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arxiv: 2606.02247 · v1 · pith:JHOQDMBDnew · submitted 2026-06-01 · 📊 stat.ML · cs.LG

ShaplEIG: Bayesian Experimental Design for Shapley Value Estimation

David Rundel , Fabian Fumagalli , Maximilian Muschalik , Bernd Bischl , Matthias Feurer This is my paper

Pith reviewed 2026-06-28 12:35 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords Shapley valuesBayesian experimental designGaussian processexpected information gainfeature attributiondata valuationmodel interpretabilitycoalition sampling
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The pith

ShaplEIG improves Shapley value estimation by adaptively selecting coalitions via closed-form expected information gain from a Gaussian process surrogate of the value function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Shapley values can be estimated more accurately when evaluations of the value function are expensive and strictly limited in number. It models the value function with a Gaussian process and chooses the next coalition to evaluate by maximizing the expected information gain about the Shapley values. Linearity of the Shapley values in the value function makes this information gain available in closed form, and the paper supplies a polynomial-time scheme using elementary symmetric polynomials to compute the required sums. Experiments on feature attribution, data valuation, and hyperparameter importance tasks demonstrate higher sample efficiency than non-adaptive baselines when the evaluation budget is small.

Core claim

By modeling the value function with a Gaussian process, ShaplEIG computes the expected information gain about the Shapley values in closed form due to their linearity, and selects coalitions to evaluate that maximize this gain, with an efficient polynomial-time scheme based on elementary symmetric polynomials, resulting in improved approximation accuracy under limited evaluation budgets.

What carries the argument

Expected information gain for Shapley values computed from a Gaussian process surrogate of the value function, made tractable by linearity and elementary symmetric polynomials.

If this is right

  • The method achieves higher accuracy with the same number of value function evaluations compared to non-adaptive sampling.
  • It applies to any setting where Shapley values are estimated from costly value function queries, such as feature attribution, data valuation, and hyperparameter importance.
  • The computational scheme scales polynomially rather than exponentially with the number of players.
  • Consistent improvements are observed across multiple diverse costly applications in the low-budget regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The linearity argument may extend the same closed-form treatment to other attribution scores that are linear functions of the value function.
  • Replacing the Gaussian process with a different surrogate could be tested while retaining the information-gain selection rule.
  • The polynomial reduction opens the door to applying the method to games with dozens of players that were previously intractable for information-gain methods.

Load-bearing premise

A Gaussian process surrogate of the value function is accurate enough that selecting coalitions by expected information gain yields better Shapley estimates than non-adaptive sampling.

What would settle it

An experiment on a real costly value function in which ShaplEIG selections produce higher Shapley estimation error than random or stratified sampling after the same small number of evaluations.

Figures

Figures reproduced from arXiv: 2606.02247 by Bernd Bischl, David Rundel, Fabian Fumagalli, Matthias Feurer, Maximilian Muschalik.

Figure 1
Figure 1. Figure 1: Mean squared error (MSE) between estimated and ground-truth Shapley values across all tasks and evaluation budgets, averaged over repetitions for ShaplEIG and the SV approximation baselines, with standard error of the mean (SEM) indicated. ShaplEIG (Ours) Regression MSR Leverage SHAP Kernel SHAP Permutation Sampling limited in these cases. Such budget constraints can also arise for repeated model evaluatio… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the behavior of EIG-based coalition selection given the initial design DT0+1 in symmetric and asymmetric games. The line width of the colored edges represents the magnitude of the covariance between coalition pairs according to the GP surrogate, while the size of the colored nodes represents the EIG scores of the candidate coalitions. one player, and due to identical lengthscales, differenc… view at source ↗
Figure 3
Figure 3. Figure 3: Mean squared error (MSE) between estimated and ground-truth Shapley values across all tasks and evaluation budgets, averaged over repetitions for ShaplEIG and the ablation baselines, with standard error of the mean (SEM) indicated. ShaplEIG (Ours) GP + Leverage Score Sampling GP + US GP + Random 40 [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean squared error (MSE) between estimated and ground-truth Shapley values across all tasks and evaluation budgets, averaged over repetitions for ShaplEIG and the SV approximation and ablation baselines, with standard error of the mean (SEM) indicated. ShaplEIG (Ours) Regression MSR Leverage SHAP Kernel SHAP Permutation Sampling GP + Leverage Score Sampling GP + US GP + Random 41 [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 5
Figure 5. Figure 5: Computational cost (in seconds) of GP hyperparameter fitting across all tasks and evaluation budgets for ShaplEIG, averaged over repetitions and with standard error of the mean (SEM) indicated. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Computational cost (in seconds) of vectorized EIG evaluation across all tasks and evaluation budgets for ShaplEIG, averaged over repetitions and with the standard error of the mean (SEM) indicated. 44 [PITH_FULL_IMAGE:figures/full_fig_p044_6.png] view at source ↗
read the original abstract

Shapley values are a principled attribution measure widely used in interpretable machine learning, but their exact computation scales exponentially with the number of players, motivating a wide range of approximation methods based on value function evaluations of sampled coalitions. This raises the question of whether approximation accuracy can be improved by adaptively selecting coalitions for evaluation based on previous evaluations. This is particularly relevant in settings where the value function is costly and the number of evaluations is severely limited, such as retraining-based feature importance, data valuation, and hyperparameter importance. For this purpose, we propose ShaplEIG, a Bayesian experimental design approach that approximates the expensive value function using a Gaussian process surrogate and adaptively selects coalitions based on their expected information gain about the Shapley values. By the linearity of the Shapley values in the value function, we show that the expected information gain is available in closed form. Furthermore, we propose an efficient computation scheme that reduces the complexity from exponential to polynomial in the number of players via elementary symmetric polynomials. In extensive experiments across diverse costly applications, our method consistently improves sample efficiency in the low-budget regime over state-of-the-art baselines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper introduces ShaplEIG, a Bayesian experimental design approach for approximating Shapley values under limited evaluations of a costly value function. It models the value function with a Gaussian process surrogate and adaptively selects coalitions to maximize expected information gain (EIG) about the Shapley values. The central technical claims are that linearity of Shapley values in the value function yields a closed-form EIG expression, and that elementary symmetric polynomials reduce the computational complexity from exponential to polynomial in the number of players. Experiments across costly applications are reported to show consistent gains in sample efficiency over baselines in the low-budget regime.

Significance. If the closed-form EIG derivation and the elementary-symmetric reduction hold without hidden approximations, the work provides a principled and scalable way to improve coalition selection for Shapley estimation when evaluations are expensive. The polynomial-time scheme is a concrete strength that addresses a practical barrier for larger player sets. The empirical demonstration of better low-budget performance, if supported by robust controls, would be relevant for feature attribution, data valuation, and hyperparameter importance tasks.

major comments (1)
  1. [Experiments (as summarized in abstract)] The headline empirical claim (consistent improvement via EIG-driven selection) is load-bearing on the assumption that the GP posterior on the value function produces predictive correlations that reliably rank coalitions better than non-adaptive designs. No calibration diagnostics, kernel sensitivity analysis, or ablation on the free GP hyperparameters are referenced, leaving open the possibility that reported gains depend on favorable surrogate mismatch rather than the EIG criterion itself.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the single major comment below and will strengthen the empirical section accordingly.

read point-by-point responses

  1. Referee: [Experiments (as summarized in abstract)] The headline empirical claim (consistent improvement via EIG-driven selection) is load-bearing on the assumption that the GP posterior on the value function produces predictive correlations that reliably rank coalitions better than non-adaptive designs. No calibration diagnostics, kernel sensitivity analysis, or ablation on the free GP hyperparameters are referenced, leaving open the possibility that reported gains depend on favorable surrogate mismatch rather than the EIG criterion itself.

    Authors: We agree that the current manuscript lacks explicit calibration diagnostics, kernel sensitivity checks, and hyperparameter ablations, which leaves the source of the reported gains open to the interpretation raised. In the revised manuscript we will add: (i) posterior predictive calibration diagnostics (e.g., coverage of 95% credible intervals on held-out coalitions and PIT histograms); (ii) results under both RBF and Matérn kernels; and (iii) an ablation table varying length-scale and signal variance over a grid of plausible values while keeping the EIG acquisition fixed. These additions will demonstrate that the ranking advantage of EIG persists across reasonable GP configurations. We note that all compared methods in the experiments already share the same GP surrogate, so any systematic mismatch would affect the baselines equally; the differential performance is therefore attributable to the choice of acquisition function. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained.

full rationale

The paper's core derivation states that linearity of Shapley values in the value function yields closed-form EIG, with polynomial reduction via elementary symmetric polynomials. This follows directly from standard properties of Shapley values and Gaussian processes without defining the target in terms of the output or fitting parameters that are then relabeled as predictions. No self-citation chains, ansatzes smuggled via prior work, or uniqueness theorems from the same authors are invoked as load-bearing steps in the provided text. The GP surrogate assumption is external and falsifiable via experiments, not a definitional loop. This is the normal case of an independent mathematical reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of Shapley values and Gaussian processes; no new entities are introduced. The Gaussian process hyperparameters are the primary free parameters.

free parameters (1)
  • Gaussian process kernel hyperparameters
    Fitted to observed value-function evaluations as part of standard GP surrogate modeling.
axioms (1)
  • standard math Shapley values are linear in the value function
    Invoked to obtain closed-form expected information gain.

pith-pipeline@v0.9.1-grok · 5744 in / 1247 out tokens · 29676 ms · 2026-06-28T12:35:30.727519+00:00 · methodology

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