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arxiv: 2605.28251 · v1 · pith:RSAEG66Vnew · submitted 2026-05-27 · 📊 stat.ML · cs.CY· cs.LG

Counterfactually Fair Regression via Optimal Transport

M. Generali Lince , S. Gaucher , J-J. Vie , P. Loiseau This is my paper

Pith reviewed 2026-06-29 09:59 UTC · model grok-4.3

classification 📊 stat.ML cs.CYcs.LG
keywords counterfactual fairnessoptimal transportregressiondemographic paritypost-processingfinite-sample boundscausal machine learning
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The pith

Counterfactual fairness equals demographic parity conditional on the latent variable, giving a closed-form optimal fair regressor via barycentric quantile map.

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

The paper shows that counterfactual fairness, defined via resampled noise, is equivalent to demographic parity conditional on the latent variable. This equivalence supplies a closed-form expression for the optimal fair regressor through a barycentric quantile map. For continuous latent variables a discretized post-processing estimator is introduced that attains high-probability finite-sample fairness guarantees with unfairness decaying at rate Õ(n^{-1/3}) together with a matching risk bound. The results extend to relaxed counterfactual fairness.

Core claim

Counterfactual fairness is equivalent to satisfying demographic parity conditional on the latent variable. This equivalence yields a closed-form expression of the optimal fair regressor via a barycentric quantile map. The discretized post-processing method provides finite-sample fairness guarantees at rate Õ(n^{-1/3}) with matching risk bounds and a matching lower bound on excess risk for almost fair predictions.

What carries the argument

Barycentric quantile map obtained from optimal transport that enforces conditional demographic parity while minimizing regression risk.

If this is right

  • The optimal fair regressor admits an explicit closed-form expression via the barycentric quantile map.
  • The estimator satisfies high-probability finite-sample fairness at rate Õ(n^{-1/3}).
  • A matching lower bound holds for the excess risk of predictors that are almost counterfactually fair.
  • The same guarantees extend to relaxed counterfactual fairness.

Where Pith is reading between the lines

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

  • The discretization step may be reusable for other post-processing fairness methods that involve continuous conditioning variables.
  • The cubic dependence on sample size suggests practical limits on how small the fairness tolerance can be made without very large datasets.
  • Similar optimal-transport reductions could apply to other causal fairness definitions that involve latent noise resampling.

Load-bearing premise

The underlying distributions satisfy mild regularity conditions that justify the convergence of the empirical maps and the discretization approximation.

What would settle it

On data drawn from distributions satisfying the regularity conditions, observe whether the post-processed estimator's conditional demographic parity violation decays slower than n to the power of minus one third with high probability.

Figures

Figures reproduced from arXiv: 2605.28251 by J-J. Vie, M. Generali Lince, P. Loiseau, S. Gaucher.

Figure 1
Figure 1. Figure 1: Structural Causal Model. Illustration. Throughout the paper, we consider the fol￾lowing example: S is a gender; V is a student’s intrinsic ability; X aggregates coursework signals (e.g., homework, class participation of previous year) downstream of (V, S); and Y is the future year course grade, downstream of X. Both X and Y reflect systemic biases [29]. Two students with the same V can have different Y due… view at source ↗
Figure 2
Figure 2. Figure 2: Synthetic Dataset. Left: Unfairness drops and stabilizes near the theoretical L ∗ th. Middle: Our method (purple) dominates WFR (gray) on the CF tradeoff frontier. Right: WFR exhibits a better DP tradeoff frontier. 7 Experiments In the main text, we evaluate on a synthetic and real-world Law School (LSAC) dataset, comparing our post-processing method against two baselines: Counterfactual Fairness (Fair K) … view at source ↗
Figure 3
Figure 3. Figure 3: LSAC: Analysis of fairness and risk trade-offs. (Left) Unfairness vs. number of intervals L, showing the bias-variance trade-off with the theoretical L ∗ th indicated by the vertical dashed line. (Center) Pareto frontier of risk vs. unfairness for varying relaxation parameter α, demonstrating smooth interpolation. (Right) Pareto frontier of risk vs. Demographic Parity for different α. Shaded regions repres… view at source ↗
Figure 4
Figure 4. Figure 4: evaluates this empirically on synthetic data. The log-log plot demonstrates that empirical unfairness decays significantly faster than the theoretical bound, reaching near-zero levels (U(fb L∗ ) < 10−5 ) with just n ≈ 2000 samples. Furthermore, because our post-processor operates entirely on 1D scalars rather than high￾dimensional features, it is highly scalable [PITH_FULL_IMAGE:figures/full_fig_p060_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: confirms this on synthetic data for K ∈ {3, 5, 10}: as K increases and data sparsity becomes extreme, the theoretical L ∗ th dynamically adapts to locate the empirical minimum [PITH_FULL_IMAGE:figures/full_fig_p061_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Discretization Trade-off on the Law School Dataset for black-box models not satisfying Properties 4.1 and 4.2. Applied to Gradient Boosting (red squares) and Random Forest (teal circles). The impact of non-smoothness dissipates as L increases, proving the method works effectively without the Lipschitz assumption on the black-box model. Minority Group Imbalance. To evaluate robustness to demographic scarcit… view at source ↗
Figure 8
Figure 8. Figure 8: Robustness to minority group imbalance (wmin). The method maintains stable, near-zero unfairness down to wmin ≈ 0.05. At extreme scarcity, the inability to re￾liably estimate the minority empirical c.d.f. causes a sharp spike in approximation bias, while global RMSE drops as the majority group dominates the metric. 10−1 100 101 Error of estimation (c = Lbcdf /Lcdf ) 10.00 15.00 20.00 25.00 Unfairness ( U( … view at source ↗
Figure 10
Figure 10. Figure 10: Robustness to unbiased proxy noise. Dual axes show that as noise σ in￾creases, unfairness rises mildly (peaking below 0.05) while RMSE decreases. 0.0 0.2 0.4 0.6 0.8 1.0 Proxy leakage (λ) 0.00 1.00 2.00 3.00 Unfairness ( U(fL∗ )) (×10−1 ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 RMSE [PITH_FULL_IMAGE:figures/full_fig_p063_10.png] view at source ↗
read the original abstract

We consider the problem of learning a counterfactually fair regressor. We adopt a causal uncertainty view in which counterfactual fairness is defined with resampled noise. We focus on obtaining theoretical fairness guarantees for a new post-processing estimator. We begin by showing that counterfactual fairness is equivalent to satisfying demographic parity conditional on the latent variable. This allows us to provide a closed-form expression of the optimal fair regressor via a barycentric quantile map. In order to handle continuous latent variables, we propose a discretized post-processing method. Then, under mild regularity assumptions, we prove high-probability finite-sample fairness guarantees for our estimator, providing an unfairness decay at rate $\tilde O(n^{-1/3})$, and establishing a matching risk bound of order $\tilde O(n^{-1/3})$. We provide a matching lower bound on the excess risk of almost fair predictions. Finally, we extend our results to the setting of relaxed counterfactual fairness. We validate our approach on real-world and synthetic data.

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

0 major / 2 minor

Summary. The paper considers learning a counterfactually fair regressor under a resampled-noise definition of counterfactual fairness. It establishes equivalence to demographic parity conditional on the latent variable, yielding a closed-form optimal fair regressor via a barycentric quantile map. A discretized post-processing estimator is proposed for continuous latent variables. Under mild regularity assumptions, high-probability finite-sample fairness guarantees are proved with unfairness decaying at rate ilde O(n^{-1/3}), along with a matching excess-risk bound of the same order, a matching lower bound on excess risk for almost-fair predictors, and an extension to relaxed counterfactual fairness. The approach is validated on synthetic and real-world data.

Significance. If the derivations hold, the work makes a solid contribution by linking counterfactual fairness to conditional demographic parity and optimal transport, delivering an explicit closed-form solution and finite-sample rates with a matching lower bound. These elements provide concrete, falsifiable guarantees that go beyond typical heuristic post-processing in fair ML, and the use of standard one-dimensional OT regularity conditions keeps the assumptions mild and interpretable.

minor comments (2)
  1. [Abstract] The abstract and main text use \tilde O without an explicit definition on first appearance; adding a short clarification (e.g., hiding polylog factors) would improve readability.
  2. The description of the discretization step for continuous latents would benefit from a brief remark on how the grid size is chosen in practice and its effect on the stated rates.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our contributions, the assessment of significance, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins by establishing an equivalence between counterfactual fairness (under the resampled-noise definition) and conditional demographic parity given the latent variable; this equivalence is derived rather than assumed by definition. The optimal fair regressor is then expressed via the barycentric quantile map from one-dimensional optimal transport, which is an external property applied to the equivalence. Finite-sample high-probability fairness guarantees at rate Õ(n^{-1/3}) and the matching excess-risk bound are obtained from concentration arguments and discretization under explicitly stated mild regularity conditions on the distributions. A separate lower bound on excess risk is provided for grounding. No steps reduce by construction to fitted inputs, self-citations, or renamed ansatzes; the central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard causal model assumptions for counterfactuals defined via resampled noise and on regularity conditions for convergence; no new entities are postulated and no free parameters are explicitly fitted beyond the discretization choice.

axioms (1)
  • domain assumption Mild regularity assumptions on the underlying distributions and latent variable
    Invoked to establish the high-probability finite-sample bounds and the quality of the discretized approximation.

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