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What Data Enables Optimal Decisions? An Exact Characterization for Linear Optimization

Omar Bennouna, Amine Bennouna, Saurabh Amin, Asuman Ozdaglar

TL;DR

The paper addresses how informative a dataset is for solving a task-specific decision problem formulated as a linear program with cost uncertainty. It develops a sharp geometric characterization, showing that data sufficiency is equivalent to spanning the key directions $\Delta(\mathcal{X},\mathcal{C})$ or, equivalently, to the span of differences between reachable optimal solutions $\mathrm{dir}(\mathcal{X}^*(\mathcal{C}))$, and provides an algorithm to construct a minimal, sufficient dataset. By linking these geometric objects to practical data collection, the authors propose an iterative MILP-based method to build small datasets that preserve task-optimality and demonstrate the approach on a hiring-interviews scenario. The results offer a principled foundation for offline, task-aware data selection that can substantially reduce data-collection and computation costs while guaranteeing optimal decisions within the given uncertainty model.

Abstract

We study the fundamental question of how informative a dataset is for solving a given decision-making task. In our setting, the dataset provides partial information about unknown parameters that influence task outcomes. Focusing on linear programs, we characterize when a dataset is sufficient to recover an optimal decision, given an uncertainty set on the cost vector. Our main contribution is a sharp geometric characterization that identifies the directions of the cost vector that matter for optimality, relative to the task constraints and uncertainty set. We further develop a practical algorithm that, for a given task, constructs a minimal or least-costly sufficient dataset. Our results reveal that small, well-chosen datasets can often fully determine optimal decisions -- offering a principled foundation for task-aware data selection.

What Data Enables Optimal Decisions? An Exact Characterization for Linear Optimization

TL;DR

The paper addresses how informative a dataset is for solving a task-specific decision problem formulated as a linear program with cost uncertainty. It develops a sharp geometric characterization, showing that data sufficiency is equivalent to spanning the key directions or, equivalently, to the span of differences between reachable optimal solutions , and provides an algorithm to construct a minimal, sufficient dataset. By linking these geometric objects to practical data collection, the authors propose an iterative MILP-based method to build small datasets that preserve task-optimality and demonstrate the approach on a hiring-interviews scenario. The results offer a principled foundation for offline, task-aware data selection that can substantially reduce data-collection and computation costs while guaranteeing optimal decisions within the given uncertainty model.

Abstract

We study the fundamental question of how informative a dataset is for solving a given decision-making task. In our setting, the dataset provides partial information about unknown parameters that influence task outcomes. Focusing on linear programs, we characterize when a dataset is sufficient to recover an optimal decision, given an uncertainty set on the cost vector. Our main contribution is a sharp geometric characterization that identifies the directions of the cost vector that matter for optimality, relative to the task constraints and uncertainty set. We further develop a practical algorithm that, for a given task, constructs a minimal or least-costly sufficient dataset. Our results reveal that small, well-chosen datasets can often fully determine optimal decisions -- offering a principled foundation for task-aware data selection.

Paper Structure

This paper contains 26 sections, 17 theorems, 42 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Active Learning, Bandits and Adaptive Experimental Design.
  3. Blackwell's Informativeness Theory.
  4. Influence Functions and Robust Statistics.
  5. Contributions.
  6. Problem Formulation
  7. Characterizing Sufficient Datasets
  8. Characterization Under No Prior Knowledge
  9. Characterization under Convex Uncertainty Sets
  10. An Algorithmically Tractable Characterization
  11. A Data Collection Algorithm: Finding Minimal Sufficient Datasets
  12. Application: Hiring Interviews
  13. Impact of $\mathcal{C}$.
  14. Impact of $\mathcal{X}$.
  15. Conclusion and Limitations
  16. ...and 11 more sections

Key Result

Proposition 1

Let $\mathcal{C}$ be an open convex set and $\mathcal{D}:=\{q_1,\dots,q_N\}$ a dataset. The following are equivalent:

Figures (2)

  • Figure 1: Optimality cones relative to $\mathcal{X}$ (left), relative to the origin (middle) and examples of the uncertainty sets ($\mathcal{C}$ and $\mathcal{C}'$) relative to the optimality cones (right).
  • Figure 2: Candidates to be interviewed (in red) to make an optimal hiring decision. Number of candidates to interview from left to right for top and bottom row respectively: $8,24,31,52$ and $8,28,43,70$.

Theorems & Definitions (40)

  • Definition 1: Sufficient Decision Dataset
  • Proposition 1: One vs All Optimal Solutions
  • Proposition 2
  • Proposition 3: Noisy Observations
  • Proposition 4
  • Definition 2: Extreme Points
  • Proposition 5: Feasible and Extreme Directions
  • Proposition 6: Optimality Cones
  • Definition 3: Relevant Extreme Directions
  • Theorem 1
  • ...and 30 more