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On the Impact of the Utility in Semivalue-based Data Valuation

Mélissa Tamine, Benjamin Heymann, Patrick Loiseau, Maxime Vono

TL;DR

The paper tackles robustness of semivalue-based data valuation to the choice of utility, a practical concern when utilities reflect downstream goals that may vary. It introduces a geometric framework via a dataset spatial signature, embedding data points into a low-dimensional space so that any linear combination of two base utilities corresponds to a projection, and defines a p-swap robustness metric $R_p$ that quantifies ranking stability as the utility direction varies on the unit circle $\mathcal{S}^1$. Empirical evaluations across multiple public datasets and semivalues (Shapley, $(4,1)$-Beta Shapley, Banzhaf) show that ranking stability as measured by $R_p$ aligns with Kendall rank-correlation results, with Banzhaf achieving higher robustness due to near-collinearity of the spatial embedding. The framework provides practitioners with a computationally efficient tool to assess whether semivalue-based data valuation offers reliable guidance under varying utility definitions and to anticipate potential retraining costs when utilities shift.

Abstract

Semivalue-based data valuation uses cooperative-game theory intuitions to assign each data point a value reflecting its contribution to a downstream task. Still, those values depend on the practitioner's choice of utility, raising the question: How robust is semivalue-based data valuation to changes in the utility? This issue is critical when the utility is set as a trade-off between several criteria and when practitioners must select among multiple equally valid utilities. We address it by introducing the notion of a dataset's spatial signature: given a semivalue, we embed each data point into a lower-dimensional space where any utility becomes a linear functional, making the data valuation framework amenable to a simpler geometric picture. Building on this, we propose a practical methodology centered on an explicit robustness metric that informs practitioners whether and by how much their data valuation results will shift as the utility changes. We validate this approach across diverse datasets and semivalues, demonstrating strong agreement with rank-correlation analyses and offering analytical insight into how choosing a semivalue can amplify or diminish robustness.

On the Impact of the Utility in Semivalue-based Data Valuation

TL;DR

The paper tackles robustness of semivalue-based data valuation to the choice of utility, a practical concern when utilities reflect downstream goals that may vary. It introduces a geometric framework via a dataset spatial signature, embedding data points into a low-dimensional space so that any linear combination of two base utilities corresponds to a projection, and defines a p-swap robustness metric that quantifies ranking stability as the utility direction varies on the unit circle . Empirical evaluations across multiple public datasets and semivalues (Shapley, -Beta Shapley, Banzhaf) show that ranking stability as measured by aligns with Kendall rank-correlation results, with Banzhaf achieving higher robustness due to near-collinearity of the spatial embedding. The framework provides practitioners with a computationally efficient tool to assess whether semivalue-based data valuation offers reliable guidance under varying utility definitions and to anticipate potential retraining costs when utilities shift.

Abstract

Semivalue-based data valuation uses cooperative-game theory intuitions to assign each data point a value reflecting its contribution to a downstream task. Still, those values depend on the practitioner's choice of utility, raising the question: How robust is semivalue-based data valuation to changes in the utility? This issue is critical when the utility is set as a trade-off between several criteria and when practitioners must select among multiple equally valid utilities. We address it by introducing the notion of a dataset's spatial signature: given a semivalue, we embed each data point into a lower-dimensional space where any utility becomes a linear functional, making the data valuation framework amenable to a simpler geometric picture. Building on this, we propose a practical methodology centered on an explicit robustness metric that informs practitioners whether and by how much their data valuation results will shift as the utility changes. We validate this approach across diverse datasets and semivalues, demonstrating strong agreement with rank-correlation analyses and offering analytical insight into how choosing a semivalue can amplify or diminish robustness.

Paper Structure

This paper contains 49 sections, 4 theorems, 70 equations, 11 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Related works.
  4. Notations.
  5. Background
  6. Utility functions.
  7. Semivalues.
  8. A methodology to assess data valuation robustness to the utility choice
  9. Utility trade-off scenario.
  10. Multiple-valid-utility scenario.
  11. A unified geometric modeling of the two scenarios
  12. A robustness metric derived from the geometric representation
  13. Link between robustness metric and rank correlation
  14. Empirical evaluation of robustness and discussion
  15. Conclusion
  16. ...and 34 more sections

Key Result

Proposition 3.1

Let $\mathcal{D}$ be any dataset of size $n$ and let $\omega \in \mathbb{R}^n$ be a semivalue weight vector. Then there exists a map $\psi_{\omega,\mathcal{D}}:\mathcal{D} \longrightarrow \mathbb{R}^2$ such that for every utility $u_\alpha=\alpha_1 u_1 +\alpha_2 u_2$, $\phi\bigl(z; \omega, u_\alpha\

Figures (11)

  • Figure 1: Spatial signature of the wind dataset for three semivalues (a) Shapley, (b) $(4,1)$-Beta Shapley, and (c) Banzhaf. Each cross marks the embedding $\psi_{\omega,\mathcal{D}}(z)$ of a data point (with $u_1=\lambda$, $u_2=\gamma$), the dashed circle is the unit circle $\mathcal{S}^1$, and the filled dot indicates one utility direction $\bar{\alpha}$.
  • Figure 2: Ranking regions induced by utilities on the unit circle $\mathcal{S}^1$ for two example spatial signatures. Each colored arc on the unit circle corresponds to one of the open arcs $A_k$. Within any single arc, the projection order (and hence the data‐point ranking) remains unchanged.
  • Figure 3: Mean $p$-robustness $R_p$ (error bars = standard errors over $5$ Monte Carlo approximations) plotted against $p \in \{500,1000,1500\}$ for each dataset and semivalue. Each plot corresponds to one dataset, with Shapley (blue), $(4,1)$-Beta Shapley (pink), and Banzhaf (green) curves. Higher $R_p$ indicates greater ranking stability under utility shifts.
  • Figure 4: Mean $r_j$ (error bars = standard errors over $5$ semivalue approximations) for breast (blue) and titanic (red) vs. coalition size $j$, with semivalue weights $\omega$ overlaid.
  • Figure 5: Spatial signature of the breast dataset for three semivalues (a) Shapley, (b) $(4,1)$-Beta Shapley, and (c) Banzhaf. Each cross marks the embedding $\psi_{\omega,\mathcal{D}}(z)$ of a data point (with $u_1=\lambda$, $u_2=\gamma$), the dashed circle is the unit circle $\mathcal{S}^1$, and the filled dot indicates one utility direction $\bar{\alpha}$.
  • ...and 6 more figures

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • Definition 3.2: Robustness metric $R_p$
  • proof
  • Proposition B.1: Extension of Proposition \ref{['claim:spatial-signature']} to $K \geq 2$ base utilities
  • proof
  • Definition B.2: Region of a hyperplane arrangement
  • Proposition B.3: Regions counts
  • ...and 8 more