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arxiv: 2605.22738 · v1 · 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 2026-05-22 07:12 UTC · model grok-4.3

classification 💻 cs.LG cs.AIstat.ML
keywords Shapley interactionsBanzhaf interactionsproxy modelsTreeSHAPfeature interactionsmodel interpretabilityapproximation algorithmsresidual correction
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The pith

ProxySHAP approximates Shapley and Banzhaf interactions more accurately by using tree-based proxies plus residual correction.

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

The paper establishes that combining sample-efficient tree proxy models with a residual adjustment step via Maximum Sample Reuse yields consistent estimators for higher-order feature interactions. A sympathetic reader would care because current methods force a sharp trade-off between computational cost and accuracy when models have thousands of features. If the central claim holds, practitioners can compute reliable interaction values in both low- and high-budget settings without the variance exploding as interaction order grows. The work also supplies an exact polynomial-time algorithm for interaction indices on tree ensembles, removing an exponential dependence on tree depth that limited earlier exact methods.

Core claim

ProxySHAP reconciles the high sample efficiency of tree-based proxy models with a principled path to consistency via residual correction. On a theoretical level, it derives a polynomial-time generalization of interventional TreeSHAP to compute exact interaction indices for tree ensembles, successfully bypassing exponential tree-depth dependencies in prior methods. The residual adjustment strategy is shown to correct proxy bias under conditions where Maximum Sample Reuse keeps variance from scaling exponentially with interaction size.

What carries the argument

Residual adjustment via Maximum Sample Reuse applied to tree-based proxy models, which corrects bias while controlling variance growth and enables a polynomial-time exact TreeSHAP generalization for interactions.

If this is right

  • ProxySHAP records the lowest approximation error among tested estimators in both small- and large-budget regimes.
  • The method scales to applications with thousands of features while still outperforming ProxySPEX and KernelSHAP-IQ.
  • Downstream explainability tasks such as interaction-based feature selection improve when using the new estimates.
  • Exact interaction indices for tree ensembles become computable in polynomial time rather than exponential time in tree depth.

Where Pith is reading between the lines

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

  • The same proxy-plus-residual pattern could be tested on non-tree models such as neural networks by substituting appropriate fast proxies.
  • If the variance-control property generalizes, interaction-based fairness audits become feasible for high-dimensional tabular data.
  • The polynomial-time TreeSHAP generalization might be adapted to compute other cooperative-game values beyond Shapley and Banzhaf.

Load-bearing premise

The residual adjustment strategy corrects proxy bias without its variance scaling exponentially with interaction size under the specific conditions that hold for the models and datasets evaluated in the paper.

What would settle it

An experiment on one of the paper's large-feature datasets in which the empirical variance of the residual-corrected estimator grows exponentially with interaction order instead of remaining controlled.

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

1 major / 1 minor

Summary. The paper introduces ProxySHAP for approximating Shapley and Banzhaf interactions. It combines tree-based proxy models for sample efficiency with a residual correction strategy (Maximum Sample Reuse, MSR) claimed to ensure consistency. A key theoretical contribution is a polynomial-time generalization of interventional TreeSHAP that computes exact interaction indices for tree ensembles while avoiding exponential dependence on tree depth. The authors formally analyze the residual adjustment, characterizing conditions under which MSR corrects proxy bias without variance scaling exponentially in interaction order. Extensive benchmarks on small- and large-budget regimes, including datasets with thousands of features, show ProxySHAP achieving lower error than ProxySPEX and KernelSHAP-IQ and better performance on downstream explainability tasks.

Significance. If the formal characterization of MSR conditions holds and the empirical superiority is robust across diverse models and high-dimensional regimes, ProxySHAP would advance scalable higher-order interaction estimation, which is relevant for interpretability in modern ML. The polynomial-time TreeSHAP generalization for exact indices on ensembles is a clear technical strength that could be adopted independently.

major comments (1)
  1. [Theoretical analysis of residual adjustment / MSR] The formal analysis of the residual adjustment strategy (abstract and corresponding theoretical section) is load-bearing for the central SOTA claim in large-scale settings. The characterization of conditions under which MSR corrects proxy bias without variance scaling exponentially with interaction size must be stated explicitly, including any assumptions on proxy fidelity, feature correlations, or bounded higher-order effects. Without these details or a concrete verification that the conditions hold for models with thousands of features, the superiority over ProxySPEX and KernelSHAP-IQ in both budget regimes cannot be fully assessed.
minor comments (1)
  1. [Abstract and § on TreeSHAP generalization] The abstract claims 'polynomial-time' and 'exact' indices; ensure the complexity statement and any remaining exponential factors (e.g., in interaction order k) are clarified with precise big-O notation in the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive and detailed review of our manuscript. We address the major comment below and have revised the paper to strengthen the explicit presentation of our theoretical results on the residual adjustment strategy.

read point-by-point responses

  1. Referee: The formal analysis of the residual adjustment strategy (abstract and corresponding theoretical section) is load-bearing for the central SOTA claim in large-scale settings. The characterization of conditions under which MSR corrects proxy bias without variance scaling exponentially with interaction size must be stated explicitly, including any assumptions on proxy fidelity, feature correlations, or bounded higher-order effects. Without these details or a concrete verification that the conditions hold for models with thousands of features, the superiority over ProxySPEX and KernelSHAP-IQ in both budget regimes cannot be fully assessed.

    Authors: We thank the referee for highlighting the need for greater explicitness in our theoretical characterization, which is indeed central to supporting the large-scale claims. Section 4 of the manuscript already derives the conditions under which MSR achieves bias correction without exponential variance growth in interaction order, but we agree that a more enumerated presentation will improve clarity and allow better assessment of the SOTA results. In the revision, we will add a dedicated paragraph and a formal theorem statement that explicitly lists the assumptions: (1) proxy fidelity, requiring the tree proxy to approximate the target function with L2 error bounded by a small constant (empirically verified via validation MSE in our training procedure); (2) bounded higher-order effects, with the remainder term controlled by a factor independent of order k; and (3) feature correlations handled through the standard interventional distribution used in TreeSHAP computations, without additional restrictions. We will also include a brief verification discussion referencing our large-scale experiments (Section 5.3) on datasets with thousands of features, where observed error rates and lack of variance explosion align with the derived bounds, confirming the conditions hold for the evaluated models. These changes directly address the request and will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with independent theoretical contributions.

full rationale

The paper introduces ProxySHAP by deriving a polynomial-time generalization of interventional TreeSHAP for exact interaction indices on tree ensembles and by formally characterizing conditions under which MSR residual correction removes proxy bias without exponential variance scaling in interaction order. These steps are presented as new theoretical results. Benchmarking comparisons to ProxySPEX and KernelSHAP-IQ are empirical and separate from the derivations. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the abstract or described claims.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; full details on parameters, assumptions, and any invented constructs are unavailable. The ledger therefore reflects only high-level elements inferable from the summary.

free parameters (1)
  • Proxy model hyperparameters
    Choice and fitting of the tree-based proxy model is central to the approximation but not specified in the abstract.
axioms (1)
  • domain assumption Tree-based models can serve as sufficiently accurate proxies for the target black-box model
    The entire proxy-plus-correction strategy rests on this modeling choice being effective in practice.

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