TowerDebias: A Novel Unfairness Removal Method Based on the Tower Property

Aditya Mittal; Norman Matloff

REVIEW 3 major objections 4 minor 40 references

TowerDebias: A Novel Unfairness Removal Method Based on the Tower Property

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that averaging a black-box model's predictions over a sensitive attribute never increases that attribute's correlation with the predictions.

desk verdict 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. read the letter →

arxiv 2411.08297 v2 pith:GATIPWZ3 submitted 2024-11-13 cs.LG cs.AImath.PRstat.APstat.ML

Norman Matloff , Aditya Mittal This is my paper · ORCID

classification cs.LGcs.AImath.PRstat.APstat.ML MSC 60A1062J0568T05

keywords algorithmicfairnessTowerpropertypost-processingdebiasingblack-boxmodelssensitiveattributesPearsoncorrelationk-nearestneighborsmachineunlearning

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

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 $S$.

What carries the argument

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 $X$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The 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.

What'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.

The 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.

Even 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.

I 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.

Best.

Referee Report

3 major / 4 minor

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.

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 (3)

[Section 4.4] 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 4.4] 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 4.4] 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.

minor comments (4)

[Title and throughout] The method name is written inconsistently as both 'towerDebias' and 'TowerDebias'; the title uses 'TowerDebias', and the abstract and body should match.
[Section 4.2.2] 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 4.4] 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 5.6] 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.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central fairness theorem is mathematically false, not self-referential or fitted.

full rationale

The claimed derivation chain is not circular. Section 4.4 attempts to prove Eq. (1) using L2 projection facts and the Tower Property. The proof is invalid because the asserted inequality ||E(Y|X)|| >= ||E(Y|X,S)|| is the reverse of the Pythagorean inequality for nested conditional-expectation subspaces, and the numerator identity E[S E(Y|X)] = E[S E(Y|X,S)] does not follow from the Tower Property. These are false mathematical premises, not a reduction of the conclusion to the assumptions by construction. The tDB output is a k-NN average of the black-box predictions and is not fitted to the reported fairness measure; no parameter is calibrated to rho(E(Y|X),S), and the empirical evaluation uses holdout data and external baselines. The only self-citation (Matloff and Zhang, 2022) appears in a related-work list and is not load-bearing. Therefore no step equates a 'prediction' with an input or imports a conclusion via a self-citation chain; the paper's issue, if any, is correctness of the theorem, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The method itself introduces no free parameters beyond k, no novel entities, and relies on standard probability. The two ad hoc axioms are false mathematical claims that the central theorem depends on.

free parameters (1)

k (number of nearest neighbors) = not fixed; user-chosen
The k-NN averaging step in tDB averages E(Y|X,S) over the k closest cases. The paper states k controls the bias-variance trade-off and indirectly the fairness-utility trade-off, but gives no principled selection rule.

assumptions (4)

domain assumption Basic Assumption (BA): (Y,X,S) follows a multivariate Gaussian distribution
Invoked in Section 4 to justify the closed-form correlation reduction in 4.1 and the claim in 4.3 that uncorrelatedness implies independence.
standard math Tower Property: E(Y|X) = E[E(Y|X,S)|X]
Used as the foundation of tDB, Section 4.2.1 Equation (4.2).
ad hoc to paper Projection norm inequality ||E(Y|X)|| >= ||E(Y|X,S)||
Asserted in Sections 4.2.2 and 4.4 to compare denominators in the correlation ratio. It is false: conditioning on more variables yields a projection onto a larger subspace with larger or equal norm.
ad hoc to paper Numerator identity E[S E(Y|X)] = E[S E(Y|X,S)]
Asserted in Section 4.4 to equate the numerators. It does not follow from the Tower Property and is false in general.

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Cite this review

Pith. "Pith review of TowerDebias: A Novel Unfairness Removal Method Based on the Tower Property." pith.science (2026). https://pith.science/paper/GATIPWZ3

@misc{pith2026241108297,
  author       = {Pith},
  title        = {Pith review of: TowerDebias: A Novel Unfairness Removal Method Based on the Tower Property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GATIPWZ3}},
  note         = {Machine review of arXiv:2411.08297}
}

read the original abstract

Decision-making processes have increasingly come to rely on sophisticated machine learning tools, raising critical concerns about the fairness of their predictions with respect to sensitive groups. The widespread adoption of commercial "black-box" models necessitates careful consideration of their legal and ethical implications for consumers. When users interact with such black-box models, a key challenge arises: how can the influence of sensitive attributes, such as race or gender, be mitigated or removed from its predictions? We propose towerDebias (tDB), a novel post-processing method designed to reduce the influence of sensitive attributes in predictions made by black-box models. Our tDB approach leverages the Tower Property from probability theory to improve prediction fairness without requiring retraining of the original model. This method is highly versatile, as it requires no prior knowledge of the original algorithm's internal structure and is adaptable to a diverse range of applications. We present a formal fairness improvement theorem for tDB and showcase its effectiveness in both regression and classification tasks using multiple real-world datasets.

Figures

Figures reproduced from arXiv: 2411.08297 by the authors.

**Figure 1.** Effect of towerDebias on MAPE increase on predicting wage income and the correlation reductions with gender in the SVCensus dataset. 5.2 Law School Admissions The Law School Admissions dataset (1991) surveys U.S. law students, capturing demographics and academics like age, GPA, income, gender, and race for admitted applicants. The goal is to predict LSAT scores (Y ) in a regression task, with race (S)—categorized a… view at source ↗

**Figure 2.** Effect of towerDebias on MAPE increase on predicting LSAT Score and correlation reduction with race in the LSA dataset. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Effect of towerDebias on Misclassification rate increase on recidivism and the correlation reductions with race in the COMPAS dataset. Similar fairness patterns emerge across racial groups, with varying baseline correlations. Applying tDB to traditional machine learning models consistently reduces the correlation between predicted recidivism probability and each racial group by 25–50%. For instance, the neural netwo… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Effect of towerDebias on Misclassification rate increase on customer churn and the correlation reductions with gender & age in the IranChurn dataset. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Effect of towerDebias on Misclassification rate increase on employment status and correlation reduction with sex in the DutchCensus dataset. Results from figure (5) are consistent with previous datasets, demonstrating fairness gains with some accuracy loss. Notably, co…

**Figure 6.** Figure 6: Effect of towerDebias on MAPE increase on predicting wage income and corresponding correlation reduction with gender in the SVCensus dataset. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: Effect of towerDebias on MAPE increase on predicting LSAT Score and correlation reduction with race in the LSA dataset. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: Effect of towerDebias on Misclassification rate increase on recidivism and correlation reduction with race in the COMPAS dataset. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Effect of towerDebias on Misclassification rate increase on customer churn and correlation reduction with gender & age in the IranChurn dataset. FairML versus towerDebias: DutchCensus Results [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: Effect of towerDebias on Misclassification rate increase on employment status and correlation reduction with sex in the DutchCensus dataset. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

Discussion (0). Continue with ORCID to comment.

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Reviewed August 12, 2026 · model on record in the stance chip above.

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