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Multicalibration yields better matchings

Riccardo Colini Baldeschi, Simone Di Gregorio, Simone Fioravanti, Federico Fusco, Ido Guy, Daniel Haimovich, Stefano Leonardi, Fridolin Linder, Lorenzo Perini, Matteo Russo, Niek Tax

TL;DR

This work tackles choosing a maximum-weight matching when edge weights are stochastic and only available via predictions. It introduces multicalibration as a post-processing tool to calibrate a predictor across protected context sets, proving that a suitably multicalibrated predictor $\hat{\gamma}$ makes the max-over-$\mathcal{C}$ decision on $\gamma$ competitive with the best rule on $\hat{\gamma}$, up to an additive error $\varepsilon$. The authors provide a constructive algorithm with worst-case sample complexity $\tilde{O}(n^{8}/\varepsilon^{4} \log|\mathcal{C}|)$, and show how the dependence on the initial predictor’s error $r$ can tighten the bound. They also extend the framework to other linear optimization problems (e.g., best-action, rejection, and matroid bases), outlining how multicalibration-based post-processing can improve ex-post decisions in data-augmented algorithm design.

Abstract

Consider the problem of finding the best matching in a weighted graph where we only have access to predictions of the actual stochastic weights, based on an underlying context. If the predictor is the Bayes optimal one, then computing the best matching based on the predicted weights is optimal. However, in practice, this perfect information scenario is not realistic. Given an imperfect predictor, a suboptimal decision rule may compensate for the induced error and thus outperform the standard optimal rule. In this paper, we propose multicalibration as a way to address this problem. This fairness notion requires a predictor to be unbiased on each element of a family of protected sets of contexts. Given a class of matching algorithms $\mathcal C$ and any predictor $γ$ of the edge-weights, we show how to construct a specific multicalibrated predictor $\hat γ$, with the following property. Picking the best matching based on the output of $\hat γ$ is competitive with the best decision rule in $\mathcal C$ applied onto the original predictor $γ$. We complement this result by providing sample complexity bounds.

Multicalibration yields better matchings

TL;DR

This work tackles choosing a maximum-weight matching when edge weights are stochastic and only available via predictions. It introduces multicalibration as a post-processing tool to calibrate a predictor across protected context sets, proving that a suitably multicalibrated predictor makes the max-over- decision on competitive with the best rule on , up to an additive error . The authors provide a constructive algorithm with worst-case sample complexity , and show how the dependence on the initial predictor’s error can tighten the bound. They also extend the framework to other linear optimization problems (e.g., best-action, rejection, and matroid bases), outlining how multicalibration-based post-processing can improve ex-post decisions in data-augmented algorithm design.

Abstract

Consider the problem of finding the best matching in a weighted graph where we only have access to predictions of the actual stochastic weights, based on an underlying context. If the predictor is the Bayes optimal one, then computing the best matching based on the predicted weights is optimal. However, in practice, this perfect information scenario is not realistic. Given an imperfect predictor, a suboptimal decision rule may compensate for the induced error and thus outperform the standard optimal rule. In this paper, we propose multicalibration as a way to address this problem. This fairness notion requires a predictor to be unbiased on each element of a family of protected sets of contexts. Given a class of matching algorithms and any predictor of the edge-weights, we show how to construct a specific multicalibrated predictor , with the following property. Picking the best matching based on the output of is competitive with the best decision rule in applied onto the original predictor . We complement this result by providing sample complexity bounds.

Paper Structure

This paper contains 14 sections, 2 theorems, 12 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Beyond Matching.
  4. Technical Challenges.
  5. Related Work
  6. Preliminaries
  7. Learning Matchings with Multicalibrated Predictions
  8. Optimization Guarantees
  9. Sample Complexity Analysis
  10. Discussion of the sample complexity.
  11. Other Models
  12. Finding the best action.
  13. Learning with Rejection.
  14. Max-Weight Base.

Key Result

Theorem 1

Let $\varepsilon \in (0,1)$ be any fixed precision and set the multicalibration parameter $\alpha =\varepsilon/2m$. If $\hat{\gamma}$ is $(\mathcal{W}, \alpha)$-MC, it holds that:

Figures (1)

  • Figure 1: Proposed pipeline to obtain a matching: (1) the initial predictor $\gamma$ is post-processed to a multicalibrated $\hat{\gamma}$, (2) the final matching is chosen according to $c^{\star}$.

Theorems & Definitions (6)

  • Definition 1: Weighted Multicalibrated predictor
  • Theorem 1
  • proof
  • Definition 2: CHECK function
  • Theorem 2
  • proof