VARSHAP: Addressing Global Dependency Problems in Explainable AI with Variance-Based Local Feature Attribution

Adam Karczmarz; Mateusz Gajewski; Miko{\l}aj Morzy; Piotr Sankowski

REVIEW 3 major objections 3 minor 25 references

VARSHAP: Addressing Global Dependency Problems in Explainable AI with Variance-Based Local Feature Attribution

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proposes VARSHAP, a Shapley-based attribution method in which a feature's local importance is the amount by which fixing its value reduces the variance of the model's prediction around the explained instance.

desk verdict Interesting local variance-based Shapley idea, but the central definition is self-contradictory (Eq. 1 vs. Appendix A.3), as the stress-test note correctly identifies. read the letter →

arxiv 2506.07229 v1 pith:E4LHN26H submitted 2025-06-08 cs.LG

Mateusz Gajewski , Miko{l}aj Morzy , Adam Karczmarz , Piotr Sankowski This is my paper

classification cs.LG

keywords featureattributionexplainableAIShapleyvaluesvariancereductionlocalexplanationsmodel-agnosticinterpretabilitydistributionshift

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that local feature attributions should be built from variance reduction rather than expected-value shifts. It defines VARSHAP, which treats the model's output variance under local perturbations as the game value in a Shapley value computation, so the attribution of each feature is its marginal contribution to lowering uncertainty about the prediction at the explained point. The paper also argues that variance is the only dispersion measure compatible with three natural axioms for attribution functions, and that this choice makes explanations insensitive to global distribution shifts that mislead SHAP. Experiments on synthetic and real datasets rank VARSHAP near the top among model-agnostic methods, ahead of KernelSHAP and LIME variants on faithfulness, robustness, and complexity metrics.

What carries the argument

The load-bearing object is the variance game $v(S) = \mathrm{Var}_{\Omega}(S)$, evaluated under a local Gaussian perturbation $\Pi(x) = \mathcal{N}(x, \mathrm{diag}(\alpha \hat{\sigma}_i^2))$ centered on the explained instance, with $\alpha$ controlling the size of the neighborhood. The Shapley kernel $\omega(|S|) = |S|!(k - |S| - 1)! / k!$ averages the marginal variance reductions over all coalitions, converting many local perturbation samples into a single axiom-compatible attribution vector.

What would settle it

On a synthetic dataset where $X_2 = X_1 + \text{noise}$ and $Y$ depends only on $X_1$, compute VARSHAP attributions with the diagonal Gaussian perturbation: any substantial nonzero attribution to $X_2$ would show that the independent-perturbation game is not modeling the real feature relationships at the explained point.

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Extended reading notes

Core claim

The central claim is that a feature matters locally to the extent that knowing its value reduces the variance of the model output when all other out-of-coalition features are perturbed around the explained instance. Formally, VARSHAP defines the attribution for feature $j$ as $\Phi_j(\Omega, \Pi, x) = \sum_{S \subseteq F \setminus \{j\}} \omega(|S|)\,(\mathrm{Var}_{\Omega}(S \cup \{j\}) - \mathrm{Var}_{\Omega}(S))$, where $\mathrm{Var}_{\Omega}(S)$ is the output variance under the local perturbation distribution with features in $S$ fixed. Because this is a Shapley value of the variance game, it satisfies efficiency, symmetry, and the null-player property, and for additive models the attribution of feature $i$ collapses to $w_i^2 \mathrm{Var}(X_i)$. The paper further proves, under axioms of shift invariance, sign independence, and additivity, that the squared deviation is the unique attribution function, which it presents as the theoretical justification for variance-based local importance.

Load-bearing premise

The method assumes that independent Gaussian noise scaled by each feature's global variance, centered on the explained point, faithfully represents the local neighborhood; if the true local structure is correlated, skewed, or non-Gaussian, the variance reductions are computed over unrealistic points and the attributions may mislead.

Editorial extensions

If this is right

Explanations for a fixed instance stay stable when the global data distribution changes far away, because only the local perturbation matters.
The attributions inherit Shapley's efficiency, symmetry, and null-player guarantees, including zero importance for features that do not affect local output variance.
For additive models, VARSHAP exactly recovers $w_i^2 \mathrm{Var}(X_i)$, giving a closed-form interpretation of attribution as each feature's share of output variance.
On the benchmark suite, VARSHAP with $\sigma = 0.6$ and $\sigma = 1.0$ ranks first among all compared methods, ahead of KernelSHAP and all LIME variants.
The uniqueness proof positions variance not as one heuristic choice, but as the only shift-invariant, sign-independent, additive dispersion measure with $d(1) = 1$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: replacing the diagonal Gaussian with a learned manifold or conditional perturbation would likely extend the same variance-Shapley machinery to correlated features, a direction the paper lists as future work.
Inference: the variance-reduction view suggests a direct calibration test: compare VARSHAP attributions to actual leave-one-feature-out variance drops at the explained point; the paper does not run this exact check.
Inference: for classification, output variance could be replaced by predictive entropy or interval width, yielding a variance-analog for non-regression targets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The core idea is worth taking seriously: define local Shapley attributions using the reduction in predictive variance under instance-centered Gaussian perturbations. That is a clean response to the global-dependence critique of SHAP, and the first case study — SHAP's attribution changing for a point whose local behavior is identical across datasets — is a nice illustration. The appendix also gives a tidy characterization result: variance is the only shift-invariant, zero-property, sign-independent, additive, normalized dispersion measure, and the proof is straightforward.

The problem is that the paper as written does not define the method consistently. Eq. (1) in the main text defines the marginal contribution as Var(S∪{j}) − Var(S). Appendix A.3, Eq. (36) uses the opposite sign, Var(S) − Var(S∪{j}), and goes on to derive positive attributions for linear models, whereas Section 2.3 states the linear-model attribution is −w_i² Var(X_i). The efficiency claim sums to −Var(∅), which only makes sense with the second convention. So the reader is left with two incompatible versions of the same quantity. That is not a stylistic quibble; it changes the sign and meaning of every attribution and every reported experimental result. The stress-test note is correct.

The other soft spots are secondary. 'Proposition 1' is framed as a proposal, not a theorem; the additivity axiom does most of the work and practically forces variance, so the axiomatic story is a characterization rather than an independent derivation. The experiments compare with KernelSHAP and LIME but not with SAGE or other variance-based Shapley methods, and the ranking advantage relies on grid-selected σ and fine-tuned metric hyperparameters, with no error bars or released code. That said, the experimental setup is not fraudulent; it just does not yet support the claimed superiority.

If I were the editor, I would not desk-reject this. The idea is original and the sign issue seems fixable — likely a propagated typo. But the current version is not publishable. It needs a corrected definition, a re-run of the experiments under the chosen convention, and stronger empirical discipline. I would send it to review only with the clear expectation of heavy revision; a referee would be right to reject this version.

Referee Report

3 major / 3 minor

Summary. The paper proposes VARSHAP, a model-agnostic local feature attribution method that defines the characteristic function of the Shapley game as the variance of the model output under local Gaussian perturbations of out-of-coalition features, with perturbation scale tied to the instance and global per-feature variances. The authors claim that the resulting attributions satisfy the Shapley axioms, are shift-invariant, are resilient to global distribution shifts, and outperform KernelSHAP and LIME on synthetic and benchmark tasks. The appendix provides an axiomatic derivation of variance as the unique local dispersion measure and a linearity result for additive models.

Significance. The local-variance-reduction viewpoint is a reasonable and potentially useful alternative to expectation-based SHAP, and if the consistency issues were resolved, the method could be a meaningful contribution to local XAI. The paper deserves credit for identifying a concrete failure mode of SHAP in the synthetic case studies and for evaluating the method with standard faithfulness, robustness, and complexity metrics. However, as written, the central definition is internally inconsistent, so the claimed theoretical guarantees and experimental results are not currently verifiable.

major comments (3)

[§2.3, Eq. (1); Appendix A.3, Eq. (36)] Equation (1) defines Φ_j = Σ_S ω(|S|)(Var(S∪{j}) − Var(S)), while Eq. (36) in Appendix A.3 defines Φ_j = Σ_S ω(|S|)(Var(S) − Var(S∪{j})). These are opposite definitions of the same quantity. The surrounding text is also split: §2.3 states that for a linear model the attribution is precisely −w_i² Var(X_i^Π), whereas A.3 Eq. (45) derives +w_j² Var(X_j). In the same section, the efficiency bullet says the sum of attributions equals Var(F)−Var(∅)=−Var(∅), yet it describes this as the total change in variance from the unknown state to the known state, which would have the opposite sign. Because the implemented sign cannot be inferred from the manuscript, the efficiency statement, the case-study interpretations, and the benchmark comparisons are not well-defined. The authors must adopt a single consistent convention, then update all prose, equations, and experimental claims accordingly.
[Appendix A.2, Proposition 1 and Theorem 1] The claimed uniqueness of variance is conditional on assumptions that essentially build variance into the result. The proof assumes from the outset that the characteristic function has the form v(S)=E[d(Ω−E[Ω])] and imposes the additivity axiom E[d(A+B)] = E[d(A)] + E[d(B)] for independent random variables A and B. The latter axiom is precisely the property that variance satisfies, so the theorem shows that among even, continuous, normalized, and additive dispersion functionals of centered outputs, the only choice is d(x)=x². It does not independently justify variance as the unique attribution metric from more basic desiderata. The manuscript should present these conditions as explicit definitional choices rather than as an independent axiomatic derivation.
[Section 4, Tables 1 and 2] The aggregated ranking in Table 1 is the main evidence for the claimed superiority, but the differences between VARSHAP (σ=0.6 and σ=1.0) and KernelShap (data sampling) are small, no confidence intervals or significance tests are reported, and the text says metric hyperparameters were 'fine-tuned to ensure an appropriate distribution of scores' (Section 4 and Appendix A.6). The ranking procedure is therefore not transparent enough to support a strong superiority claim. In addition, the perturbation scale σ is a configuration choice of VARSHAP; the paper should justify the selected values or demonstrate stability across a range rather than selecting them from the same benchmarks used for evaluation.

minor comments (3)

[Section 5] The limitation that independent Gaussian perturbations may not capture complex feature interactions is acknowledged, but because this assumption is load-bearing for the locality claim, the manuscript should explain more explicitly how it affects the synthetic and benchmark conclusions rather than leaving it only as future work.
[Section 3.1, Figures 1 and 2] The stability claims about VARSHAP and LIME in Case Study 1 are based on visual inspection of normalized attribution bars; providing numerical attribution values or a quantitative similarity measure would make the comparison verifiable.
[Notation and proofreading] There are several presentation issues: Figure 8's caption contains the typo 'moel', Table 2 repeats 'V ARSHAP (σ=0.6σ= 0.6σ=0.6)', Eq. (1) says 'k total features' while Appendix A.3 uses |F| for the same quantity, and the proof of Theorem 1 switches notation from d to f without explanation. A careful proofread and notation harmonization are needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: VARSHAP's derivation is an internal axiomatic characterization; the paper's sign inconsistency is a correctness defect, not a circular step.

full rationale

VARSHAP's derivation is self-contained in the circularity sense. Equation (1) explicitly defines the attribution as a Shapley value of the game v(S)=Var_Omega(S), so the claimed efficiency, symmetry, and null-player properties follow by construction from Shapley's theorem rather than by importing an unverified or self-cited result. Appendix A.2 contains a genuine functional-equation argument: from d(0)=0, d(-x)=d(x), E[d(A+B)]=E[d(A)]+E[d(B)] for independent zero-mean A,B, and d(1)=1, the proof derives d(x)=x^2. The additivity axiom is strong and can be viewed as chosen so that variance emerges, but the proof does not assume the conclusion, so this is a characterization theorem, not a circular reduction. The paper contains no self-citations; the impossibility result of Bilodeau et al. is external, and the benchmark comparisons are external and report the tested hyperparameter variants rather than hiding a fitted parameter. The serious defect is internal consistency, not circularity: Section 2.3 Eq. (1) defines Phi_j = sum omega(|S|)(Var(S union {j}) - Var(S)), while Appendix A.3 Eq. (36) defines the same quantity with the opposite sign, Var(S) - Var(S union {j}), and derives Phi_j = Var(Omega_j(X_j)) >= 0; Section 2.3 separately states that the linear-model attribution is exactly -w_i^2 Var(X_i). The abstract's 'variance reduction' narrative is therefore true only for one of the two definitions, and the method is undefined as written. That is a correctness/consistency flaw to be fixed, not an equivalence-by-construction or self-citation chain; hence the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the variance uniqueness theorem (which itself rests on posited axioms, especially additivity), on the Gaussian diagonal perturbation model, and on experimental choices (σ, metric hyperparameters) that are tuned rather than justified independently.

free parameters (2)

perturbation scale α/σ = σ = 0.3, 0.6, 1.0; best at 0.6 and 1.0
Controls how far the local Gaussian perturbation reaches; selected by grid search for best benchmark ranking, with no independent calibration or validation criterion.
evaluation metric hyperparameters = subset size 6, Lipschitz std 0.2, MaxSensitivity bound 0.02, EffectiveComplexity epsilon 0.05
The authors state these were fine-tuned to ensure an appropriate distribution of scores (Appendix A.6), so the ranking that supports the empirical claim is not a fixed, pre-specified protocol.

assumptions (5)

ad hoc to paper Additivity for independent random variables: E[d(A+B)] = E[d(A)] + E[d(B)]
Appendix A.2 posits this property to characterize d; it is essentially the defining property of variance, so the uniqueness result is built into the assumption.
ad hoc to paper The characteristic function v has the form E[d(Ω(x_S, X_{-S}) - E[Ω])]
Proposition 1 in Section 2.1 asserts this form without proof; it excludes other possible importance measures and is a modeling choice.
domain assumption Zero property and sign independence for d
Reasonable desiderata but still posited; together with additivity they force d(x)=x^2.
domain assumption Gaussian perturbation with diagonal covariance and E[X] = x
Section 2.2; the locality claim depends on this distribution, but it is acknowledged as a simplification in Section 5.
standard math Shapley axioms: efficiency, symmetry, null player, additivity
Standard Shapley value theory [19]; these properties themselves are not in question.

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Cite this review

Pith. "Pith review of VARSHAP: Addressing Global Dependency Problems in Explainable AI with Variance-Based Local Feature Attribution." pith.science (2026). https://pith.science/paper/E4LHN26H

@misc{pith2026250607229,
  author       = {Pith},
  title        = {Pith review of: VARSHAP: Addressing Global Dependency Problems in Explainable AI with Variance-Based Local Feature Attribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E4LHN26H}},
  note         = {Machine review of arXiv:2506.07229}
}

read the original abstract

Existing feature attribution methods like SHAP often suffer from global dependence, failing to capture true local model behavior. This paper introduces VARSHAP, a novel model-agnostic local feature attribution method which uses the reduction of prediction variance as the key importance metric of features. Building upon Shapley value framework, VARSHAP satisfies the key Shapley axioms, but, unlike SHAP, is resilient to global data distribution shifts. Experiments on synthetic and real-world datasets demonstrate that VARSHAP outperforms popular methods such as KernelSHAP or LIME, both quantitatively and qualitatively.

Figures

Figures reproduced from arXiv: 2506.07229 by the authors.

**Figure 2.** Feature attributions for GTMs trained on Dataset 1 (left) and Dataset 2 (right) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Neural Network Model This limitation is further confirmed by [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Feature attributions trained on Dataset 3 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 8.** Figure 8: Histogram for metric faithfulnesscorrelation for NN moel for Parkinson dataset [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: Decision surface visualization of NNMs trained on Dataset 1 (left) and Dataset 2 (right). [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: Decision surface visualization of GTMs trained on Dataset 1 (left) and Dataset 2 (right). [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Agarwal, N

C. Agarwal, N. Johnson, M. Pawelczyk, S. Krishna, E. Saxena, M. Zitnik, and H. Lakkaraju. Rethinking stability for attribution-based explanations.arXiv preprint arXiv:2203.06877, 2022

arXiv 2022

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Alvarez Melis and T

D. Alvarez Melis and T. Jaakkola. Towards robust interpretability with self-explaining neural networks. InAdvances in Neural Information Processing Systems, volume 31, 2018

work page 2018

[3] [3]

Bhatt, A

U. Bhatt, A. Weller, and J. M. Moura. Evaluating and aggregating feature-based model ex- planations. InProceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, pages 3016–3022. International Joint Conferences on Artificial Intelligence Or- ganization, 2020

work page 2020

[4] [4]

Bilodeau, N

B. Bilodeau, N. Jaques, P. W. Koh, and B. Kim. Impossibility theorems for feature attribution. Proceedings of the National Academy of Sciences, 121(2):e2304406120, 2024

work page 2024

[5] [5]

Chalasani, J

P. Chalasani, J. Chen, A. R. Chowdhury, X. Wu, and S. Jha. Concise explanations of neural networks using adversarial training. InProceedings of the 37th International Conference on Machine Learning, pages 1383–1391. PMLR, 2020

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[7] [7]

Guidotti, A

R. Guidotti, A. Monreale, S. Ruggieri, F. Turini, F. Giannotti, and D. Pedreschi. A survey of methods for explaining black box models.ACM computing surveys (CSUR), 51(5):1–42, 2018

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work page 2020