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REVIEW 3 major objections 5 minor 18 references

Prediction via Shapley Value Regression

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A single trained network can simultaneously produce accurate predictions and exact Shapley-value explanations, because the prediction is defined as the sum of the Shapley outputs.

desk verdict The joint predictor-and-explainer idea is genuinely new and the experiments are solid, but Theorem 3.4 overclaims for the default link-equipped models and the explanation benchmark is partly self-referential. read the letter →

arxiv 2505.04775 v2 pith:Z4BVGFX4 submitted 2025-05-07 cs.LG

Amr Alkhatib , Roman Bresson , Henrik Boström , Michalis Vazirgiannis This is my paper · ORCID
classification cs.LG
keywords ShapleyvaluesexplainableAIKolmogorov-ArnoldNetworkspredictionbysummationfeatureattributionFastSHAPtabulardataCIFAR-10
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 proposes ViaSHAP, a training scheme in which a single network outputs a matrix $\phi_{\mathrm{Via}}(x;\theta)$ of per-feature attributions and the prediction is defined by summing those attributions column-wise. This turns the Shapley explanation into the prediction itself, so no post-hoc explainer is needed at inference time. The paper claims that at a global optimum of its dual objective, the learned attributions are exactly the Shapley values of the model's own pre-link predictions, and that the KAN-based implementation is competitive with XGBoost and Random Forests on 25 tabular datasets while producing more accurate Shapley-value approximations than FastSHAP on both tabular data and CIFAR-10. A reader should care because interpretability usually costs extra compute or accuracy; here explanation and prediction are the same artifact, and the reported results suggest that cost can be largely avoided.

What carries the argument

The load-bearing object is the attribution matrix $\phi_{\mathrm{Via}}(x;\theta)$ together with the additive identity $1^\top \phi_{\mathrm{Via}}(x;\theta)$ that defines the prediction before the link function. Since Shapley values are the unique additive attribution satisfying local accuracy, missingness, and consistency, the argument reduces to showing the trained $\phi_{\mathrm{Via}}$ satisfies those three properties at the global optimum; the Shapley-weighted least-squares loss in equation (6) is the device that enforces them. The Kolmogorov-Arnold Network (KAN) implementation, built from learnable univariate spline functions, is the architecture that makes the tabular and image results work in practice, while the MLP variant rests on the universal approximation theorem.

What would settle it

Train ViaSHAP on a synthetic linear model with independent features, where the Shapley value of each feature under baseline removal is known in closed form, and compare the learned $\phi_{\mathrm{Via}}$ to those values; if, after convergence, the attributions do not satisfy efficiency and match the closed-form values within the paper's bound $2\sqrt{L_\phi(\theta)}$, the global-optimum guarantee is not being attained by gradient training.

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Extended reading notes

Core claim

ViaSHAP trains $\phi_{\mathrm{Via}}: \mathcal{X}\to\mathbb{R}^{n\times d}$ so that the model's prediction is $y=\sigma(1^\top \phi_{\mathrm{Via}}(x;\theta))$, with $\sigma$ a link function. The training loss is the prediction loss plus a scaled Shapley-weighted least-squares term that matches the prediction on a masked input $x_S$ against the sum $1_S^\top \phi_{\mathrm{Via}}(x;\theta)$ of the attributions for the features in $S$. The paper proves three properties of the global minimizer---local accuracy, missingness, and consistency---and then invokes Young's uniqueness theorem to conclude in Theorem 3.4 that $\phi_{\mathrm{Via}}(x;\theta^*)$ computes the exact Shapley values of ViaSHAP's predictions. The exactness is established for the raw additive sum; the default classifiers place the sigmoid or softmax after the sum, so the theorem's guarantee applies to the pre-link logits. Empirically, the Kolmogorov-Arnold implementation ranks first among the compared predictors on tabular AUC, is statistically indistinguishable from XGBoost, exceeds 0.96 AUC on CIFAR-10, and its explanations are closer to converged KernelSHAP values than FastSHAP's are.

Load-bearing premise

The exact-Shapley theorem assumes the training loss reaches a global optimum and that the prediction is the raw additive sum; the reported classifiers apply a sigmoid or softmax after the sum, so the guarantee applies to the pre-link logits, and the explanation benchmark additionally takes KernelSHAP's values for those logits as the ground truth.

Editorial extensions

If this is right

  • A single forward pass returns the prediction and its full Shapley explanation, cutting explanation latency from seconds per thousand instances with KernelSHAP to milliseconds with ViaSHAP.
  • On the 25 tabular datasets tested, KANVia's AUC ranking is statistically indistinguishable from XGBoost and Random Forests, so Shapley-clean explanations need not come at the cost of accuracy.
  • On CIFAR-10, ViaSHAP variants exceed 0.96 AUC, and their top-feature inclusion/exclusion curves show more faithful attributions than FastSHAP applied to the same models.
  • Because the prediction is the sum of the attributions, local accuracy holds by construction, so the explanation cannot drift from the model as post-hoc explainers can.
  • The Shapley regression objective can improve prediction itself: the same KAN architecture trained with the Shapley loss significantly outperforms the architecture trained without it.

Reading between the lines

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

  • Because the exact-Shapley theorem covers the raw additive output, a natural extension is to define the game on the probability output after the link function and train with a corresponding objective; the paper leaves this unexplored.
  • Since prediction and explanation share the same parameters, an adversarial perturbation that changes the explanation must also change the prediction; this suggests concrete adversarial-attack and explanation-robustness tests the paper does not run.
  • The empirical ground truth is KernelSHAP's Shapley values of ViaSHAP's own pre-link outputs, so the explanation benchmark measures internal consistency with the model rather than alignment with human judgement; a user study would be a separate test.
  • A direct finite-sample check is to train ViaSHAP on a linear model with independent features, where closed-form Shapley values are known, and verify the learned attributions against them; this would expose how far the global-optimum guarantee holds under gradient training.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes ViaSHAP, a method that trains a neural network to output both predictions and Shapley-value attributions in a single forward pass. The model produces an n×d matrix of attributions φVia(x;θ); predictions are formed as the column sum 1^T φVia(x;θ), optionally transformed by a link function (sigmoid/softmax). Training uses a dual objective (Eq. 7) that combines a weighted least-squares Shapley loss (Eq. 6) with a prediction loss. The authors prove (Lemmas 3.1–3.3 and Theorem 3.4) that a global minimizer of the Shapley loss computes exact Shapley values of the ViaSHAP prediction, provide epsilon-bounded extensions in Appendices B and C, and evaluate four tabular implementations (KANVia, KANVia-rho, MLPVia, MLPVia-theta) on 25 datasets plus image variants on CIFAR-10. They report competitive tabular AUC versus XGBoost and higher similarity of explanations to KernelSHAP ground truth than FastSHAP.

Significance. If the exact-Shapley claim is established for the actual deployed models, ViaSHAP is a notable contribution: it eliminates post-hoc explanation cost and offers an internal Shapley consistency property. The paper's strengths include a large-scale empirical setup (25 datasets, five seeds), public code, and a careful ablation study covering the link function, the efficiency constraint, and the β hyperparameter. The epsilon-bounded results in Appendices B and C are a useful step toward practice. However, the central theoretical claim currently applies only to the no-link, idealized optimum, so the significance of the work hinges on the revisions described below.

major comments (3)
  1. [§3.1, Eq. (6), Theorem 3.4, Appendix J.3] Theorem 3.4 states that φVia(x;θ*) computes the exact Shapley values of the predictions of ViaSHAP(x), but the proof (Lemma 3.1, Appendix A) relies on the identity ViaSHAP(x) = 1^T φVia(x;θ). Section 3.1 explicitly allows a link function σ, and the default experimental configuration applies sigmoid/softmax, as confirmed by Appendix J.3, which compares 'KANVia (default settings)' with 'KANVia without a link function'. Under a nonlinear link, the prediction is σ(1^T φVia(x;θ)), so local accuracy fails: 1^T φVia(x;θ) ≠ σ(1^T φVia(x;θ)). Since Shapley values are not equivariant under monotone transformations, φVia is not the Shapley value of the game v(S) = σ(1^T φVia(x_S;θ)). Consequently, the exact-Shapley guarantee does not cover the models whose predictive performance is reported in Table 1. The paper's own Table 12 shows that the default link-equipped models have markedly lower cosine similarity and R² to the KernelSHAP ground truth than the no-link variants, which is consistent with this gap. Please either present the no-link variant as the main method and state Theorem 3.4 for that setting, or provide a separate analysis for the link case and clarify that the guarantee applies to the pre-link logits only.
  2. [§3.2, Eq. (7), Theorem 3.4] The lemmas and Theorem 3.4 are proved for a global minimizer of the Shapley loss L_φ(θ) in Eq. (6), but training uses the dual objective L(θ) in Eq. (7), which adds a prediction-loss term with no guarantee that its optimum coincides with a minimizer of Eq. (6). A model that minimizes Eq. (7) may not satisfy efficiency, missingness, or consistency exactly, and the paper does not quantify the gap between the two optima. Please state the theoretical claim in terms of the actual training objective, or present Theorem 3.4 as an idealized result and add an analysis (theoretical or empirical) of the effect of the prediction term on the Shapley properties.
  3. [§4.3, Eq. (6), Tables 2–3] The explanation evaluation computes ground truth by running unbiased KernelSHAP on ViaSHAP's own predictions, but Eq. (6) minimizes the same weighted-least-squares objective for which KernelSHAP is an estimator (Section 2.3, Eqs. (3)–(4)). High cosine similarity to this ground truth therefore partly reflects agreement with the training objective rather than an independent validation of the faithfulness of the explanations. For the no-link variant, the reported R² values approach 1.0 on many datasets (Table 12), which may largely follow from construction. A non-self-referential evaluation would use a separate black-box model (e.g., an independently trained XGBoost) as the explained model and compare ViaSHAP's attributions to exact Shapley values of that model, or report removal-based fidelity curves as in the image experiments (Section 4.4).
minor comments (5)
  1. [Appendix B] The final step of the epsilon-bounded missingness proof writes |φVia_i(x;θ)| ≤ 2L_φ(θ) after defining the loss as attaining value ϵ²; since L_φ(θ) = ϵ², the correct bound is 2√L_φ(θ). The conclusion is unaffected, but the displayed inequality is dimensionally inconsistent.
  2. [Abstract and §3.1] The abstract states that predictions are 'derived directly by summation,' which is inconsistent with the default use of a sigmoid/softmax link function; please align the wording with the actual default model.
  3. [Table 5] The column headers 'Inclusion AUC' and 'Exclusion AUC' are described as being computed using top-1 accuracy; please clarify whether these are areas under the inclusion/exclusion curves or simply the top-1 accuracy values at a fixed threshold, since the numeric range is consistent with accuracy rather than AUC.
  4. [§4.5] The phrase 'Appendex J' should be 'Appendix J'; additionally, the caption of Table 3 omits MLPVia-theta although the table includes four columns.
  5. [Appendix G, Eq. (10)] The displayed formula for the relaxed loss appears to have an unbalanced parenthesis in the notation; please check the bracket structure so that the argument of the outer sum is unambiguous.

Circularity Check

2 steps flagged · score 6.0 of 10

The explanation benchmark is self-referential (KernelSHAP ground truth is the optimum of the same loss used to train ViaSHAP), and Theorem 3.4's exact-Shapley guarantee holds only for the raw-sum definition, not for the default link-equipped models.

  1. fitted input called prediction [Section 4.3 (Explainability Evaluation); Eq. (6) vs. Eq. (3)]
    "The explainability of the various ViaSHAP implementations is evaluated by measuring the similarity of ViaSHAP's Shapley values (ϕVia(x;θ) to the ground truth Shapley values (ϕ), computed by the unbiased KernelSHAP ... taking ViaSHAP as the black-box model. ... Lϕ(θ)=Σ_x Σ_j E_p(S)[(ViaSHAP_j(x_S)−ViaSHAP_j(0)−1_S^T ϕVia_j(x;θ))^2]."

    KernelSHAP's objective (3) is argmin_ϕ E_p(S)[(v_x(S)−v_x(0)−1_S^T ϕ)^2]; with v_x := ViaSHAP this is exactly the Shapley-loss term in (6) that trains ϕVia. The 'ground truth' used in Section 4.3 is therefore the minimizer of the same weighted least-squares objective the model is trained on. High cosine similarity, Spearman rank, and R² to this target largely measure how well the training loss (6) was minimized rather than agreement with an independent explanation standard. The reduction is partial because (7) adds a prediction-loss term and finite-sample effects exist, but the explanation-accuracy claim is self-referential by construction.

  2. self definitional [Section 3.1, Lemma 3.1, Theorem 3.4; Appendix A]
    "ViaSHAP(x) = 1⊤ϕVia(x;θ) i.e., summing column-wise. A link function σ can be applied to accommodate a valid range of outputs (y =σ(1⊤ϕVia(x;θ)) ... By definition of ViaSHAP: ViaSHAP(x) = 1⊤ϕVia(x;θ) ... This is the definition of local accuracy."

    Lemma 3.1 and hence Theorem 3.4 establish 'exact Shapley values of the predictions of ViaSHAP(x)' by defining the prediction to be the raw sum 1^T ϕVia(x;θ). However, Section 3.1 explicitly allows, and the default models use, a link σ such as sigmoid or softmax; ablation J.3 compares 'KANVia (default settings)' against 'KANVia without a link function', confirming the reported models apply the link. For linked models the prediction is σ(1^T ϕVia), so local accuracy 1^T ϕVia = σ(1^T ϕVia) fails, and Shapley values are not equivariant under monotone transformations. The exactness result is thus true by the raw-sum definition only and does not cover the default models whose predictive accuracy is reported; it is not derived for the actual prediction function.

full rationale

The predictive-performance comparison against XGBoost, Random Forests, and TabNet is independent and not circular: it evaluates AUC on test data against externally trained baselines, and the KAN-vs-MLP comparisons are also external. No load-bearing self-citation chain is used; the Shapley uniqueness argument invokes Young (1985) and Lundberg & Lee (2017), which are external standard results. The circularity is concentrated in the explanation-accuracy claim. ViaSHAP trains ϕVia by minimizing the Shapley-kernel-weighted squared loss (6), and the 'ground truth' in Section 4.3 is the unbiased KernelSHAP solution of exactly the same objective applied to ViaSHAP itself; consequently the reported similarity metrics partially measure in-sample fit of the training objective rather than agreement with an external explanation benchmark. Additionally, Theorem 3.4's proof defines local accuracy as 1^T ϕVia(x;θ), while the default classifiers apply sigmoid/softmax after the sum, so the exact-Shapley guarantee is valid only for the pre-link logits and is true by definition for that object, not for the probability predictions whose performance is reported. These two issues make the central explanation claims partially circular, giving a score of 6 rather than higher because the predictive-performance contribution retains independent empirical content.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central method rests on standard Shapley theory plus realizability assumptions for neural networks. No invented entities are introduced. The Shapley loss targets an internal consistency property rather than an externally testable quantity, which is the main caveat.

free parameters (4)
  • beta = 10
    Scaling hyperparameter for the Shapley loss in Eq (7), chosen by quasi-random search; it controls the trade-off between prediction loss and Shapley consistency.
  • number of sampled coalitions per instance = 32 (tabular), 4 (images)
    The number of coalitions used in the Shapley loss; chosen by the authors, with an ablation showing low sensitivity.
  • KAN architecture widths = 64-128-64 hidden layers
    KAN and MLP layer widths were chosen by the authors; the MLP variant was later widened to match KAN parameter counts.
  • baseline vector for removal = zero vector after normalization (approximate training mean)
    The baseline removal value function is defined with a fixed baseline, and Shapley values are baseline-dependent; the choice affects all explanation results.
assumptions (5)
  • standard math Shapley values are the unique additive attribution satisfying efficiency, missingness, and consistency (Young 1985).
    Invoked in Section 2.1 and used to turn Lemmas 3.1 to 3.3 into Theorem 3.4.
  • domain assumption The chosen neural network families (MLP and KAN) can represent the target Shapley value function and can reach a global minimizer of the non-convex loss (6).
    Theorem 3.4 is about the global optimizer; the empirical success requires that Stochastic Gradient Descent finds a near-optimal solution in the chosen architecture.
  • domain assumption Baseline removal with a fixed baseline is the appropriate value function for defining feature coalitions.
    Used in the default configuration; Appendix H shows that marginal expectations perform much worse, so this choice is load-bearing.
  • domain assumption Unbiased KernelSHAP converges to exact Shapley values and can serve as ground truth.
    The explanation evaluation relies on KernelSHAP as ground truth, cited to Covert and Lee 2021 and Jethani et al. 2022.
  • standard math Universal approximation and Kolmogorov-Arnold representation theorems justify the use of MLP and KAN architectures.
    Section 3.3 motivates the two implementations, but the theorems are existence results and do not guarantee trainability.

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Pith. "Pith review of Prediction via Shapley Value Regression." pith.science (2026). https://pith.science/paper/Z4BVGFX4

@misc{pith2026250504775,
  author       = {Pith},
  title        = {Pith review of: Prediction via Shapley Value Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4BVGFX4}},
  note         = {Machine review of arXiv:2505.04775}
}
read the original abstract

Shapley values have several desirable, theoretically well-supported, properties for explaining black-box model predictions. Traditionally, Shapley values are computed post-hoc, leading to additional computational cost at inference time. To overcome this, a novel method, called ViaSHAP, is proposed, that learns a function to compute Shapley values, from which the predictions can be derived directly by summation. Two approaches to implement the proposed method are explored; one based on the universal approximation theorem and the other on the Kolmogorov-Arnold representation theorem. Results from a large-scale empirical investigation are presented, showing that ViaSHAP using Kolmogorov-Arnold Networks performs on par with state-of-the-art algorithms for tabular data. It is also shown that the explanations of ViaSHAP are significantly more accurate than the popular approximator FastSHAP on both tabular data and images.

Figures

Figures reproduced from arXiv: 2505.04775 by the authors.

Figure 1
Figure 1. ViaSHAP generates predictions by first estimating the Shapley values, whose summation produces the final outcome. ViaSHAP computes the Shapley values prior to each predic￾tion formulation, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The optimization of ViaSHAP is conducted using a dual-objective loss function that aims to learn an optimal function for generating the Shapley values while minimizing the prediction loss. a continuous mapping function from the input dataset to the true targets, which also applies to the true Shapley values as a continuous function. (Liu et al., 2024) recently proposed Kolmogorov–Arnold Networks (KAN), as an alterna… view at source ↗
Figure 4
Figure 4. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (17 more)
Figure 3
Figure 3. Figure 3: The average rank of the 7 predictors on the 25 datasets with respect to the AUC (the lower rank is better). The critical difference (CD) is the largest statistically insignificant difference. 4.3. Explainability Evaluation The explainability of the various ViaSHAP impl…
Figure 5
Figure 5. Figure 5: The explanations for the predicted class generated by ViaSHAP and FastSHAP using two randomly selected images from the CIFAR-10 dataset. 4.5. Ablation Study The ablation study was conducted after the empirical evalua￾tion to ensure that no prior knowledge of the data o…
Figure 6
Figure 6. Figure 6: The average rank of KANVia , KANVia ϱ , MLPVia, and MLPVia θ on the 25 datasets with respect to the Spearman correlation between the ground truth Shapley values and the values obtained from the compared models. A lower rank is better and the critical difference (CD) re…
Figure 7
Figure 7. Figure 7: The explanations generated by ViaSHAP models and FastSHAP using two randomly selected images from the CIFAR-10 dataset. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: The inclusion and exclusion curves of ViaSHAP implementations as well as their FastSHAP explainers. We show how the top-1 accuracy of the predictive model changes as we exclude or include an increasing share of the important features, where the important features are d…
Figure 9
Figure 9. Figure 9: The explanations of ResNet18Via for 10 randomly selected predictions on the CIFAR-10 dataset. Each column corresponds to a CIFAR-10 class, and the predicted probability by ResNet18Via displayed beneath each image. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: The effect of different values of β on the predictive performance (AUC), alignment with the true Shapley values (cosine similarity), and the similarity in the order of features to the ground truth (Spearman rank). 27 [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: The effect of different values of β on the predictive performance (AUC), alignment with the true Shapley values (cosine similarity), and the similarity in the order of features to the ground truth (Spearman rank). J.2. The Number of Samples We assessed the impact of t…
Figure 12
Figure 12. Figure 12: The effect of different number of samples on the predictive performance (AUC), alignment with the true Shapley values (cosine similarity), and the similarity in the order of features to the ground truth (Spearman rank). 28 [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: The effect of different number of samples on the predictive performance (AUC), alignment with the true Shapley values (cosine similarity), and the similarity in the order of features to the ground truth (Spearman rank). 29 [PITH_FULL_IMAGE:figures/full_fig_p029_13.png]
Figure 14
Figure 14. Figure 14: The effect of β value on the progress of the training and the validation loss values. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_14.png]
Figure 15
Figure 15. Figure 15: The effect of β value on the progress of the training and the validation loss values. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_15.png]
Figure 16
Figure 16. Figure 16: The effect of β value on the progress of the training and the validation loss values. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_16.png]
Figure 17
Figure 17. Figure 17: The effect of β value on the progress of the training and the validation loss values. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_17.png]
Figure 18
Figure 18. Figure 18: The effect of β value on the progress of the training and the validation loss values. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_18.png]
Figure 19
Figure 19. Figure 19: The effect of β value on the progress of the training and the validation loss values. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_19.png]
Figure 20
Figure 20. Figure 20: The training time and prediction time on 1000 data instance of KANVia and MLPVia . 43 [PITH_FULL_IMAGE:figures/full_fig_p043_20.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.