{"id":"5a180c49-7b7e-4cfe-9a01-556c82c9ebfb","arxiv_id":"2411.08297","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"TowerDebias averages predictions over the sensitive attribute using the Tower Property, but its claimed fairness-improvement theorem is not proven and is false as stated.","lead":"This paper proposes towerDebias, a post-processing method that averages a black-box model's predictions over a sensitive attribute to reduce its influence. The method is simple and model-agnostic, but the paper's formal proof that this always improves fairness relies on two incorrect mathematical statements.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The fairness-improvement theorem Eq. (1) is false: a Gaussian example with Var(X)=Var(S)=1, Cov(X,S)=0.9, and Y=X-0.9S gives ρ(E(Y|X),S)=0.9 but ρ(E(Y|X,S),S)=0, so the central guarantee fails.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing flaw: both supporting premises of Eq. (1) in Section 4.4 are false. The projection-norm inequality is backwards, and the numerator equality is not implied by the Tower Property. My independent Gaussian counterexample sharpens the point because it lies inside the paper's own multivariate-normal basic assumption, so the theorem cannot be rescued by restricting to the stated setting. The formal fairness-improvement guarantee is the central claimed contribution of the paper; once it fails, the post-processing method remains an averaging heuristic with no proven guarantee of correlation reduction. Empirical plots in Section 5 suggest the heuristic is often effective, and the authors do present reproducible-looking methodology and multiple datasets, but these observations do not repair the broken theorem. For these reasons the reader's REJECT verdict is appropriate, and no change to the verdict is needed.","tokens_in":12162,"tokens_out":10261,"duration_ms":97923,"concrete_test":"Analytically evaluate the Gaussian counterexample: take (X,S) jointly Gaussian with mean 0, Var(X)=Var(S)=1, Cov(X,S)=0.9, and Y=X−0.9S. Compute both correlations directly: ρ(E(Y|X),S)=0.9 and ρ(E(Y|X,S),S)=0, contradicting Eq. (1). As a robustness check, add independent zero-mean noise to Y; the conditional expectations, and therefore the violation, remain unchanged.","verdict_should_be":"REJECT","load_bearing_attack":"Section 4.4's proof of Eq. (1) rests on two claims, and both are false. First, it asserts ||E(Y|X)|| ≥ ||E(Y|X,S)||. Since σ(X) ⊆ σ(X,S), the reverse inequality holds: ||E(Y|X,S)||² = ||E(Y|X)||² + ||E(Y|X,S) − E(Y|X)||², by the Pythagorean property of nested conditional-expectation projections. Second, the numerator identity E[S E(Y|X)] = E[S E(Y|X,S)] would require E[E(Y|X)|S] = E(Y|X,S), which is not a consequence of the Tower Property and is false in general. A concrete Gaussian counterexample within the paper's own basic assumption refutes the theorem directly: let X and S be centered Gaussian with Var(X)=Var(S)=1 and Cov(X,S)=0.9, and set Y = X − 0.9S. Then E(Y|X,S)=X−0.9S has zero covariance with S, so ρ(E(Y|X,S),S)=0, while E(Y|X)=0.19X has covariance 0.171 with S and standard deviation 0.19, so ρ(E(Y|X),S)=0.9. Thus Eq. (1) is false even under the stated multivariate normal assumption, and the formal claim that tDB guarantees no correlation increase is broken. The empirical results may still support the method as a heuristic, but they cannot substitute for the invalidated theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes towerDebias (tDB), a post-processing fairness method for black-box models. The method estimates E(Y|X) by averaging a black-box model's predictions over the sensitive attribute S, using a k-nearest-neighbor approximation of the Tower Property. The authors claim a formal fairness-improvement theorem, Eq. (1) in Section 4.4, stating that the correlation between predictions and S never increases after applying tDB. They support this with a proof based on L2 projections and conditional expectations, and they report empirical results on five datasets for regression and classification tasks, comparing tDB against several machine learning baselines and fair-regression baselines.","tokens_in":12555,"tokens_out":4128,"duration_ms":41679,"significance":"If the formal theorem were correct, tDB would be an appealing and simple model-agnostic post-processing method: it requires no access to the model's internals, no retraining, and is easy to implement from a textbook identity. The empirical study is broad, covering multiple datasets, model classes, and both regression and classification, and the paper is commendable for making its central claim explicit and falsifiable. However, the central theoretical guarantee is invalid. Both premises of the proof in Section 4.4 are false, and the theorem is contradicted by a simple counterexample within the paper's own multivariate Gaussian assumption. The empirical results may indicate that tDB is a useful heuristic in some settings, but they cannot substitute for the broken formal claim, which is presented as a key contribution of the paper.","major_comments":[{"comment":"The proof's denominator claim is backwards. The paper states that ||E(Y|X)|| ≥ ||E(Y|X,S)||, but since σ(X) ⊆ σ(X,S), the conditional expectations E(Y|X) and E(Y|X,S) are projections onto nested subspaces. The Pythagorean relation ||E(Y|X,S)||² = ||E(Y|X)||² + ||E(Y|X,S) − E(Y|X)||² applies, so the norm of E(Y|X,S) is never smaller than the norm of E(Y|X). The variance comparison used to compare denominators in Eq. (1) therefore fails in the opposite direction.","section":"Section 4.4"},{"comment":"The numerator identity is false in general. The proof writes E[S E(Y|X)] = E{E[S E(Y|X)|S]}, and then identifies the inner term with S E(Y|X,S). But E[S E(Y|X)|S] = S E[E(Y|X)|S], and equality with S E(Y|X,S) would require E[E(Y|X)|S] = E(Y|X,S). This is not a consequence of the Tower Property and is not generally true; the Tower Property gives E[E(Y|X,S)|X] = E(Y|X), not the reverse relation used here.","section":"Section 4.4"},{"comment":"Eq. (1) is false even under the paper's basic assumption. Let X and S be centered Gaussian with Var(X)=Var(S)=1 and Cov(X,S)=0.9, and set Y = X − 0.9S. Then E(Y|X) = 0.19X and E(Y|X,S) = X − 0.9S. Direct computation gives ρ(E(Y|X),S) = 0.9 and ρ(E(Y|X,S),S) = 0, contradicting the claimed inequality ρ(E(Y|X),S) ≤ ρ(E(Y|X,S),S). This counterexample lies entirely within the multivariate Gaussian setting assumed in Section 4, so the formal fairness-improvement theorem is not merely unproven; it is incorrect.","section":"Section 4.4"}],"minor_comments":[{"comment":"The method name is written inconsistently as both 'towerDebias' and 'TowerDebias'; the title uses 'TowerDebias', and the abstract and body should match.","section":"Title and throughout"},{"comment":"The property '||H1|| ≤ ||G||' stated for H1 = H − E(H|G) cannot be right in general: if H is independent of G with large variance, ||H1|| can exceed ||G||. This looks like a typo, likely intended to be an inequality involving ||H||, but as written it is false.","section":"Section 4.2.2"},{"comment":"The theorem and the fairness discussion use ρ without absolute values, while Section 2 defines the fairness measure as the absolute Pearson correlation. Eq. (1) should be stated with |ρ| or with an explicit clarification about signed correlation.","section":"Section 4.4"},{"comment":"There are several typographical issues, including 'Choose an appropriatek' missing a space, 'the initial correlation approach zero' (subject-verb agreement), and inconsistent capitalization of 'Law School Admission Results' versus 'Law School Admissions.'","section":"Section 5.6"}],"recommendation":"reject","confidential_remarks":"The paper's central theoretical claim is false, and the counterexample is simple and within the paper's own assumptions. The empirical study is reasonably broad and may support tDB as a heuristic, but as written the manuscript presents the theorem as a main contribution and the proof is load-bearing. I do not see a route to publication in this venue without replacing the theorem with a substantially different and correct analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline is that the main theorem doesn't hold up. The claim in Eq. (1), that rho(E(Y|X),S) <= rho(E(Y|X,S),S), is refuted even within the paper's own Gaussian setting. Take X and S centered with Var(X)=Var(S)=1 and Cov(X,S)=0.9, and set Y = X - 0.9S. Then E(Y|X,S) is uncorrelated with S, giving rho=0, while E(Y|X)=0.19X gives rho=0.9. So Eq. (1) is false. The proof in Section 4.4 rests on two incorrect premises. First, it asserts ||E(Y|X)|| >= ||E(Y|X,S)||, but the reverse inequality holds for nested conditional expectations. Second, it claims E[S E(Y|X)] = E[S E(Y|X,S)], which would require E(Y|X,S) to be a function of S alone—not implied by the Tower Property.\n\nWhat's creditable: the paper is clearly written and identifies a practical problem—debias a black-box model's predictions without retraining. The heuristic, averaging the black-box output over S given X, is sensible, and the empirical plots suggest it lowers correlations on several datasets. The k-NN implementation is a reasonable practical choice.\n\nThe soft spots beyond the broken theorem: the empirical results are reported graphically without error bars, code, or data, so robustness is hard to assess. The comparisons with FRRM/FGRRM lack detail. And the claim of \"removing\" S's influence is overstated; the method reduces linear correlation, not all dependence.\n\nEven if the theorem were correct, the novelty would be incremental—this is essentially the Tower Property. With the theorem false, the paper is just empirical observations about a known technique. That is not enough for publication.\n\nI would not send this to peer review in its current form. The authors could revise it as an empirical study of a debiasing heuristic, but they must withdraw the false guarantee. As submitted, it is not viable.\n\nBest.","headline":"The paper's central guarantee is false: the proof of Eq. (1) uses an incorrect projection inequality and a false identity, and a simple Gaussian example breaks the claimed result.","tokens_in":13030,"tokens_out":2727,"would_cite":false,"duration_ms":28259,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60A10","62J05","68T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that averaging a black-box model's predictions over a sensitive attribute never increases that attribute's correlation with the predictions.","keywords":["algorithmic fairness","Tower property","post-processing debiasing","black-box models","sensitive attributes","Pearson correlation","k-nearest neighbors","machine unlearning"],"falsifier":"A direct check is to construct a synthetic Gaussian model with a known conditional mean (for instance $Y = X + S$ with $X$ and $S$ dependent), compute both sides of inequality (1), and see whether the left-hand correlation ever exceeds the right-hand one; if it does, the claimed guarantee fails.","tokens_in":11951,"feed_emoji":"⚖️","tokens_out":11299,"duration_ms":94120,"temperature":0.7,"pith_summary":"towerDebias (tDB) is a post-processing method for removing the influence of a sensitive attribute $S$ from a black-box predictor. The paper argues that replacing the model's original predictions $E(Y|X,S)$ with the averaged predictions $E(Y|X)$, obtained from the Tower Property, does not increase the Pearson correlation between predictions and $S$, a commonly used fairness measure. This would deliver fairness gains without retraining or internal access to the model. The paper supports the claim with a formal theorem bounding the correlation of the averaged predictor by that of the original predictor, and with experiments across regression and classification datasets. If the theorem is right, any vendor model could be debiased by averaging its outputs over $S$.","feed_headline":"Averaging predictions over a sensitive trait can cut its influence","feed_subtitle":"towerDebias debiases any black-box model by averaging outputs over the sensitive attribute, no retraining needed.","key_machinery":"The Tower Property, $E(Y|X) = E[E(Y|X,S)|X]$, which says that the $S$-free regression function is the $S$-averaged $S$-conditional regression function. The argument works in the $L^2$ space of mean-zero random variables with inner product $\\langle U,V \\rangle = E(UV)$, so variance is squared norm and conditional expectation is orthogonal projection. For finite data, tDB approximates the inner expectation with $k$-nearest-neighbour averaging: average the black-box predictions over the $k$ training cases whose $X$ values are closest to the new $X$.","core_discovery":"The paper's central claim is inequality (1), $\\rho(E(Y|X),S) \\le \\rho(E(Y|X,S),S)$: replacing a black-box model's predictions with their $S$-averaged conditional expectation never increases the Pearson correlation between predictions and the sensitive attribute. The proof treats conditional expectations as vectors in an $L^2$ space of mean-zero random variables, applies the Tower Property to equate the two numerator terms, and compares denominators by the norms of the projections. The paper therefore claims a universal, model-independent fairness-improvement guarantee for tDB at the population level, with only the $k$-NN averaging step introducing finite-sample approximation error.","pith_inferences":["Editorial: the same averaging operation could be applied to any prediction that is a conditional expectation, such as quantile or risk estimates, because the Tower Property does not depend on how the black-box model was trained.","Editorial: a direct synthetic-data test with non-Gaussian dependence would show whether the inequality holds beyond the Gaussian setting the paper uses for its closed-form correlation reduction.","Editorial: if the variance and cross-moment identities in the proof hold only under special conditions, the universal guarantee would reduce to a conditional one, so checking those identities empirically on real model outputs is a natural next step."],"forward_implications":["Any black-box model — linear, tree, random forest, or neural network — can have $S$ influence reduced by averaging its outputs over $S$, with no retraining and no access to weights or training data.","Correlation-based fairness metrics improve with tDB, so fairness gains can compound when tDB is applied after existing fair-training methods such as FRRM/FGRRM.","Because tDB produces the best $S$-free predictor under squared loss, any accuracy loss beyond the gap between $E(Y|X,S)$ and $E(Y|X)$ is a bias-variance artifact of the $k$-NN approximation, controlled by $k$.","In classification, applying tDB to predicted probabilities $P(Y=1|X,S)$ yields a debiased probability estimate with the claimed fairness guarantee, not just a relabeled decision."],"supporting_citations":[{"why":"Provides the Tower Property statement used to justify averaging $E(Y|X,S)$ over $S$.","marker":"Wolpert, 2009"},{"why":"Supplies the $L^2$ space and conditional-expectation projection facts used in the proof.","marker":"Durrett, 2019"},{"why":"Gives the inner-product projection inequalities invoked to compare variances of the two predictors.","marker":"Axler, 2015"},{"why":"Establishes the multivariate-normal conditional-mean linearity used in the correlation-reduction formula.","marker":"Johnson and Wichern, 1993"},{"why":"The prior linear fair-regression method whose steps Section 4.3 retraces in a more precise form.","marker":"Komiyama et al., 2018"},{"why":"The fair ridge baselines that tDB is compared against in the FairML versus towerDebias experiments.","marker":"Scutari, 2023"}],"fun_headline_variants":["Tower Property guarantees fairer black-box predictions","Averaging over sensitive trait provably cuts bias","No retraining needed: Tower Property debiases any model","New proof: averaging over S reduces sensitive correlation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central guarantee depends on two identities about the original model's outputs: the $S$-free averaged predictions must have no smaller variance than the original predictions, and the average of $S$ times the averaged prediction must equal the average of $S$ times the original prediction.","fun_headline_variants_meta":{"raw":{"variants":["Tower Property guarantees fairer black-box predictions","Averaging over sensitive trait provably cuts bias","No retraining needed: Tower Property debiases any model","New proof: averaging over S reduces sensitive correlation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1335,"prompt_tokens":851,"completion_tokens":484,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":421}},"tokens_in":467,"tokens_out":484,"duration_ms":5267,"temperature":1.0,"reasoning_tokens":421,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:44:42.162931+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check is to construct a synthetic Gaussian model with a known conditional mean (for instance $Y = X + S$ with $X$ and $S$ dependent), compute both sides of inequality (1), and see whether the left-hand correlation ever exceeds the right-hand one; if it does, the claimed guarantee fails.","supporting_citations":[{"cited_title":"Institute of statistics and decision sciences, 2009","cited_arxiv_id":null,"evidence_quote":"Provides the Tower Property statement used to justify averaging $E(Y|X,S)$ over $S$."},{"cited_title":"Applied Multivariate Statistical Analysis","cited_arxiv_id":null,"evidence_quote":"Establishes the multivariate-normal conditional-mean linearity used in the correlation-reduction formula."},{"cited_title":"Nonconvex optimization for regression with fairness constraints","cited_arxiv_id":null,"evidence_quote":"The prior linear fair-regression method whose steps Section 4.3 retraces in a more precise form."},{"cited_title":"fairml: A Statistician's Take on Fair Machine Learning Modelling","cited_arxiv_id":"2305.02009","evidence_quote":"The fair ridge baselines that tDB is compared against in the FairML versus towerDebias experiments."}],"review_version":1}