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Robust Generalization with Adaptive Optimal Transport Priors for Decision-Focused Learning

Haixiang Sun, Andrew L. Liu

TL;DR

A Prototype-Guided Distributionally Robust Optimization (PG-DRO) framework that learns class-adaptive priors from abundant base data via hierarchical optimal transport and embeds them into the Sinkhorn DRO formulation, which achieves stronger robust generalization in few-shot scenarios.

Abstract

Few-shot learning requires models to generalize under limited supervision while remaining robust to distribution shifts. Existing Sinkhorn Distributionally Robust Optimization (DRO) methods provide theoretical guarantees but rely on a fixed reference distribution, which limits their adaptability. We propose a Prototype-Guided Distributionally Robust Optimization (PG-DRO) framework that learns class-adaptive priors from abundant base data via hierarchical optimal transport and embeds them into the Sinkhorn DRO formulation. This design enables few-shot information to be organically integrated into producing class-specific robust decisions that are both theoretically grounded and efficient, and further aligns the uncertainty set with transferable structural knowledge. Experiments show that PG-DRO achieves stronger robust generalization in few-shot scenarios, outperforming both standard learners and DRO baselines.

Robust Generalization with Adaptive Optimal Transport Priors for Decision-Focused Learning

TL;DR

A Prototype-Guided Distributionally Robust Optimization (PG-DRO) framework that learns class-adaptive priors from abundant base data via hierarchical optimal transport and embeds them into the Sinkhorn DRO formulation, which achieves stronger robust generalization in few-shot scenarios.

Abstract

Few-shot learning requires models to generalize under limited supervision while remaining robust to distribution shifts. Existing Sinkhorn Distributionally Robust Optimization (DRO) methods provide theoretical guarantees but rely on a fixed reference distribution, which limits their adaptability. We propose a Prototype-Guided Distributionally Robust Optimization (PG-DRO) framework that learns class-adaptive priors from abundant base data via hierarchical optimal transport and embeds them into the Sinkhorn DRO formulation. This design enables few-shot information to be organically integrated into producing class-specific robust decisions that are both theoretically grounded and efficient, and further aligns the uncertainty set with transferable structural knowledge. Experiments show that PG-DRO achieves stronger robust generalization in few-shot scenarios, outperforming both standard learners and DRO baselines.
Paper Structure (24 sections, 5 theorems, 48 equations, 3 figures, 6 tables, 1 algorithm)

This paper contains 24 sections, 5 theorems, 48 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related works
  4. Few-shot Learning
  5. Optimal Transport
  6. Robust Optimization
  7. Preliminary
  8. Methodology
  9. Adaptive Optimal Transport Priors.
  10. Prototype-Guided DRO (PG-DRO)
  11. Decision rule and training objective.
  12. Experiments
  13. Simulation
  14. Few-shot simulation
  15. Classification
  16. ...and 9 more sections

Key Result

Lemma 3.2

$V_D(\lambda)$ is convex in $\lambda$ on $[0,\infty)$. Moreover, unless $\ell$ is $\nu$-a.s. constant, $V_D(\lambda)$ is strictly convex on $(0,\infty)$ and thus admits a unique minimizer $\lambda^\star>0$. In the degenerate case where $\ell$ is $\nu$-a.s. constant, the unique minimizer is attained

Figures (3)

  • Figure 1: Comparison of source and target domain with 2D projection
  • Figure 2: Normalized weights $\tilde{w}_{bc}$ for adaptive OT from source to target domain
  • Figure 3: Comparison of robustness under prior shift: PG-DRO consistently improves both average accuracy and worst-case performance over ERM and OT baselines.

Theorems & Definitions (9)

  • Definition 3.1: Sinkhorn DRO
  • Lemma 3.2: Convexity and uniqueness of the dual minimizer; cf. wang2025sinkhorn, Thm. 1(III) and Lemma 3
  • Corollary 4.1: Multi-class convexity
  • Theorem 4.2: Contraction of $V$ under adaptive OT
  • Theorem 4.3: Consistency
  • proof
  • Lemma B.5
  • proof
  • proof