{"id":"af882785-1d0d-4994-b7ea-434dba4fb548","arxiv_id":"2412.13866","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"SHAP scores can assign zero importance to a relevant feature and nonzero importance to an irrelevant feature, even for Lipschitz-continuous and arbitrarily differentiable regression models.","lead":"This paper proves that SHAP scores, a popular tool for explaining machine learning predictions, can be badly misleading even for simple, smooth regression models, not just for classifiers. The authors construct explicit examples where an irrelevant feature gets a nonzero importance score while a relevant feature gets zero, and they extend this to Lipschitz-continuous and arbitrarily differentiable functions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 9's arbitrary-differentiability construction is an unproven sketch: the stated rectangle-average constraint does not preserve the slice expectation E[rho | x2=1], and the non-differentiability along x2=1 is not removed.","rationale":"I agree with the reader that the paper's central and most defensible result is Proposition 8: the explicit Lipschitz regression model rho3 is verifiable and correctly yields SHAP scores in which the relevant feature has score 0 and the irrelevant feature has nonzero score. I verified by direct integration that the expected values in Table 1 are correct and that piecewise-affine continuity on a compact domain implies Lipschitz continuity. The reader's weakest-assumption concern about the Boolean constructions is real but not fatal: for the specific conditions in Proposition 1, explicit small functions exist, so the induction is plausibly repairable. However, the arbitrary-differentiability section is materially weaker than Proposition 8. It is admittedly only a 'rationale,' and it omits the crucial preservation of the conditional expectation E[rho | x2=1] as well as the joint smoothing of the x2=1 kink. Since the abstract promises this final result, the paper is not ready for full acceptance. The reader's CONDITIONAL verdict remains appropriate; I would keep it unchanged, with the primary required revision being a complete proof or explicit construction for Proposition 9.","tokens_in":1375,"tokens_out":1282,"duration_ms":228848,"concrete_test":"Use a computer algebra system for alpha=1, eps=0.1, and n=2. Enumerate the polynomial family on [1-eps,1+eps] x [0,1] satisfying the stated value and derivative constraints on x1 at the three listed lines and glued to the interpolated polynomial on [1-eps,1+eps] x [1,2]. Check whether any member simultaneously preserves both (i) the average over the rectangle [1-eps,1+eps] x [0,2] and (ii) the average over the slice x2=1, namely E[rho3|x2=1]=1+alpha=2, while also having continuous x2-derivative at x2=1. If no member satisfies all three, the construction as stated cannot deliver an n-times differentiable model with unchanged SHAP scores. Alternatively, an explicit construction of the patched function would settle the issue positively.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The core Lipschitz claim (Proposition 8) is sound: rho3 in Figure 2 is continuous piecewise affine on [0,2]^2, hence Lipschitz, and direct integration shows the four conditional expectations match Table 1 for every nonzero alpha, giving ScE(1)=0 and ScE(2)=alpha while feature 1 is relevant and feature 2 is irrelevant. The Boolean existence gaps flagged by the reader are likely repairable because explicit sub-functions satisfying the stated conditions exist (e.g., for m=2, kappa1=1 on neighbours of 11 and f=1 only at 11). The load-bearing weakness is the 'Arbitrary Differentiability' section. The paper explicitly says it only gives a rationale, not a proof. Its polynomial patch fixes values and derivatives on x1 in {1-eps, 1, 1+eps}, then picks a polynomial with the same average value over the rectangle [1-eps,1+eps] x [0,2]. But SHAP scores depend on four conditional expectations: E[rho], E[rho|x1=1], E[rho|x2=1], and E[rho|x1=1,x2=1]. The construction fixes E[rho|x1=1] and E[rho|x1=1,x2=1] by construction, and the rectangle-average only fixes one scalar, not E[rho|x2=1], which is an integral over the slice x2=1 not determined by the rectangle average. Moreover, the kink across x2=1 is not smoothed: the construction says to use a polynomial in x1 for x2<=1 and a linearly interpolated polynomial for x2>=1, but these will generally have different x2-derivatives at x2=1. The claim that 'the same can be done with the other non-derivable line' does not address compatibility at the intersection (1,1) or joint preservation of all relevant expected values. Therefore Proposition 9, and the abstract's final claim about arbitrarily differentiable regression models, are not established by the manuscript as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21821,"tokens_out":11498,"duration_ms":107357,"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":[{"comment":"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.","section":"Regression – Arbitrary Differentiability (Proposition 9)"},{"comment":"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.","section":"Supplemental Materials, Propositions 1–5 (Classification – Boolean Domains)"}],"minor_comments":[{"comment":"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'.","section":"Regression – Arbitrary Differentiability"},{"comment":"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).","section":"Supplemental Materials, Proposition 10"},{"comment":"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.","section":"Figure 2"},{"comment":"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.","section":"Example 9"},{"comment":"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.","section":"Preliminaries, Equation (14)"}],"recommendation":"major_revision","confidential_remarks":"The core Lipschitz counterexample rho3 is a solid and useful contribution, and the paper is likely fixable. The Boolean-domain claims appear repairable by supplying explicit sub-function families for arbitrary m, and the smoothness section could be rewritten as a conjecture or replaced with a fully verified construction. My main reservation is that the current version overstates what is proved, especially in the abstract and Proposition 9; the revision should align the claims with the actual proofs. The use of the authors' own AXp/CXp framework as the normative standard is a judgment call and, in my view, not circular, but the paper should cite that this is the same framework used in the prior work being extended."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result is real. This paper extends the known SHAP-score failure from Boolean classifiers to regression, and its central example ρ3 is a genuine Lipschitz-continuous function on [0,2]^2 whose conditional expectations reproduce Table 1, so ScE(1)=0 and ScE(2)=α while feature 1 is relevant and feature 2 is not. I checked the integrals; they work. That alone is a real contribution: it removes the hope that smoothness or Lipschitz continuity rescues SHAP scores. The similarity predicate is a clean device for treating classification and regression uniformly, and the ρ2 example with uncountable codomain is also correct.\n\nThe soft spots are where the paper reaches beyond what it proves. Proposition 9, the 'arbitrarily differentiable' claim, is explicitly presented as a rationale, not a proof. The stress-test is right: the polynomial patching fixes values and derivatives along vertical lines, but it only fixes one scalar average over the rectangle, which does not control E[ρ|x2=1]; and the construction does not remove the kink along x2=1. So the claim that SHAP scores fail for arbitrarily differentiable regression models is not established by this manuscript. A referee should ask for either a complete construction or a weakened statement.\n\nThe Boolean 'arbitrarily many' propositions (1–5) have a different weakness: they assert the existence of sub-functions κ1, f, etc., with very specific properties, without giving a construction for every m. I agree with the reader that this is likely repairable—small explicit functions exist—but it is still a gap in the proof as written. It should be fixed in revision.\n\nThe citation pattern is fine; the paper builds directly on Huang and Marques-Silva 2023 and says so. The newness is genuine: prior work only covered classifiers with numeric classes.\n\nBottom line: the central Lipschitz example is correct and worth publishing. The differentiability section is currently a sketch and should not be stated as a theorem. This paper deserves peer review; send it to a venue that does formal XAI, and tell the reviewer to focus on Proposition 9 and the Boolean existence steps. If those are fixed, this is a strong paper.","headline":"The Lipschitz counterexample is real and checkable; the arbitrary-differentiability claim is a sketch, so the paper needs revision before it can be accepted.","tokens_in":22426,"tokens_out":3260,"would_cite":true,"duration_ms":26925,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["SHAP scores","Shapley values","explainable AI","feature attribution","Lipschitz continuity","regression models","Boolean classifiers","formal explanations"],"falsifier":"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.","tokens_in":21189,"feed_emoji":"📐","tokens_out":8875,"duration_ms":74426,"temperature":0.7,"pith_summary":"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.","feed_headline":"Lipschitz-smooth regression models still break SHAP scores","feed_subtitle":"Exact SHAP scores assign zero to the relevant feature and nonzero to an irrelevant one.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines SHAP scores as Shapley values for model predictions; this is the object the paper shows can be misleading.","marker":"(Lundberg and Lee 2017)"},{"why":"Established the first theoretical inadequacy examples for classifiers, which this paper extends to regression and Lipschitz models.","marker":"(Huang and Marques-Silva 2023)"},{"why":"Showed exact SHAP score computation is tractable on Boolean circuits, enabling rigorous assessment of the scores.","marker":"(Arenas et al. 2021)"},{"why":"Supplies the formal explanation framework and AXp/CXp duality used to define feature relevance.","marker":"(Marques-Silva and Ignatiev 2022)"},{"why":"Contributes complexity and formal notions for binary classifier explanations that ground the abductive formulation.","marker":"(Wäldchen et al. 2021)"},{"why":"Provides the standard definition of Lipschitz continuity used to classify the regression models.","marker":"(O'Searcoid 2006)"},{"why":"Original Shapley value axioms underlying the score definition.","marker":"(Shapley 1953)"}],"fun_headline_variants":["SHAP scores fail for Lipschitz regression models","Even smooth models break SHAP scores","SHAP gives zero to relevant feature in smooth models","Lipschitz doesn't fix SHAP's feature misranking","SHAP scores unreliable for differentiable regressors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SHAP scores fail for Lipschitz regression models","Even smooth models break SHAP scores","SHAP gives zero to relevant feature in smooth models","Lipschitz doesn't fix SHAP's feature misranking","SHAP scores unreliable for differentiable regressors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000223,"raw_usage":{"total_tokens":1455,"prompt_tokens":943,"completion_tokens":512,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":438}},"tokens_in":559,"tokens_out":512,"duration_ms":4931,"temperature":1.0,"reasoning_tokens":438,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:43:09.287627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}