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REVIEW 2 major objections 5 minor 28 references

SHAP scores fail pervasively even when Lipschitz succeeds

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper constructs Lipschitz-continuous regression models whose exact SHAP scores reverse feature relevance: the only relevant feature scores zero, an irrelevant feature scores nonzero.

desk verdict The Lipschitz counterexample is real and checkable; the arbitrary-differentiability claim is a sketch, so the paper needs revision before it can be accepted. read the letter →

arxiv 2412.13866 v1 pith:EHLZI5IW submitted 2024-12-18 cs.LG cs.AI

Olivier Letoffe , Xuanxiang Huang , Joao Marques-Silva This is my paper · ORCID
classification cs.LGcs.AI
keywords SHAPscoresShapleyvaluesexplainableAIfeatureattributionLipschitzcontinuityregressionmodelsBooleanclassifiersformalexplanations
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 is trying to establish that the known failures of SHAP scores are pervasive rather than rare. It builds Boolean classifiers of any size whose exact SHAP scores contradict feature relevance, then moves to regression: finite codomains, uncountable codomains, and finally Lipschitz-continuous regression models. The central exhibit is a two-feature piecewise-linear function whose conditional expectations reproduce one small table, from which the Shapley arithmetic gives $Sc_E(1)=0$ for the only relevant feature and $Sc_E(2)=\alpha\neq 0$ for an irrelevant feature. The paper concludes that Lipschitz continuity, a regularity property associated with adversarial robustness, does not rescue SHAP scores from producing misleading attributions, and argues the same holds for arbitrarily differentiable models. A sympathetic reader would care because these are exact computations, not sampling artifacts, targeting models with the smoothness properties the field treats as desirable.

What carries the argument

The load-bearing mechanism is the Shapley characteristic function $\upsilon_e(S;E)=E[\tau(x)\mid x_S=v_S]$ inserted into $Sc_E(i)=\sum_{S\subseteq F\setminus\{i\}}\varsigma(S)\,\Delta_i(S;E,\upsilon_e)$. The argument designs models so these expectations match the fixed table (Table 1): $\upsilon_e(\emptyset)=1-\alpha$, $\upsilon_e(\{1\})=1$, $\upsilon_e(\{2\})=1+\alpha$, $\upsilon_e(\{1,2\})=1$. That table forces $Sc_E(1)=0$ and $Sc_E(2)=\alpha$. Relevance is decided separately through a similarity predicate $\sigma(x;E)=[|\rho(x)-\rho(v)|\le\delta]$, from which weak abductive and contrastive explanations are formed; for the examples, $\{1\}$ is the unique AXp and CXp. The Lipschitz step is showing that the piecewise-linear $\rho_3$ achieves those expectations while all slopes stay bounded; the differentiability step is a polynomial-gluing argument intended to preserve the same averages.

What would settle it

Compute the four integrals $E[\rho_3(x)\mid x_S=v_S]$ for $S=\emptyset,\{1\},\{2\},\{1,2\}$; if any value differs from Table 1, or if the slope of $\rho_3$ across a seam exceeds a finite Lipschitz constant, the central counterexample fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Proposition 8: there exist regression models respecting Lipschitz continuity for which each feature is either irrelevant with a non-zero SHAP score or relevant with a SHAP score of zero. The witness is the function $\rho_3$ of Figure 2, defined on $[0,2]^2$, with target sample $((1,1),1)$; its four conditional expectations are exactly those of Table 1, so the same calculation yields $Sc_E(1)=0$ and $Sc_E(2)=\alpha\neq 0$. Feature 1 is the unique abductive and contrastive explanation, so the score ordering is fully reversed. The paper also claims this failure is guaranteed for arbitrarily many Boolean variables (Propositions 1–5), for regression over uncountable codomains (Proposition 7), and for arbitrarily differentiable regression models (Proposition 9).

Load-bearing premise

The load-bearing premise is that $\rho_3$ is genuinely Lipschitz-continuous and that its four conditional expectations match Table 1; the entire score reversal is computed from that table.

Editorial extensions

If this is right

  • If Proposition 8 is correct, a Lipschitz-continuous regression model—the kind trained with robustness constraints—can have exact SHAP scores that rank an irrelevant feature above the only relevant one.
  • For Boolean classifiers, the failures are not isolated: for each $n\ge 3$ there are $n$-variable functions exhibiting at least one of the listed issues, so no finite set of exceptions can be patched away.
  • The failures also occur for regression models with uncountable codomains and for arbitrarily differentiable functions, so requiring smoothness does not by itself certify SHAP scores.
  • Negating a classifier flips the sign of every SHAP score but preserves all six issues, so the phenomenon is not an artifact of choosing prediction 1 versus 0 as the target class.

Reading between the lines

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

  • The counterexamples depend only on a four-entry conditional-expectation table, so the same recipe could be applied to other regularity classes—for instance, monotonically constrained or convex regression models—by building a function in that class with the same table.
  • A practical reader can test the phenomenon directly: on $\rho_3$, any Monte Carlo approximation of SHAP would converge to the claimed zero and non-zero scores, meaning the failure would show up in the tool SHAP itself, not only in the exact definition.
  • The proof strategy suggests that attribution methods defined by averaging over feature subsets suffer a structural vulnerability: if the conditional expectations are pre-chosen, the function realizing them is almost irrelevant to the scores, so no amount of function smoothness can force the scores to align with relevance.
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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

2 major / 5 minor

Summary. The paper argues that the well-known theoretical inadequacies of SHAP scores are not limited to artificial numeric classifiers. It claims (i) for Boolean classifiers, arbitrarily many features admit functions and samples exhibiting each of six failure modes I1–I6; (ii) regression models with uncountable codomain can exhibit irrelevant features with nonzero SHAP scores and relevant features with zero SHAP scores; (iii) the same phenomenon persists for Lipschitz-continuous regression models, witnessed by an explicit two-dimensional piecewise-affine function rho3 (Figure 2) with target sample ((1,1),1); and (iv) the same failures are guaranteed for arbitrarily differentiable regression models. The normative yardstick is the AXp/CXp notion of feature relevance under a uniform distribution and a thresholded similarity predicate for regression.

Significance. If fully established, the Lipschitz-continuity result would be a genuine strengthening of existing negative results for SHAP scores: Lipschitz continuity is a widely used regularity property, and the constructed rho3 is a simple, explicit, alpha-parameterized family giving uncountably many counterexamples. The expected-value computations in Tables 1 and 5 are correct and can be verified by hand, and the SHAP scores for M1/M3 are indeed ScE(1)=0 and ScE(2)=alpha, so the core example in Proposition 8 is sound. However, the paper's broader claims for Boolean classifiers and for C-infinity regression models are not backed by complete proofs. The Boolean proofs assume the existence of auxiliary sub-functions with very specific properties without giving constructions, and the arbitrary-differentiability section is explicitly only a rationale. The choice of AXp/CXp relevance as the benchmark is a defensible modeling decision, not a circular derivation, since the SHAP scores are computed independently from the Shapley formula.

major comments (2)
  1. [Regression – Arbitrary Differentiability (Proposition 9)] Proposition 9 is not established by the text. The section explicitly says 'we just give the rationale for constructing the model,' and the sketched construction has two specific gaps. First, the rectangle-average constraint fixes only one scalar, namely the average of the replacement polynomial over [1-epsilon,1+epsilon]x[0,2]; it does not determine the conditional expectation E[rho | x2=1], which is an integral over the slice x2=1 and is one of the four quantities (Table 1) that determine the SHAP scores. Second, the proposed repair matches values and x1-derivatives on the lines x1 in {1-epsilon,1,1+epsilon}, but a function of two variables that is n-times differentiable must have continuous derivatives in all directions; the interpolation between the polynomial on x2<=1 and the linearly interpolated polynomial on x2>=1 will generally have a jump in the x2-derivative at x2=1. The claim that 'the same can be done with the other non-derivable line' does not resolve this compatibility issue. Since Proposition 9 is the sole support for the abstract's claim about arbitrarily differentiable regression models, it must either be replaced by a rigorous construction or clearly downgraded to a conjecture.
  2. [Supplemental Materials, Propositions 1–5 (Classification – Boolean Domains)] The proofs of Propositions 1, 2, 3, and 5 rely on unproved existence assumptions about auxiliary Boolean sub-functions. For example, in the proof of Proposition 1 the text states that 'the non-constant sub-functions kappa1 and f ... satisfy the following conditions' (kappa1 != kappa1 v f, kappa1 ^ f = 0, both predict a specific point to 0, and the CXp sets of kappa1 and kappa1 v f are identical), but no construction or existence argument is supplied for arbitrary m. Proposition 4 is different in that it gives an explicit example for its kappa1, but the others do not. The assertion that the issues occur 'for any n >= 3' (or odd/even n, as applicable) is therefore conditional on an unproved premise. Since this underpins the paper's claim that there are arbitrarily many Boolean classifiers with unsatisfactory SHAP scores, the proofs need explicit sub-function constructions or a general lemma establishing the required properties for all m.
minor comments (5)
  1. [Regression – Arbitrary Differentiability] The text contains several typos: 'polunomials' should be 'polynomials', 'infinitly' should be 'infinitely', 'necessarly' should be 'necessarily', and '1-degree 1 polynomials' should be 'degree-1 polynomials'.
  2. [Supplemental Materials, Proposition 10] The proof of Lipschitz continuity of rho3 is given only as a sketch in the supplemental materials, and the main text refers to it as 'the proof.' Since rho3 is continuous piecewise affine on the compact domain [0,2]^2, the claim is true, but a rigorous proof should check continuity across the four pieces and give a finite Lipschitz constant (or a bound on the subgradients).
  3. [Figure 2] The definition of rho3 uses the conditions alpha x1 <= alpha and alpha x1 >= alpha, which depend on the sign of alpha and swap the two regions when alpha < 0. Rewriting the cases as x1 <= 1 and x1 >= 1 (with the appropriate formulas using |alpha|) would make the piecewise structure immediately transparent for all nonzero alpha.
  4. [Example 9] The claim that the AXps and CXps of E3 are exactly {{1}} is asserted without the threshold analysis that was given for E1. A short argument showing that a suitably small delta makes the similarity predicate depend only on whether x1 = 1 would improve readability.
  5. [Preliminaries, Equation (14)] The notation E[kappa1 | xS = vS] is used before the paper states that conditionings of Boolean functions are to be interpreted as the expected value of the 0/1 indicator; a one-line clarification would avoid ambiguity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: SHAP scores and AXp/CXp feature relevance are computed by independent formulas; self-citations frame the benchmark but do not force any derivation step.

full rationale

Walked the full derivation chain. SHAP scores (Eq. 7) are computed from conditional expectations υe (Eq. 4) via the Shapley formula, while relevance (Eq. 12) is defined through AXps/CXps (Eqs. 8-11), which the paper states in full rather than importing. The two quantities are computed independently for every example, so no equation defines one in terms of the other (no pattern 1-2). For the regression examples, ρ2 and ρ3 are given explicitly (Example 5 and Figure 2); the supplemental integrates them to obtain the Table 1 / Table 5 expectations, and Table 3 arithmetic yields ScE(1)=0 and ScE(2)=α≠0. The remark that ρ3 is "devised such that the expected values ... are exactly the ones shown in Table 1" (Example 10) is a standard existence-proof construction: the SHAP scores are derived consequences of the explicitly given function, not fitted outputs, and the expectations are verified by direct integration rather than assumed. Relevance for these models is also verified inline (Example 9: "it is plain to reach the conclusion that the set of AXps is {{1}}"), not cited. Self-citations are present: the benchmark framework (formal explanations and the issue taxonomy I1-I6) comes from Huang and Marques-Silva 2023 and Marques-Silva and Ignatiev 2022, works involving the current senior author. However, the definitions are restated in the manuscript and the examples' AXps/CXps are computed directly, so these citations frame the evaluation but are not load-bearing proof steps, and the central Lipschitz claim resists reduction to them. Two flagged limitations are correctness risks rather than circularity: Proposition 9 is explicitly only a rationale (footnote 10: "we just give the rationale for constructing the model"), and the Boolean proofs assert existence of sub-functions κ1, f without constructing them (Supplemental Proposition 1: "The non-constant sub-functions κ1 and f ... satisfy the following conditions"), with Proposition 10's Lipschitz proof likewise a sketch. No step reduces by construction to its own input, and no fitted parameter is renamed as a prediction; the honest finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation depends on two hand-chosen numeric parameters (α and δ), one unproven existential assumption about auxiliary Boolean functions, and an informal polynomial-gluing step for the smooth case. No physical entities or new mediators are introduced. The uniformity assumption on the input distribution is a modeling choice that affects both SHAP scores and relevance judgments.

free parameters (2)
  • α = arbitrary non-zero real
    Parameterizes models M1 and M3. The SHAP score magnitudes scale with α, but the qualitative pattern (zero for the relevant feature, nonzero for the irrelevant one) is independent of α as long as α≠0. It is chosen by hand, not fitted to data.
  • δ = e.g., < 1/4
    Similarity threshold for regression explanations in M2 and M3. It determines which features are deemed relevant via the similarity predicate, but the SHAP score computation itself does not depend on δ. It is a hand-picked parameter that must be small enough for the examples to work.
assumptions (4)
  • domain assumption Uniform and independent input distribution over features
    Invoked in the 'Distributions, expected value' section for all expected-value computations. Both the SHAP scores and the relevance judgments are computed under this distribution, so changing the distribution could change the conclusions.
  • ad hoc to paper Existence of sub-functions κ1, f, g with the stated conditions in Propositions 1-5
    The proofs of the Boolean 'arbitrarily many' claims (Supplemental Propositions 1-5) assert that such functions exist and satisfy conditions like κ1≠κ1∨f, κ1∧f=0, and identical CXp sets, but no concrete construction is given. This is load-bearing for the claim that counterexamples exist for every n≥3.
  • ad hoc to paper Polynomial gluing with prescribed values and derivatives preserves Lipschitz continuity and the required expected values
    The arbitrary-differentiability section assumes that one can replace the relevant part of ρ3 with polynomials that match the first n derivatives at the boundary and also match the average values needed to keep the SHAP scores unchanged. This is argued informally, not proven.
  • standard math Gluing of Lipschitz-continuous functions yields a Lipschitz-continuous function
    Used in the proof sketch of Proposition 10 (Lipschitz continuity of ρ3). This is standard and not problematic.

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Pith. "Pith review of SHAP scores fail pervasively even when Lipschitz succeeds." pith.science (2026). https://pith.science/paper/EHLZI5IW

@misc{pith2026241213866,
  author       = {Pith},
  title        = {Pith review of: SHAP scores fail pervasively even when Lipschitz succeeds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHLZI5IW}},
  note         = {Machine review of arXiv:2412.13866}
}
read the original abstract

The ubiquitous use of Shapley values in eXplainable AI (XAI) has been triggered by the tool SHAP, and as a result are commonly referred to as SHAP scores. Recent work devised examples of machine learning (ML) classifiers for which the computed SHAP scores are thoroughly unsatisfactory, by allowing human decision-makers to be misled. Nevertheless, such examples could be perceived as somewhat artificial, since the selected classes must be interpreted as numeric. Furthermore, it was unclear how general were the issues identified with SHAP scores. This paper answers these criticisms. First, the paper shows that for Boolean classifiers there are arbitrarily many examples for which the SHAP scores must be deemed unsatisfactory. Second, the paper shows that the issues with SHAP scores are also observed in the case of regression models. In addition, the paper studies the class of regression models that respect Lipschitz continuity, a measure of a function's rate of change that finds important recent uses in ML, including model robustness. Concretely, the paper shows that the issues with SHAP scores occur even for regression models that respect Lipschitz continuity. Finally, the paper shows that the same issues are guaranteed to exist for arbitrarily differentiable regression models.

Figures

Figures reproduced from arXiv: 2412.13866 by the authors.

Figure 1
Figure 1. Tabular representation (TR) of a simple function. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Example of regression model that is Lipschitz cont [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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Reference graph

Works this paper leans on

28 extracted references · 26 canonical work pages

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    Choose this specific m-dimensional point v1..m and extend it with vn = 1

    The set of CXps for both κ1 and κ1 ∨ f with respect to the point v1..m are identical. Choose this specific m-dimensional point v1..m and extend it with vn = 1 . This means κ0(v) = κ1(v) = 0 , and therefore κ(v) = 0 . For any S ⊆ F \ { n}, we have ∆ n(S; E, υe) = 1 2 · (E[κ1|xS ...

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