arxiv: 2605.15154 · v1 · submitted 2026-05-14 · 📊 stat.ML · cs.LG

Recognition: 2 theorem links

· Lean Theorem

RoSHAP: A Distributional Framework and Robust Metric for Stable Feature Attribution

Lanxin Xiang , Liang Shi , Youhui Ye , Boyu Jiang , Dawei Zhou , Feng Guo

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:50 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords feature attributionSHAProbust metricdistributional frameworkbootstrapstabilitymachine learning interpretabilityfeature ranking

0 comments

The pith

RoSHAP summarizes the distribution of SHAP values to rank features by their activity, strength, and stability simultaneously.

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

The paper introduces a distributional framework to handle the stochastic variation in feature attribution methods like SHAP, which can change with different data splits or random seeds. It models these variations using bootstrap resampling and kernel density estimation. Under mild regularity conditions, the aggregated scores are shown to be asymptotically Gaussian, simplifying the estimation process. RoSHAP then condenses this distribution into a single metric for robust feature ranking. Experiments show it identifies important features more reliably and allows models to use fewer features without losing predictive power.

Core claim

The central discovery is that by modeling the full distribution of SHAP attribution scores rather than relying on point estimates, one can derive a robust ranking criterion called RoSHAP that accounts for feature activity, strength, and stability, leading to more consistent and reliable feature selections in machine learning models.

What carries the argument

The RoSHAP metric, which integrates the distributional properties of SHAP values estimated via bootstrap and kernel density estimation into a criterion that rewards active, strong, and stable features.

If this is right

RoSHAP-selected features enable models with substantially fewer predictors while maintaining comparable predictive performance.
The framework reduces the computational cost of estimating attribution distributions through the asymptotic Gaussianity result.
Feature rankings become more stable across different train-test splits and random seeds.
Signal features are identified more accurately than with standard single-run SHAP measures.

Where Pith is reading between the lines

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

Applying similar distributional approaches to other attribution techniques could improve their reliability in practice.
This might particularly benefit high-stakes applications where inconsistent explanations could lead to flawed decisions.
Future work could explore exact conditions under which the Gaussian approximation holds for specific model classes.

Load-bearing premise

The aggregated feature attribution scores are asymptotically Gaussian under mild regularity conditions on the data and model.

What would settle it

Observing bootstrap samples of aggregated SHAP scores that exhibit clear non-Gaussian behavior, such as skewness or multimodality, for standard datasets and models would challenge the asymptotic result.

Figures

Figures reproduced from arXiv: 2605.15154 by Boyu Jiang, Dawei Zhou, Feng Guo, Lanxin Xiang, Liang Shi, Youhui Ye.

**Figure 2.** Figure 2: Overall framework for distributional feature attribution estimation. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Empirical distributions of robust importance estimates over 1000 bootstrap runs with fitted [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Distribution of top 10 XGBoost feature contribution estimates for the Golub dataset [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Feature-selection performance on the Golub data using XGBoost. Methods are SHAP [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Distribution of top 15 CatBoost feature attribution estimates for the Musk dataset. The Musk experiment studies feature-level attribution using a CatBoost classifier [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: Feature-selection performance on the Musk (Version 2) data using CatBoost. Methods [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 8.** Figure 8: Distribution of top 15 LightGBM feature attribution estimates for the UJIIndoorLoc dataset. This experiment studies feature-level attribution using a LightGBM regression model [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗

**Figure 9.** Figure 9: Feature-selection performance on the UJIIndoorLoc data using LightGBM. Methods are [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 10.** Figure 10: CIFAR-10 ship example using 3 bootstrap rounds. The proposed framework provides additional insight into attribution patterns behind image predictions [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗

**Figure 11.** Figure 11: CIFAR-10 explanations after 50 ViT-base bootstrap runs. Columns show the original [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗

**Figure 12.** Figure 12: Golub gene selection performance comparison using CatBoost. Feature selection methods [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗

**Figure 13.** Figure 13: Golub gene selection performance comparison using Gradient Boosting. Feature selection [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗

**Figure 14.** Figure 14: Golub gene selection performance comparison using LightGBM. Feature selection [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗

**Figure 15.** Figure 15: Golub gene selection performance comparison using Logistic Regression. Feature [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗

**Figure 16.** Figure 16: Golub gene selection performance comparison using Random Forest. Feature selection [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗

**Figure 17.** Figure 17: Musk (Version 2) variable selection performance comparison using XGBoost. Feature [PITH_FULL_IMAGE:figures/full_fig_p015_17.png] view at source ↗

**Figure 18.** Figure 18: Musk (Version 2) variable selection performance comparison using Gradient Boosting. [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗

**Figure 19.** Figure 19: Musk (Version 2) variable selection performance comparison using LightGBM. Feature [PITH_FULL_IMAGE:figures/full_fig_p015_19.png] view at source ↗

**Figure 20.** Figure 20: Musk (Version 2) variable selection performance comparison using Logistic Regression. [PITH_FULL_IMAGE:figures/full_fig_p015_20.png] view at source ↗

**Figure 21.** Figure 21: Musk (Version 2) variable selection performance comparison using Random Forest. [PITH_FULL_IMAGE:figures/full_fig_p015_21.png] view at source ↗

**Figure 22.** Figure 22: UJIIndoorLoc variable selection performance comparison using CatBoost. Feature [PITH_FULL_IMAGE:figures/full_fig_p016_22.png] view at source ↗

**Figure 23.** Figure 23: UJIIndoorLoc variable selection performance comparison using Gradient Boosting. [PITH_FULL_IMAGE:figures/full_fig_p016_23.png] view at source ↗

**Figure 24.** Figure 24: UJIIndoorLoc variable selection performance comparison using XGBoost. Feature [PITH_FULL_IMAGE:figures/full_fig_p016_24.png] view at source ↗

**Figure 25.** Figure 25: UJIIndoorLoc variable selection performance comparison using Linear Regressio (Ridge). [PITH_FULL_IMAGE:figures/full_fig_p017_25.png] view at source ↗

**Figure 26.** Figure 26: Prediction and bootstrap results for image 8. [PITH_FULL_IMAGE:figures/full_fig_p017_26.png] view at source ↗

**Figure 27.** Figure 27: Prediction and bootstrap results for image 11. [PITH_FULL_IMAGE:figures/full_fig_p017_27.png] view at source ↗

**Figure 28.** Figure 28: Prediction and bootstrap results for image 20. [PITH_FULL_IMAGE:figures/full_fig_p017_28.png] view at source ↗

**Figure 29.** Figure 29: Prediction and bootstrap results for image 36. [PITH_FULL_IMAGE:figures/full_fig_p018_29.png] view at source ↗

**Figure 30.** Figure 30: Prediction and bootstrap results for image 41. [PITH_FULL_IMAGE:figures/full_fig_p018_30.png] view at source ↗

**Figure 31.** Figure 31: Prediction and bootstrap results for image 66. [PITH_FULL_IMAGE:figures/full_fig_p018_31.png] view at source ↗

**Figure 32.** Figure 32: Prediction and bootstrap results for image 85. [PITH_FULL_IMAGE:figures/full_fig_p018_32.png] view at source ↗

read the original abstract

Feature attribution analysis is critical for interpreting machine learning models and supporting reliable data-driven decisions. However, feature attribution measures often exhibit stochastic variation: different train--test splits, random seeds, or model-fitting procedures can produce substantially different attribution values and feature rankings. This paper proposes a framework for incorporating stochastic nature of feature attribution and a robust attribution metric, RoSHAP, for stable feature ranking based on the SHAP metric. The proposed framework models the distribution of feature attribution scores and estimates it through bootstrap resampling and kernel density estimation. We show that, under mild regularity conditions, the aggregated feature attribution score is asymptotically Gaussian, which greatly reduces the computational cost of distribution estimation. The RoSHAP summarizes the distribution of SHAP into a robust feature-ranking criterion that simultaneously rewards features that are active, strong, and stable. Through simulations and real-data experiments, the proposed framework and RoSHAP outperform standard single-run attribution measures in identifying signal features. In addition, models built using RoSHAP-selected features achieve predictive performance comparable to full-feature models while using substantially fewer predictors. The proposed RoSHAP approach improves the stability and interpretability of machine learning models, enabling reliable and consistent insights for analysis.

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.

Desk Editor's Note private letter to a colleague

RoSHAP gives a distributional view of SHAP values with a new ranking metric that rewards stability, and the experiments show practical gains in feature selection, but the asymptotic normality claim rests on unverified conditions for real ML models.

read the letter

This paper's main move is to treat SHAP attributions as having their own distribution rather than single numbers, estimate that distribution with bootstrap plus KDE, and then define RoSHAP as a summary that ranks features by a combination of activity, strength, and stability. The asymptotic Gaussianity result is offered as a shortcut that cuts the cost of full distribution estimation under mild regularity conditions. Experiments on simulations and real data indicate that RoSHAP picks signal features more reliably than single-run SHAP and supports simpler models with comparable predictive performance. That part is useful for applied settings where rankings shift across seeds or splits. The distributional framing and the specific RoSHAP criterion are new relative to standard SHAP work, and the empirical results give a concrete sense of the stability improvement. The soft spot is the central asymptotic claim. The regularity conditions are not listed or checked against the nonlinear functionals that arise in typical models like trees or neural nets, and there is no direct assessment of how KDE bandwidth selection holds up as feature count grows. The stress-test concern about Lindeberg-type conditions and high-dimensional behavior is reasonable and not addressed in the provided details. If the full proofs and checks are in the paper they need to be explicit; otherwise the computational advantage stays theoretical. This is for practitioners who need more stable feature rankings for model simplification or downstream decisions. A reader working with noisy attributions in applied ML would get value from trying the metric. I would send it to referees. The practical problem is real, the experiments are promising, and the gaps are fixable with more detail on the conditions and verification.

Referee Report

3 major / 3 minor

Summary. The paper proposes a distributional framework for SHAP-based feature attribution that models stochastic variation via bootstrap resampling and kernel density estimation. It establishes that the aggregated attribution score is asymptotically Gaussian under mild regularity conditions, thereby reducing the cost of full distribution estimation. The RoSHAP metric is introduced as a robust summary that ranks features by simultaneously rewarding activity, strength, and stability. Simulations and real-data experiments indicate that RoSHAP identifies signal features more reliably than single-run SHAP and yields models with comparable predictive performance using substantially fewer predictors.

Significance. If the asymptotic normality result is rigorously established and the regularity conditions hold for typical ML models, the framework would meaningfully improve the stability and reproducibility of feature attributions, an important practical concern in interpretability. The computational reduction via the Gaussian approximation and the RoSHAP ranking criterion are potentially useful contributions for feature selection tasks where consistency across resamples matters.

major comments (3)

[theoretical section on asymptotic normality] The central asymptotic Gaussianity claim (abstract and theoretical development) rests on unspecified 'mild regularity conditions.' Because SHAP values are nonlinear functionals of the fitted model, the manuscript must explicitly state the required smoothness, differentiability, or Lindeberg-type conditions and verify their plausibility for the tree-based and neural models used in the experiments; without this, the justification for replacing bootstrap+KDE with the Gaussian approximation remains incomplete.
[methodology / RoSHAP definition] The exact mathematical definition of RoSHAP is not provided with sufficient detail (methodology section). It is unclear how the distributional summary (e.g., moments or quantiles from the estimated density) is combined into the single ranking score that rewards activity, strength, and stability simultaneously; this definition is load-bearing for the claim that RoSHAP is a well-defined robust criterion.
[experiments and KDE implementation] The interaction between KDE bandwidth selection and the curse of dimensionality is not addressed (experiments and methodology). When the number of features grows, the bandwidth choice can dominate the quality of the density estimate; the paper should report sensitivity analyses or theoretical guidance on bandwidth scaling with dimension.

minor comments (3)

[abstract] The abstract and introduction contain minor grammatical inconsistencies (e.g., 'the aggregated feature attribution score is asymptotically Gaussian' appears without prior definition of the aggregation operator).
[experiments] Data preprocessing, exact train-test split procedures, and hyperparameter choices for the base models are insufficiently documented, hindering reproducibility of the reported performance gains.
[figures] Figure captions and axis labels in the simulation and real-data plots could be clarified to indicate whether the displayed distributions are bootstrap estimates or the fitted Gaussian approximations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which have helped clarify several aspects of the manuscript. We address each major comment point by point below.

read point-by-point responses

Referee: [theoretical section on asymptotic normality] The central asymptotic Gaussianity claim (abstract and theoretical development) rests on unspecified 'mild regularity conditions.' Because SHAP values are nonlinear functionals of the fitted model, the manuscript must explicitly state the required smoothness, differentiability, or Lindeberg-type conditions and verify their plausibility for the tree-based and neural models used in the experiments; without this, the justification for replacing bootstrap+KDE with the Gaussian approximation remains incomplete.

Authors: We agree that the regularity conditions should be stated explicitly rather than left as 'mild.' In the revised manuscript we will add a dedicated paragraph in the theoretical section that specifies the conditions: the SHAP functional must possess finite second moments and the model prediction map must be Lipschitz continuous with respect to small input perturbations. For tree-based models the finite number of leaves ensures the Lipschitz property holds; for neural networks we invoke standard bounded-gradient assumptions under typical training regimes. We will briefly verify plausibility against the specific models and datasets used in the experiments. revision: yes
Referee: [methodology / RoSHAP definition] The exact mathematical definition of RoSHAP is not provided with sufficient detail (methodology section). It is unclear how the distributional summary (e.g., moments or quantiles from the estimated density) is combined into the single ranking score that rewards activity, strength, and stability simultaneously; this definition is load-bearing for the claim that RoSHAP is a well-defined robust criterion.

Authors: We will supply the precise definition in the revised methodology section. RoSHAP for feature j is defined as E[|phi_j|] * P(phi_j != 0) / sqrt(Var(phi_j)), where the expectation, probability, and variance are taken with respect to the bootstrap distribution of the SHAP value phi_j. The first term captures strength, the second activity, and the third stability. This closed-form expression will be stated mathematically together with a short derivation showing how it arises from the estimated density. revision: yes
Referee: [experiments and KDE implementation] The interaction between KDE bandwidth selection and the curse of dimensionality is not addressed (experiments and methodology). When the number of features grows, the bandwidth choice can dominate the quality of the density estimate; the paper should report sensitivity analyses or theoretical guidance on bandwidth scaling with dimension.

Authors: We acknowledge that bandwidth selection becomes critical in higher dimensions. Our reported experiments use moderate-dimensional data where standard selectors (Silverman's rule) perform adequately. In the revision we will add a short sensitivity subsection that varies bandwidth by factors of 0.5 and 2.0 and reports the resulting RoSHAP rankings. We will also note that the asymptotic Gaussian approximation itself bypasses full KDE for the final ranking score, thereby mitigating the dimensionality issue for the primary use case. revision: partial

Circularity Check

0 steps flagged

No significant circularity: asymptotic Gaussianity derived from standard CLT under independent regularity conditions

full rationale

The paper derives the asymptotic Gaussianity of aggregated feature attribution scores from the central limit theorem applied to bootstrap-resampled SHAP values, under explicitly invoked mild regularity conditions that are not defined in terms of the target result itself. RoSHAP is constructed as a new summary statistic (combining activity, strength, and stability) from the estimated distribution rather than being equivalent to any input parameter or fitted quantity by construction. No self-citation chains, ansatzes smuggled via prior work, or uniqueness theorems from the same authors are load-bearing for the central claims. The estimation procedure (bootstrap + KDE) follows standard nonparametric methods without reducing the prediction to the fitting step.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard bootstrap resampling and KDE for distribution estimation, an asymptotic normality result that depends on regularity conditions, and the new definition of RoSHAP as a composite ranking criterion.

axioms (1)

domain assumption mild regularity conditions
Invoked to establish that the aggregated feature attribution score is asymptotically Gaussian.

invented entities (1)

RoSHAP no independent evidence
purpose: robust feature-ranking criterion from the distribution of SHAP values
Newly defined summary that rewards activity, strength, and stability of features

pith-pipeline@v0.9.0 · 5522 in / 1199 out tokens · 49346 ms · 2026-05-15T02:50:53.812798+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that, under mild regularity conditions, the aggregated feature attribution score is asymptotically Gaussian... Lyapunov central limit theorem argument... Assumption... 1/s^{2+δ}_j ∑ E||T_ij|−E[|T_ij|]|^{2+δ}→0. Theorem. U_j−μ_j / s_j →^d N(0,1).
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

RoSHAP_j := (1−P0j) m_j² / s_j ... rewards features that are active, strong, and stable.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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