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arxiv: 2605.28233 · v1 · pith:RMS6QLWAnew · submitted 2026-05-27 · 📊 stat.ML · cs.CY· cs.LG

Geometry of Relaxed Fair Regression: A Unified Framework for Aware and Unaware Settings

M. Generali Lince , V. Divol , R. Flamary , S. Gaucher , P. Loiseau This is my paper

Pith reviewed 2026-06-29 10:07 UTC · model grok-4.3

classification 📊 stat.ML cs.CYcs.LG
keywords fair regressiondemographic parityoptimal transportWasserstein distancetotal variationunaware fairnessfairness constraintsrelaxed fairness
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The pith

Fair regression under demographic parity can be reformulated exactly as an optimal transport problem that works in both aware and unaware settings.

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

The paper shows that adding a demographic parity penalty to a regression objective turns the problem into one of finding an optimal transport map between the distributions of predictions and targets. This single formulation covers both the aware case, where the sensitive attribute is known at training and test time, and the unaware case, where it is unavailable at prediction. The authors derive that the optimal predictors are precisely the transport maps for two concrete penalties, the squared Wasserstein-2 distance and total variation. Each penalty encodes a different notion of fairness: one spreads the adjustment across the whole population and the other meets the parity constraint exactly on a subset of points. From this geometry they obtain a practical algorithm that is easy to implement and competitive on standard benchmarks.

Core claim

Formulating regression under a demographic parity penalty as an optimal transport problem unifies both the aware and unaware settings and characterizes optimal prediction functions via optimal transport maps, under both squared Wasserstein-2 and Total Variation penalties. These results reveal that the Wasserstein penalty induces a smooth, population-wide compromise, while Total Variation enforces exact parity for a subset of individuals.

What carries the argument

the optimal transport formulation of the demographic parity penalty, whose solutions are the optimal transport maps between the relevant conditional distributions

If this is right

  • The optimal fair predictors are exactly the optimal transport maps for the chosen penalty.
  • The Wasserstein-2 penalty produces a smooth adjustment across the entire population.
  • The total variation penalty meets demographic parity exactly for a subset of individuals.
  • A simple algorithm derived from the transport characterization is computationally efficient and matches or exceeds prior methods on real benchmarks.

Where Pith is reading between the lines

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

  • The transport view could guide the design of new fairness penalties based on other cost functions between distributions.
  • The same geometry might extend to fairness notions beyond demographic parity once they are expressed as transport costs.
  • In practice, the choice between penalties could be driven by whether a smooth or a hard fairness guarantee is preferred on a given dataset.

Load-bearing premise

The demographic parity penalty admits an exact optimal transport formulation whose solutions are precisely the OT maps for the chosen penalties, without further restrictions on the regressor class or data distributions.

What would settle it

A concrete counter-example distribution and loss where the minimizer of the fairness-penalized regression objective is not an optimal transport map under either the Wasserstein-2 or total variation penalty.

Figures

Figures reproduced from arXiv: 2605.28233 by M. Generali Lince, P. Loiseau, R. Flamary, S. Gaucher, V. Divol.

Figure 1
Figure 1. Figure 1: Geometry of the OT maps. Left (Aware): 1D-to-1D transport mapping group￾conditional distributions to almost-fair predictions. Right (Unaware): 2D-to-1D transport mapping the joint distributions of (η(X), ∆(X)) under µ + η,∆ and µ − η,∆ to almost-fair predictions. Unaware Setting Aware Setting Objective OTcu (µ + η,∆, ν+) + OTcu (µ − η,∆, ν−) + λ OTcD (ν +, ν−) p + OTc2 (µ + η , ν+) + p − OTc2 (µ − η , ν−) … view at source ↗
Figure 2
Figure 2. Figure 2: Individual prediction trajectories for W2 vs. TV Relaxation. Evolution of marginal distributions and subset of individual prediction paths from Unfair (Bayes) to strictly Fair (100%-0% of initial unfairness). The lines show W2 induces proportional shift for all individuals, while TV leaves predictions unchanged (horizontal lines) until a hard threshold triggers abrupt merging. 2.4 The implicit philosophies… view at source ↗
Figure 3
Figure 3. Figure 3: Accuracy-fairness trade-offs on the Law School dataset. Relaxation trajectories from the unconstrained ERM (black dot) to the exact fair models (stars). Error bars indicate the standard deviation across 10 random data splits. MSE of the constant predictor is 10.1 × 10−3 . 0.2 0.4 0.6 KS 2.0 2.5 3.0 3.5 4.0 4.5 MSE ×10 2 MSE vs KS 0.1 0.2 0.3 W2 MSE vs W2 0.2 0.3 0.4 0.5 0.6 TV MSE vs TV FairReg OT-U W2 OT-… view at source ↗
Figure 4
Figure 4. Figure 4: Accuracy-fairness trade-offs on the Communities dataset. Relaxation trajecto￾ries from the unconstrained ERM to the exact fair models. MSE of the constant predictor is 5.5 × 10−2 . 5 Limitations and perspectives This work introduced a unified geometric framework for relaxed fair regression, where optimal predictors admit closed-form expressions via optimal transport maps. These characterizations are valuab… view at source ↗
Figure 5
Figure 5. Figure 5: Unaware relaxations. Comparison of the Bayes predictor against the optimal relaxed unaware predictors under W2 and TV penalties, matched to an 80% unfairness budget. Qualitative analysis of the aware relaxations To complement the unaware visualizations above, we empirically evaluate the exact same relaxations in the aware setting on our 1D synthetic dataset. As shown in [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 6
Figure 6. Figure 6: Aware relaxations. Comparison of the base ERM predictor against the optimal relaxed aware predictors under W2 and TV penalties. While W2 yields a smooth shift, the TV penalty applies an exact hard-thresholding, creating a discontinuous effect where predictions either perfectly merge or go back to the unfair ERM. E.2 Impact of confounding strenght and trade-offs We generate n independent samples from a cont… view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the aware marginal distributions. Evolution of the predicted marginal distributions from Unfair to Fair. The discontinuous nature of the aware TV hard￾thresholding creates fragmented, intermediate distributions, contrasting with the smooth contin￾uous shift of the W2 penalty. Sensitive attribute and features. We sample the binary sensitive attribute uniformly at random S ∼ Bernoulli(0.5), and … view at source ↗
Figure 8
Figure 8. Figure 8: Accuracy-fairness trade-offs on the synthetic dataset (γ = 0.5). Relaxation trajectories from the unconstrained empirical risk minimizer (ERM, black dot) to the strict fair models (stars). The unconstrained baseline naturally converges to the theoretical noise variance limit (MSE = 0.25). Our exact optimal transport mappings (OT-U W2 and OT-U TV) recover the optimal aware Pareto frontier and outperform bot… view at source ↗
Figure 9
Figure 9. Figure 9: Impact of confounding strength (Absolute). Evolution of the MSE (left) and W2 unfairness (right) under strict DP. As γ → 1, OT Unaware converges to the Aware model, tightly bounding the MSE degradation. FairReg suffers from optimization instability, triggering severe drops in predictive utility. 10 2 10 1 10 0 10 0 MSE / base MSE vs 10 2 10 1 10 0 10 1 10 0 W2 / b ase W2 vs ERM OT-U 2 OT-A 2 FairReg OT S 2… view at source ↗
Figure 10
Figure 10. Figure 10: Impact of confounding strength (Relative). Evolution of the MSE and W2 unfairness expressed relative to the unconstrained ERM baseline. The relative visualization highlights FairReg’s rapid deterioration in utility at moderate confounding (γ ≈ 0.1), whereas OT Unaware maintains stability and accurately tracks the theoretical aware bound. 10 2 10 1 10 0 10 0 MSE MSE vs 10 2 10 1 10 0 10 0 2 × 10 0 3 × 10 0… view at source ↗
Figure 11
Figure 11. Figure 11: Within-Group Variance Analysis. Evolution of the MSE (left) and within-group prediction variance (right) across varying confounding strengths (γ). At low γ, OT Unaware conservatively collapses the variance to act as a safe, constant predictor since the density ratio ∆ˆ (x) is uninformative. As structural bias increases, it successfully resolves the 2D geometry and recovers the variance profile of the opti… view at source ↗
Figure 12
Figure 12. Figure 12: Accuracy-fairness trade-offs on the Adult dataset. Relaxation trajectories from the unconstrained empirical risk minimizer (ERM, black dot) to the exact fair models (stars). Error bars indicate the standard deviation across 10 random data splits. MSE of the constant predictor is 1.69 × 10−2 . F.2 Evaluation of the fully constrained models Experimental setup. Tables 2 to 4 present the performance of the fu… view at source ↗
read the original abstract

Fairness-accuracy trade-offs are a central concern in the deployment of fairness-aware machine learning methods. When sensitive attributes are unavailable at inference time-the so called unawareness setting, principled methods for obtaining accurate predictions under relaxed fairness constraints are largely missing. In this work, we address this gap by formulating regression under a demographic parity penalty as an optimal transport problem. Our framework unifies both the \emph{aware} and \emph{unaware} settings and characterizes optimal prediction functions via optimal transport maps, under both squared Wasserstein-2 and Total Variation penalties. These results reveal that the choice of penalty reflects fundamentally different fairness philosophies: the Wasserstein penalty induces a smooth, population-wide compromise, while Total Variation enforces exact parity for a subset of individuals. Building on these theoretical characterizations, we propose an algorithm that is simple to implement, computationally efficient, and consistently matches or outperforms state-of-the-art baselines on real-world benchmarks.

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.

Referee Report

2 major / 2 minor

Summary. The paper formulates regression under a demographic parity penalty as an optimal transport problem. It unifies the aware and unaware settings by characterizing optimal prediction functions via OT maps, for both squared Wasserstein-2 and Total Variation penalties. The work distinguishes the fairness philosophies induced by each penalty and proposes a simple, efficient algorithm that matches or exceeds baselines on real-world data.

Significance. If the OT characterizations are rigorous, the paper supplies a geometric lens on relaxed fairness constraints that directly yields practical algorithms for both aware and unaware regimes. The explicit link between penalty choice and population-wide versus subset-exact parity is a useful conceptual contribution, and the algorithm's reported performance suggests immediate applicability.

major comments (2)
  1. [§4] §4 (unaware case): The derivation equates the demographic parity penalty to an OT problem whose solutions are OT maps, but does not explicitly enforce that the regressor must be measurable with respect to σ(X) alone. If the resulting map depends on S, the claimed unification requires an additional conditional-expectation or projection step whose effect on the attained penalty value is not shown to be zero; this step is load-bearing for the unaware claim.
  2. [Theorem 3.1] Theorem 3.1 / Eq. (12): The statement that the Wasserstein-2 penalty induces a 'smooth, population-wide compromise' is presented as following directly from the OT map, yet the proof sketch does not quantify how the map behaves under the demographic parity constraint when the regressor class is restricted; a counter-example or explicit bound would strengthen the claim.
minor comments (2)
  1. Notation for the sensitive attribute S and the predictor f is introduced inconsistently between the aware and unaware sections; a single table of symbols would improve readability.
  2. The experimental section reports aggregate metrics but omits per-dataset variance or statistical significance tests against the strongest baseline; adding these would make the performance claims easier to assess.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [§4] §4 (unaware case): The derivation equates the demographic parity penalty to an OT problem whose solutions are OT maps, but does not explicitly enforce that the regressor must be measurable with respect to σ(X) alone. If the resulting map depends on S, the claimed unification requires an additional conditional-expectation or projection step whose effect on the attained penalty value is not shown to be zero; this step is load-bearing for the unaware claim.

    Authors: We agree that the unaware case requires an explicit treatment of the σ(X)-measurability constraint. The OT map is initially derived on the joint law of (X,S,Y), but the unaware regressor must be a function of X alone. In the revision we will insert a dedicated paragraph in §4 that applies the conditional-expectation projection onto σ(X)-measurable functions and proves that this projection leaves the demographic-parity penalty unchanged: the penalty is a functional of the marginal push-forward measure of the predictions, which is invariant under conditioning on X. This step therefore does not alter the attained penalty value and preserves the claimed unification. revision: yes

  2. Referee: [Theorem 3.1] Theorem 3.1 / Eq. (12): The statement that the Wasserstein-2 penalty induces a 'smooth, population-wide compromise' is presented as following directly from the OT map, yet the proof sketch does not quantify how the map behaves under the demographic parity constraint when the regressor class is restricted; a counter-example or explicit bound would strengthen the claim.

    Authors: Theorem 3.1 characterizes the unrestricted optimal map, which illustrates the population-level geometry induced by the Wasserstein penalty. When the function class is restricted, the result serves as an ideal benchmark. We will revise the discussion following Eq. (12) to state this distinction explicitly and add a quantitative bound on the sub-optimality gap that arises from the restriction, obtained from the Lipschitz continuity of the Wasserstein-2 map with respect to the marginal constraint. This addition will make the scope of the claim precise. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard OT theory

full rationale

The paper derives its unification of aware and unaware fair regression by mapping the demographic parity penalty to an optimal transport problem and characterizing solutions via OT maps under W2 and TV penalties. This equivalence is presented as following from the mathematical definition of the penalties and standard OT results, without any reduction of predictions to fitted parameters by construction, self-definitional loops, or load-bearing self-citations. The unaware setting constraint (regressors measurable w.r.t. X only) is addressed as part of the framework setup rather than smuggled in via prior author work. No steps match the enumerated circularity patterns, and the central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; full paper would list explicit assumptions on loss functions, support of distributions, and existence of transport maps. No free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Demographic parity penalty can be exactly encoded as a cost in an optimal transport problem whose solutions are OT maps.
    Core modeling step that enables the unification and characterizations.

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discussion (0)

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    On the one hand, plugging this value into the first-order conditions yield ( y∗ 1 = h1(1+λa2)+λh2a1 1+λ(a1+a2) , y∗ 2 = h2(1+λa1)+λh1a2 1+λ(a1+a2)

    From the optimality conditions we have ( y∗ 1 −h 1 =−λa 1u, y∗ 2 −h 2 =λa 2u, Subtracting these identities givesu = (h1 −h 2) −λ (a1 + a2)u, which resolves tou = h1−h2 1+λ(a1+a2). On the one hand, plugging this value into the first-order conditions yield ( y∗ 1 = h1(1+λa2)+λh2a1 1+λ(a1+a2) , y∗ 2 = h2(1+λa1)+λh1a2 1+λ(a1+a2) . (12) On the other hand, plug...

  39. [39]

    According to Lemma 2, a solution(ν+∗, ν−∗)of (Pλ,D) is given by the marginal distributions of (y∗ 1(Z1, Z2), y∗ 2(Z1, Z2))where( Z1, Z2) ∼π ∗ and π∗ is as in(5)

    = (λa1u)2 a1 + (λa2u)2 a2 +λu 2 =λu2 (1 +λ(a 1 +a 2)) =λ (h1 −h 2)2 (1 +λ(a 1 +a 2))2 (1 +λ(a 1 +a 2)) = λ 1 +λ(a 1 +a 2)(h1 −h 2)2. According to Lemma 2, a solution(ν+∗, ν−∗)of (Pλ,D) is given by the marginal distributions of (y∗ 1(Z1, Z2), y∗ 2(Z1, Z2))where( Z1, Z2) ∼π ∗ and π∗ is as in(5). Our next goal is to show that under Assumption 1, the random v...

  40. [40]

    for allz 1,ψ(z 1) = inf z2(ϕ(z2) +C λ,c2(z1, z2)); 2.π ∗ is supported on theCu λ,c2-subdifferential of a Kantorovich potentialψ, that is the set Γ ={(z 1, z2)∈ Z × Z:ψ(z 1)−ϕ(z 2) =C λ,c2(z1, z2)}. The arguments used to prove Lemma 3 in Divol and Gaucher[12] can be reproduced to show that under Assumption 1, the Kantorovich potentialψ is differentiable in...

  41. [41]

    In particular, the two output laws become arbitrarily close inW2

    Since R h2 |d| dµ± η,∆(z) = R X± η(x)2 dχ(x) ≤ E[η(X)2] <∞ , we haveW2 2(ν∗ 1,λ, ν∗ 2,λ) → 0as λ→ ∞ . In particular, the two output laws become arbitrarily close inW2. Therefore the relaxed solutions asymptotically enforce a single common output distribution; if the unrelaxed barycenter is unique, this common limit must coincide with the unrelaxed barycen...

  42. [42]

    Thus Cλ,c0(z1, z2) = λ and thus y∗ 1(z1, z2) = h1 clearly only depends onz1

    +λ−ϕ(z 2)≥λ. Thus Cλ,c0(z1, z2) = λ and thus y∗ 1(z1, z2) = h1 clearly only depends onz1. This shows that the mapT + λ,TV is well-definedµ + η,∆-almost everywhere. Similarly, we show that the mapT− λ,TV is well-definedµ − η,∆-almost everywhere. We then conclude using Lemma 1 to show that solutions of Equation (3) take the form described in the lemma. Let ...

    2000

  43. [43]

    For the Law School dataset, the initialW2 gap is approximately0.03

    between the initial biased distributions. For the Law School dataset, the initialW2 gap is approximately0.03. The exact geometric cost to perfectly repair this disparity is therefore bounded by0.032 = 0.0009. Because the targets are scaled to [−1, 1], the base ERM risk is naturally around0.010. Adding the absolute maximum fairness penalty yields an expect...