\mathsf{H}}$. Second, the knowledge constraints, $K$, are processed, further reducing the feasible set to the subset, $\mathbb{M}_{K}$ (\ref{['set:hierachical_model']}). The optimal hierarchical model is $\mathsf{M}^{o}$ (on the boundary of $\mathbb{M}_K$), s.t. $\pi \sim \mathsf{S}^o (\pi | K)$ (\ref{['eq:hyperprior']}).
• Figure 4: The two-step principle underlying HFPD-OT. $\mathsf{S}^{o}(\pi|K)$ is a generative model (i.e. a distribution) of random transport plans, $\pi$. Realizations, $\pi^{(k)}$, of $\pi$ can be sampled from $\mathsf{S}^{o}(\pi|K)$, and these samples can then be used to estimate an expected transport plan (\ref{['eq:expected_plan']}) for downstream transport problems, via ergodic averaging. In addition, HFPD-OT enables a principled analysis of the intrinsic uncertainty in the transport problem.
• Figure 5: Schematic of an uncertain transport plan in the $\Delta_{3}$ simplex, annotating the corresponding nominal (i.e. prior-specified) row and column marginals. The $(2,2)$ entry (i.e. contract) is necessarily $\pi_{22} = 1 - \pi_{11} - \pi_{12} - \pi_{21}$.

Theorems & Definitions (16)

• definition 1: Expected transport plan
• theorem 1
• proof : Proof method
• Remark 1
• Remark 2
• definition 2: HFPD-OT hyperprior for the parametric transport plan
• Remark 3: Inference with the HFPD-OT hyperprior, $\mathsf{S}^o (\pi | K )$
• Remark 4
• Remark 5
• definition 3: Diversity index