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arxiv: 2605.22738 · v2 · pith:YDSIFA3Mnew · submitted 2026-05-21 · 💻 cs.LG · cs.AI· stat.ML

Proxy-Based Approximation of Shapley and Banzhaf Interactions

Santo M. A. R. Thies , Hubert Baniecki , R. Teal Witter , Eyke H\"ullermeier , Maximilian Muschalik , Fabian Fumagalli This is my paper

Pith reviewed 2026-05-25 06:23 UTC · model grok-4.3

classification 💻 cs.LG cs.AIstat.ML
keywords Shapley interactionsBanzhaf interactionsproxy modelsresidual correctionTreeSHAPinteraction approximationexplainable AI
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The pith

ProxySHAP combines tree-based proxy models with residual correction to approximate Shapley and Banzhaf interactions more accurately than prior estimators.

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

The paper introduces ProxySHAP to estimate higher-order Shapley and Banzhaf interactions that current methods trade off between speed and accuracy. It pairs the sample efficiency of tree-based proxy models with residual correction to reach consistency. A polynomial-time generalization of interventional TreeSHAP is derived to obtain exact interaction indices for tree ensembles without exponential tree-depth costs. The Maximum Sample Reuse strategy is analyzed to correct proxy bias without variance growing exponentially with interaction size. Benchmarking shows ProxySHAP records the lowest error across small- and large-budget regimes and outperforms ProxySPEX and KernelSHAP-IQ on downstream tasks.

Core claim

ProxySHAP reconciles the high sample efficiency of tree-based proxy models with a principled path to consistency via residual correction. It derives a polynomial-time generalization of interventional TreeSHAP to compute exact interaction indices for tree ensembles, bypassing exponential tree-depth dependencies, and formally characterizes conditions under which Maximum Sample Reuse corrects proxy bias without exponential variance scaling.

What carries the argument

ProxySHAP, which pairs tree-based proxy models with residual adjustment via Maximum Sample Reuse (MSR) to correct bias while preserving efficiency.

Load-bearing premise

The residual adjustment strategy corrects proxy bias without its variance scaling exponentially with interaction size under the conditions analyzed in the paper.

What would settle it

A test showing that ProxySHAP fails to achieve the lowest error versus ProxySPEX and KernelSHAP-IQ on models with thousands of features in either small- or large-budget regimes would disprove the superiority claim.

Figures

Figures reproduced from arXiv: 2605.22738 by Eyke H\"ullermeier, Fabian Fumagalli, Hubert Baniecki, Maximilian Muschalik, R. Teal Witter, Santo M. A. R. Thies.

Figure 1
Figure 1. Figure 1: Left: A ProxySHAP explanation of the SigLIP-2 model using only 2048 model calls. Right: In Phase 1, we fit a regression proxy model using sampled binary coalitions and game values. In Phase 2, we extract proxy interactions and, when appropriate, adjust them using residual estimates. sum of its marginal contributions across all possible subsets: ϕ p i (ν) := X T ⊆N\{i} pt(n)∆iν(T), (1) where ∆iν(T) := ν(T ∪… view at source ↗
Figure 2
Figure 2. Figure 2: Runtime improvement of extracting interactions using our Algo￾rithm 2 over Fourier-based extraction. Per-dataset speedups and the effect of tree depth on approximation quality are shown in Figures 13 and 14. Proposition 3.2 shows that the interactions of the tree proxy can be computed exactly by aggregating leaf-wise contributions. In particular, for a fixed interaction S, ex￾traction requires only a singl… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of ProxySHAP with and without MSR adjustment, measured by the MSE ratio. While MSR improves Shapley value approximation, it can degrade higher-order interaction estimates, as its variance scales as n k−1/|T | for interactions of order k (Theorem 3.3). This motivates the adjusted ProxySHAP estimator ϕˆProxySHAP S (ν; T ) = ϕ p S (ˆνT ) + ϕˆMSR S (ν − νˆT ; T ). While MSR often improves singleton … view at source ↗
Figure 4
Figure 4. Figure 4: Approximation quality (Relative MSE) for Shapley interactions of ProxySHAP across different configurations and state-of-the-art baselines. Additional results for Shapley and Banzhaf interactions on all 47 datasets can be found in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Relative MSE for pairwise Shapley interac￾tion approximation of ProxySHAP with HPO (top) and for large n (bottom). Further results in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Area between the insertion/deletion curves (AID) for explaining two CLIP ViT vari￾ants on the MS COCO dataset with ProxySHAP, ProxySPEX, and the FIxLIP baseline. Setup. Similarly to Baniecki et al. [2], we ana￾lyze the CLIP model in two vision transformer variants: ViT-32 and ViT-16. We explain a sample of 200 image–text pairs from the MS COCO dataset [33], which contain around 10– 30 text tokens per input… view at source ↗
Figure 7
Figure 7. Figure 7: Empirical variance scaling with sampling budget |T| for interaction orders |S| = k. For each order k, the plot shows the mean, minimum, and maximum empirical variance over all subsets S of size k. The black curve denotes the theoretical bound shape, namely proportional to ∥v∥ 2 ∞ log(n)/|T| for k = 1 and to ∥v∥ 2 ∞n k−1/|T| for k > 1. Since big-O bounds are defined only up to a multiplicative constant, the… view at source ↗
Figure 8
Figure 8. Figure 8: Faithfulness R2 for explaining CLIP (ViT-16) on the MS COCO dataset with ProxySHAP, ProxySPEX, and the FIxLIP baseline. D.2 XGBoost Default for Large Player Counts [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Ablation on approximating the cross-modal FIxLIP estimator. Faithfulness R2 for explaining two CLIP variants on MS COCO with ProxySHAP and the FIxLIP baseline. Motivated by this observation, we evaluate ProxySHAP (XGBoost+HPO-Informed) as an alternative default proxy for large-scale games. The results show that this configuration improves approximation quality in low-budget regimes and for games with many … view at source ↗
Figure 10
Figure 10. Figure 10: Approximation quality of two different XGBoost defaults. We show that using 2000 trees with a maximum depth of 3 improves estimation quality in low- to medium-budget regimes. low-budget regimes and for games with many players, where the standard XGBoost default may be insufficient to capture the relevant interaction structure. D.3 Runtime We evaluate runtime by translating model evaluations into wall-cloc… view at source ↗
Figure 11
Figure 11. Figure 11: Approximation quality as a function of runtime for second- and third-order interaction estimation across different per-evaluation cost regimes. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Ablations of sampling weights and residual approximators for ProxySHAP. We further investigate the effect of the resid￾ual approximator and sampling weights used in the adjustment step. Specifically, we compare SHAP-IQ [15] and KernelSHAP-IQ [16] as model-agnostic residual approxima￾tors. We also compare leverage weights, as used in LeverageSHAP [45], with KernelSHAP￾IQ weights [16]. As underlying games, … view at source ↗
Figure 13
Figure 13. Figure 13: Approximation quality (Relative MSE) of ProxySHAP and ProxySPEX using different maximum tree depth options across small, medium, and large player domains. Our method relies on the ability to efficiently extract exact cardinal-probabilistic interaction indices from the underlying tree-based model. We extend interventional TreeSHAP by Zern et al. [65] to extract the exact cardinal-probabilistic interaction … view at source ↗
Figure 14
Figure 14. Figure 14: Speedup of interventional extraction compared to Fourier extraction for extracting all interactions of order 1, 2, and 3 across different datasets. 41 [PITH_FULL_IMAGE:figures/full_fig_p041_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Predicted versus ground-truth normalized interaction values for different approximation methods and sampling budgets. Each point represents one interaction value from one dataset and one benchmark run; points closer to the diagonal indicate better agreement with the exact interaction values. Columns compare ProxySHAP, ProxySPEX, SHAPIQ, PermutationSamplingSII, and KernelSHAPIQ, while rows correspond to in… view at source ↗
Figure 16
Figure 16. Figure 16: ProxySHAP with disjoint coalition sets for proxy fitting and residual adjustment. drawn as the multiplicative interval [¯r/s, r¯ · s], corresponding to one standard deviation in log-space. Hence, values below one indicate that adjustment improves approximation quality, whereas values above one indicate that it deteriorates. For interaction indices, however, the effect of adjustment is more nuanced. While … view at source ↗
Figure 17
Figure 17. Figure 17: Selection of representative approximation curves for SII and BII at second- and third-order interactions. 45 [PITH_FULL_IMAGE:figures/full_fig_p045_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Winnermap comparing the best performing method for each dataset and budget for SII orders 2 and 3. Note that the HPO-Informed variants are considered only for datasets with more than 1000 features in this overview. 46 [PITH_FULL_IMAGE:figures/full_fig_p046_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Winnermap comparing the best performing method for each dataset and budget for BII orders 2 and 3. Note that the HPO-Informed variants are considered only for datasets with more than 1000 features in this overview. 47 [PITH_FULL_IMAGE:figures/full_fig_p047_19.png] view at source ↗
read the original abstract

Shapley and Banzhaf interactions capture the complex dynamics inherent in modern machine learning applications. However, current estimators for these higher-order interactions trade off between speed and accuracy. To overcome this limitation, we introduce ProxySHAP. ProxySHAP reconciles the high sample efficiency of tree-based proxy models with a principled path to consistency via residual correction. On a theoretical level, we derive a polynomial-time generalization of interventional TreeSHAP to compute exact interaction indices for tree ensembles, successfully bypassing exponential tree-depth dependencies in prior methods. Furthermore, we formally analyze the residual adjustment strategy, characterizing the specific conditions under which Maximum Sample Reuse (MSR) corrects proxy bias without its variance scaling exponentially with interaction size. Extensive benchmarking demonstrates that ProxySHAP sets a new state-of-the-art standard for approximation quality, including in large-scale applications with thousands of features. By achieving the lowest error in both small- and large-budget regimes, ProxySHAP significantly outperforms the prior best estimators ProxySPEX and KernelSHAP-IQ, while also delivering superior performance on downstream explainability tasks.

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

0 major / 3 minor

Summary. The manuscript introduces ProxySHAP for approximating Shapley and Banzhaf interaction indices. It derives a polynomial-time exact interventional TreeSHAP generalization for tree ensembles that avoids exponential depth dependence, proposes the Maximum Sample Reuse (MSR) residual correction strategy, and formally characterizes conditions under which MSR removes proxy bias without exponential variance growth in interaction order. Extensive experiments claim that ProxySHAP achieves the lowest approximation error across small- and large-budget regimes and outperforms ProxySPEX and KernelSHAP-IQ, with additional gains on downstream explainability tasks even for thousands of features.

Significance. If the polynomial-time exact proxy derivation and the variance-control analysis hold, the work supplies a practically useful advance in higher-order interaction estimation for tree-based models. The combination of an exact, efficient proxy with a theoretically grounded residual correction is a clear technical contribution, and the reported benchmarking, if reproducible, would support adoption in large-scale XAI applications.

minor comments (3)
  1. [§4.2] §4.2 (MSR algorithm): a short pseudocode block would make the sample-reuse logic and the exact bias-correction step easier to verify.
  2. [Table 2] Table 2 and Figure 4: standard deviations or error bars across the reported runs are missing; their addition would strengthen the SOTA claim.
  3. [§6] The paper would benefit from an explicit limitations paragraph addressing applicability outside tree ensembles and the sensitivity of MSR to the proxy-model quality.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of ProxySHAP, recognition of the polynomial-time exact interventional TreeSHAP generalization, and the variance-control analysis for MSR, as well as the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives a new polynomial-time generalization of interventional TreeSHAP for exact interaction indices on tree ensembles and provides a formal analysis of the Maximum Sample Reuse residual correction under explicitly stated conditions. These steps are presented as independent theoretical contributions rather than reductions to fitted parameters or prior self-citations. Benchmarking results are empirical comparisons against external baselines (ProxySPEX, KernelSHAP-IQ) and do not rely on any internal redefinition or self-referential prediction. The derivation chain remains self-contained with no load-bearing steps that collapse to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies insufficient detail to enumerate free parameters, axioms, or invented entities; the central claims rest on unstated modeling assumptions about tree ensembles and the validity of the residual-correction analysis.

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