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Decision-Focused Optimal Transport

Suhan Liu, Mo Liu

TL;DR

The paper introduces the decision-focused divergence (DF divergence), a coupling-aware metric for comparing distributions of stochastic optimization coefficients by measuring downstream decision impact via the SPO loss. It develops optimistic, robust, and entropy-regularized variants, establishes connections to classical metrics, and presents a tractable W_2-type reformulation with a lift to original couplings. The authors prove dimension-independent sample complexity for estimating the DF distance, and they demonstrate decision-focused interpolation and couplings on synthetic and real data (notably a Parkinson’s disease dataset), showing substantial benefits over decision-blind metrics. Overall, the DF framework provides a principled, computationally feasible approach to quantify and interpolate distributional shifts in downstream optimization problems, with broad applicability to privacy-preserving forecasting, clustering, and uncertainty set construction.

Abstract

We propose a fundamental metric for measuring the distance between two distributions. This metric, referred to as the decision-focused (DF) divergence, is tailored to stochastic linear optimization problems in which the objective coefficients are random and may follow two distinct distributions. Traditional metrics such as KL divergence and Wasserstein distance are not well-suited for quantifying the resulting cost discrepancy, because changes in the coefficient distribution do not necessarily change the optimizer of the underlying linear program. Instead, the impact on the objective value depends on how the two distributions are coupled (aligned). Motivated by optimal transport, we introduce decision-focused distances under several settings, including the optimistic DF distance, the robust DF distance, and their entropy-regularized variants. We establish connections between the proposed DF distance and classical distributional metrics. For the calculation of the DF distance, we develop efficient computational methods. We further derive sample complexity guarantees for estimating these distances and show that the DF distance estimation avoids the curse of dimensionality that arises in Wasserstein distance estimation. The proposed DF distance provides a foundation for a broad range of applications. As an illustrative example, we study the interpolation between two distributions. Numerical studies, including a toy newsvendor problem and a real-world medical testing dataset, demonstrate the practical value of the proposed DF distance.

Decision-Focused Optimal Transport

TL;DR

The paper introduces the decision-focused divergence (DF divergence), a coupling-aware metric for comparing distributions of stochastic optimization coefficients by measuring downstream decision impact via the SPO loss. It develops optimistic, robust, and entropy-regularized variants, establishes connections to classical metrics, and presents a tractable W_2-type reformulation with a lift to original couplings. The authors prove dimension-independent sample complexity for estimating the DF distance, and they demonstrate decision-focused interpolation and couplings on synthetic and real data (notably a Parkinson’s disease dataset), showing substantial benefits over decision-blind metrics. Overall, the DF framework provides a principled, computationally feasible approach to quantify and interpolate distributional shifts in downstream optimization problems, with broad applicability to privacy-preserving forecasting, clustering, and uncertainty set construction.

Abstract

We propose a fundamental metric for measuring the distance between two distributions. This metric, referred to as the decision-focused (DF) divergence, is tailored to stochastic linear optimization problems in which the objective coefficients are random and may follow two distinct distributions. Traditional metrics such as KL divergence and Wasserstein distance are not well-suited for quantifying the resulting cost discrepancy, because changes in the coefficient distribution do not necessarily change the optimizer of the underlying linear program. Instead, the impact on the objective value depends on how the two distributions are coupled (aligned). Motivated by optimal transport, we introduce decision-focused distances under several settings, including the optimistic DF distance, the robust DF distance, and their entropy-regularized variants. We establish connections between the proposed DF distance and classical distributional metrics. For the calculation of the DF distance, we develop efficient computational methods. We further derive sample complexity guarantees for estimating these distances and show that the DF distance estimation avoids the curse of dimensionality that arises in Wasserstein distance estimation. The proposed DF distance provides a foundation for a broad range of applications. As an illustrative example, we study the interpolation between two distributions. Numerical studies, including a toy newsvendor problem and a real-world medical testing dataset, demonstrate the practical value of the proposed DF distance.
Paper Structure (57 sections, 11 theorems, 119 equations, 10 figures, 5 tables)

This paper contains 57 sections, 11 theorems, 119 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Decision-focused setting
  3. Preventive healthcare and diagnostic testing.
  4. Why coupling matters.
  5. Three interpolations via couplings.
  6. Decision-focused average at age $45$.
  7. Organization of the paper.
  8. Literature review
  9. Decision-focused learning for stochastic linear optimization.
  10. Distributional distances and optimal transport.
  11. Decision-Focused Divergence
  12. Examples of decision-focused divergence
  13. Entropy-regularized DF divergences
  14. Properties of the Decision-Focused Divergence
  15. Calculation of Decision-Focused Optimal Transport
  16. ...and 42 more sections

Key Result

Proposition 1

Let $X\in\mathbb{R}^d$ be any random cost vector with distribution $\mu$. Let $\kappa$ be any nonnegative random scalar (which may depend on $X$), and define the transformed vector $Y := \kappa X$, with distribution $\nu$. Then we have $W^{\textnormal{O}}_{\textnormal{DF}}(\mu,\nu)=0$.

Figures (10)

  • Figure 1: Three couplings between age $40$ and age $50$ BMD distributions. The arrows illustrate how probability mass at age $40$ is matched to mass at age $50$. Different couplings lead to different conditional trajectories and can induce substantially different decision risks. Figure \ref{['fig:bmd_opt']} corresponds to a minimal-change alignment (small individual-level shifts), Figure \ref{['fig:bmd_worst']} corresponds to a maximal-change alignment (large individual-level shifts), and Figure \ref{['fig:bmd_iid']} corresponds to an independent scenario in which each individual shares the same age-$50$ target distribution, regardless of their BMD at age $40$.
  • Figure 2: Patient-level trajectories implied by different coupling assumptions. The figure shows plausible BMD paths for a patient with BMD $=1.7$ at age $40$ under an optimistic coupling (best-case alignment), an independent coupling, and a robust coupling (worst-case alignment).
  • Figure 3: Left: a polyhedral feasible region $S$ with four extreme points and their associated normal cones. Right: the point clouds illustrate three example groups of cost vectors. The DF distance between Groups 1 and 2 is zero, while the DF distance between Groups 2 and 3 is nonzero.
  • Figure 4: Comparison of McCann interpolation and linear density averaging for Gaussian measures. Left: McCann interpolation captures a smooth displacement from $\mu_0$ to $\mu_1$. Right: linear density averaging can be bimodal for intermediate $t$ and does not reflect a plausible transport of mass.
  • Figure 5: Three customer types have distinct discrete demand distributions and, under $\alpha=0.6$, induce distinct optimal order quantities $q^{*(1)}=9$, $q^{*(2)}=10$, and $q^{*(3)}=11$ (vertical red lines).
  • ...and 5 more figures

Theorems & Definitions (30)

  • Definition 1: Decision-focused divergence
  • Definition 2: Decision-focused distances
  • Proposition 1: Random rescaling yields zero optimistic DF distance
  • Definition 3: Entropy-regularized DF divergences
  • Proposition 2: Optimistic DF distance is Lipschitz with respect to $W_1$ distance
  • Definition 4: Push-forward measure
  • Proposition 3: Optimistic DF distance is Lipschitz with respect to $W_2$ distance
  • Proposition 4: monotonicity and sandwich inequalities for entropy-regularized DF distance
  • Lemma 1: Coupling reduction via oracle push-forward
  • Theorem 1: Quadratic OT reformulation of $W^{\textnormal{O}}_{\textnormal{DF}}$
  • ...and 20 more