Relaxed Counterfactual Fairness

• Relaxed counterfactual fairness is a framework that replaces strict invariance in causal models with bounded discrepancies to address finite data challenges and causal uncertainties.
• It employs transport-based methods, latent variable adjustments, and decision-dependent parity criteria to quantify and enforce approximate fairness in predictive models.
• The approach offers explicit fairness–utility trade-offs and flexible operationalizations that make counterfactual fairness more practical in real-world and imperfect causal settings.

Relaxed counterfactual fairness replaces exact invariance under causal interventions with bounded discrepancies, distributional alignments, or surrogate criteria that remain counterfactually motivated but are more implementable in finite data and under causal uncertainty (Lara et al., 2021). In the standard structural-causal formulation, a predictor is counterfactually fair if, for every individual $u$ and for all protected-attribute values $a,a'$, $f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$ (Robertson et al., 2024). Recent work weakens this hard equality in several ways: $\epsilon$-tolerant prediction and Wasserstein constraints (Lara et al., 2021), unfairness budgets of the form $U(f)\le \alpha U(f^*)$ (Lince et al., 27 May 2026), approximate invariance learned from synthetic supervision rather than enforced analytically (Robertson et al., 2024), minimax and Wasserstein relaxations under an imperfect structural causal model (Duong et al., 2023), and decision-dependent counterfactual parity criteria defined with respect to potential outcomes such as $Y(0)$ (Coston et al., 2019, Mishler et al., 2020).

1. Classical counterfactual fairness and the rationale for relaxation

Let $A$ denote a protected attribute, $X$ the observable features, $Y$ the outcome, and $\hat{Y}=f(X,A)$ the prediction. Classical counterfactual fairness requires invariance of the predictor under interventions on $a,a'$0 while holding fixed the latent background variable $a,a'$1. In one common formulation, for all $a,a'$2 and almost all $a,a'$3, $a,a'$4. An equivalent conditional statement is that, given factual evidence $a,a'$5, the predictive counterfactual distributions must coincide: $a,a'$6 (Lara et al., 2021).

The motivation for relaxed formulations is practical. Several papers state that the underlying causal model is usually unknown and difficult to be validated in real-world scenarios, that state-of-the-art models to compute counterfactuals are either unrealistic or unfeasible, and that exact invariance is rarely attained in practice (Lara et al., 2021, Duong et al., 2023). This has produced multiple operational notions of approximation. Some papers replace equality by a distance or probability tolerance; some enforce fairness only after conditioning on latent background variables; some learn a predictor to mimic “fair” synthetic labels; and some replace unobservable counterfactual constraints by observable group metrics under specific causal contexts (Lince et al., 27 May 2026, Robertson et al., 2024, Anthis et al., 2023).

The resulting literature does not present a single canonical relaxation. Instead, it offers a family of counterfactually motivated criteria. In some settings the relaxation is explicit, with parameters such as $a,a'$7, $a,a'$8, or $a,a'$9. In others it is implicit and empirical: fairness is assessed by reductions in counterfactual deviations, causal effects, or decision-dependent disparities rather than guaranteed as a per-individual hard constraint (Robertson et al., 2024, Coston et al., 2019).

2. Distance-based and probabilistic formulations

A direct relaxation replaces exact equality by bounded discrepancies between factual and counterfactual predictions. In the transport-based formulation, prediction-level relaxed counterfactual fairness requires

$f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$0

with $f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$1 taken as a metric such as absolute value, squared loss, or a divergence. The same framework also considers a conditional version per group, a distributional relaxation

$f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$2

and a feature-level realism constraint

$f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$3

which prevents unrealistic counterfactual shifts (Lara et al., 2021).

The same paper introduces an approximate $f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$4 counterfactual fairness rate: for all $f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$5 and $f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$6-almost every $f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$7,

$f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$8

and then aggregates the fraction of points satisfying this condition. This turns strict counterfactual fairness into a rate-based criterion that can be estimated under an empirical coupling (Lara et al., 2021).

Other work uses closely related tolerance formulations. EXOC adopts an approximate variant with tolerance $f_{A\leftarrow a}(U=u)=f_{A\leftarrow a'}(U=u)$9,

$\epsilon$0

and treats this bounded counterfactual discrepancy as its working relaxed notion of counterfactual fairness (Tian et al., 2024). In decision-dependent settings, $\epsilon$1-relaxed counterfactual equalized odds, $\epsilon$2-relaxed counterfactual demographic parity, and $\epsilon$3-relaxed counterfactual predictive parity are defined by requiring the corresponding counterfactual disparities to be at most $\epsilon$4 (Coston et al., 2019). In the binary-risk post-processing setting, the same idea appears as slack variables $\epsilon$5 and $\epsilon$6 controlling counterfactual false-positive-rate and false-negative-rate gaps (Mishler et al., 2020).

A distinct but related relaxation is path-specific. FairPFN notes that current pretraining removes all causal influence of $\epsilon$7, corresponding to a strict “block-all-paths-from-$\epsilon$8” notion in the synthetic supervision, and identifies relaxations that keep only allowed paths as a straightforward extension (Robertson et al., 2024). Counterfactual equal opportunity provides another path-restricted variant: it holds non-sensitive attributes fixed and requires decision invariance to the sensitive attribute, thereby allowing mediated effects through non-sensitive attributes and relaxing strict individual-level counterfactual fairness (Wang et al., 2019).

3. Transport-based counterfactual models

Transport-based counterfactual models define counterfactuals as couplings between observable group-conditional distributions. For two group conditionals $\epsilon$9 and $U(f)\le \alpha U(f^*)$0, a transport-based counterfactual model is a family of couplings $U(f)\le \alpha U(f^*)$1 such that the marginals match the observed group distributions, $U(f)\le \alpha U(f^*)$2 is identity on each group, and symmetry holds under coordinate swap. The central optimization is the Kantorovich optimal transport problem

$U(f)\le \alpha U(f^*)$3

with Wasserstein distance

$U(f)\le \alpha U(f^*)$4

and, in deterministic form, a Monge map $U(f)\le \alpha U(f^*)$5 satisfying $U(f)\le \alpha U(f^*)$6 (Lara et al., 2021).

The paper’s main claim is that optimal transport furnishes statistically faithful and computable counterfactuals. Because the coupling marginals equal the observed data distributions, the generated counterfactuals are in-distribution. Under exogeneity conditions in which $U(f)\le \alpha U(f^*)$7 is exogenous relative to $U(f)\le \alpha U(f^*)$8, the structural counterfactual model itself becomes a family of transport couplings between observables. Under additional invertibility and regularity assumptions, and when the structural counterfactual operator is the gradient of a convex function, the structural counterfactual operator is the unique quadratic-cost Monge map. A corollary covers linear additive structural causal models, where the counterfactual operator is an affine map and therefore the quadratic optimal transport map (Lara et al., 2021).

These results support relaxed counterfactual fairness by replacing inaccessible structural counterfactuals with learned transport counterfactuals. Given an instance $U(f)\le \alpha U(f^*)$9 from group $Y(0)$0, one defines $Y(0)$1 either as a draw from the learned coupling $Y(0)$2 or through a deterministic map $Y(0)$3. Training can then minimize empirical loss plus a prediction-level penalty, a distributional Wasserstein penalty, and optionally a feature-level displacement penalty. The practical pipeline consists of estimating empirical group measures, solving optimal transport exactly or with entropic regularization and Sinkhorn iterations, generating random or barycentric-projection counterfactuals, computing relaxed fairness penalties, and optimizing the predictor with SGD or Adam (Lara et al., 2021).

The framework also clarifies several conceptual points. Under exogeneity, counterfactual fairness implies statistical parity, but not conversely. The same implication holds for transport-based counterfactual fairness using any transport model satisfying the marginal constraints. The paper therefore warns that statistical parity alone is too weak. It also notes an ethical risk: given any statistically parity-fair classifier, one can fabricate a transport-based coupling to make it counterfactually fair under that model, so the choice of counterfactual model must itself be justified (Lara et al., 2021).

4. Latent-variable post-processing and explicit unfairness budgets

A different line of work formulates relaxed counterfactual fairness through a latent background variable. In the regression setting with resampled noise, the structural model uses a discrete sensitive attribute $Y(0)$4, an exogenous latent “ability” variable $Y(0)$5 independent of $Y(0)$6, observed features $Y(0)$7, and outcome $Y(0)$8, with the assumption that $Y(0)$9 almost surely. The counterfactual operator keeps $A$0 fixed, intervenes on $A$1, and resamples the feature noise conditional on $A$2. Under this formulation, counterfactual fairness is equivalent to a demographic-parity-type constraint conditioned on $A$3: for almost every $A$4, the conditional law of $A$5 given $A$6 must be identical across sensitive groups (Lince et al., 27 May 2026).

This equivalence yields a quantitative unfairness functional,

$A$7

where $A$8 is the conditional prediction law at latent value $A$9 and group $X$0. Exact counterfactual fairness is $X$1. The counterfactually fair optimal regressor is obtained by a barycentric quantile map: $X$2 with $X$3. In one dimension, the Wasserstein barycenter has a closed-form quantile expression, so the fair correction is a monotone rank-preserving transport to the conditional barycenter (Lince et al., 27 May 2026).

Relaxation enters through an explicit budget. The paper defines $X$4-CF by

$X$5

and proves that the unique solution is the geodesic mixture

$X$6

The same result gives an exact fairness–utility trade-off: $X$7 This is one of the clearest explicit definitions of relaxed counterfactual fairness in the recent literature (Lince et al., 27 May 2026).

For continuous $X$8, the paper proposes a discretized post-processing estimator based on partitioning $X$9, estimating empirical conditional cdfs and quantiles in each $Y$0 cell, forming interval-wise barycenters, and applying the local rank-to-barycenter map. With $Y$1, it proves a high-probability fairness guarantee

$Y$2

a matching risk bound of order $Y$3, and a minimax lower bound of order $Y$4 up to logarithmic factors (Lince et al., 27 May 2026).

5. Relaxations under unknown or imperfect structural causal models

Several approaches treat relaxed counterfactual fairness as approximate invariance learned without access to the correct structural causal model. FairPFN is a transformer-based prior-fitted network pretrained on synthetic fairness data. It receives a biased dataset as context and is trained to predict “fair” labels generated by dropping all outgoing edges from the protected attribute in synthetic structural causal models. The paper explicitly states that it does not introduce a formal $Y$5 definition; instead, it aims to eliminate the causal effect of the protected attribute and assesses approximation to counterfactual fairness empirically through causal effect reduction and the counterfactual fairness violation measure

$Y$6

Empirically, it dominates Exponentiated Gradient Reductions on 5 out of 6 synthetic case studies, is on the Pareto front in all 6, and achieves the lowest CF-MAE on both Law School and Adult (Robertson et al., 2024).

“Achieving Counterfactual Fairness with Imperfect Structural Causal Model” formulates relaxation through a latent representation $Y$7 and the conditional invariance criterion

$Y$8

Its minimax objective combines a fair-learning predictor $Y$9, a sensitive-aware predictor $\hat{Y}=f(X,A)$0, and a disparity penalty $\hat{Y}=f(X,A)$1, while a Wasserstein term regularizes the learned representation distribution toward an ideal fair one. The paper provides a worst-case generalization bound over a Wasserstein ball and presents the method as a relaxation of strong structural-causal assumptions rather than a computation of true counterfactuals in a validated SCM (Duong et al., 2023).

Generative Counterfactual Fairness Network learns the counterfactual distribution of mediators $\hat{Y}=f(X,A)$2 and enforces fairness through the counterfactual mediator regularization

$\hat{Y}=f(X,A)$3

If $\hat{Y}=f(X,A)$4 is Lipschitz in the mediator argument, then the counterfactual prediction disparity is bounded by generator error plus $\hat{Y}=f(X,A)$5. When the fairness weight $\hat{Y}=f(X,A)$6 is finite, the model balances predictive accuracy with fairness and therefore implements relaxed counterfactual fairness rather than hard equality (Ma et al., 2023).

Representation-learning approaches realize a similar logic. CF-VAE constructs a structured latent representation $\hat{Y}=f(X,A)$7 from a concept-level DAG, adds total correlation regularization and orthogonality promoting regularization, and relies on the proposition that any downstream predictor depending only on non-descendants of $\hat{Y}=f(X,A)$8 is counterfactually fair. The paper explicitly notes that it does not introduce a tolerance-based relaxation; the relaxation is implicit because exact fairness would hold only under perfect representation learning and correct domain knowledge, while the model is evaluated with a situation test that measures individual-level counterfactual instability (Xu et al., 2022). EXOC introduces an auxiliary node $\hat{Y}=f(X,A)$9, a control node $a,a'$00, and the objective

$a,a'$01

where increasing $a,a'$02 tightens fairness and can reduce accuracy. Its approximate fairness notion is again a bounded counterfactual discrepancy with tolerance $a,a'$03 (Tian et al., 2024).

6. Decision-dependent relaxations, observable proxies, and empirical patterns

In risk assessment instruments, relaxed counterfactual fairness is often defined with respect to potential outcomes under decisions rather than interventions on the protected attribute. “Counterfactual Risk Assessments, Evaluation, and Fairness” defines counterfactual analogues of TPR, FPR, PPV, calibration, demographic parity, and equalized odds using $a,a'$04, typically $a,a'$05, and estimates them with doubly robust estimators. It also defines $a,a'$06-relaxed counterfactual equalized odds, $a,a'$07-relaxed counterfactual demographic parity, and $a,a'$08-relaxed counterfactual predictive parity. Its central theoretical claim is that parity in standard observational fairness metrics generally does not imply parity in their counterfactual analogues unless strong balance or independence conditions hold; empirically, fairness corrections targeting observational metrics can worsen counterfactual fairness (Coston et al., 2019).

“Fairness in Risk Assessment Instruments: Post-Processing to Achieve Counterfactual Equalized Odds” specializes this idea to the criterion $a,a'$09. For a binary post-processed score $a,a'$10, relaxed fairness is imposed by linear constraints

$a,a'$11

where $a,a'$12 parameterizes a group-specific randomized mapping $a,a'$13. The paper shows that the resulting optimization is a linear program, that doubly robust coefficient estimators yield fast rates, and that loss and excess unfairness converge at the same rate as the coefficient estimates. In simulations, exact CEO increases loss relative to the unconstrained baseline, whereas relaxed tolerances yield an explicit fairness–performance trade-off (Mishler et al., 2020).

A separate operationalization appears in causal-context analysis. “Causal Context Connects Counterfactual Fairness to Robust Prediction and Group Fairness” argues that, under specific causal graphs and faithfulness, strict counterfactual fairness is equivalent to an observable group metric: measurement error implies equivalence to demographic parity, selection on label implies equivalence to equalized odds, and selection on predictors implies equivalence to calibration. The paper treats this as a relaxation of measurement and testing requirements rather than of the fairness notion itself: instead of observing counterfactuals directly, one can enforce or test the corresponding group fairness metric when the causal context justifies the equivalence (Anthis et al., 2023).

Across empirical studies, several common patterns recur. Optimal-transport regularization steadily improves counterfactual fairness rates with a controlled accuracy or MSE trade-off and typically also improves group fairness metrics such as parity gap or KS distance (Lara et al., 2021). In the latent-variable post-processing setting, conditional optimal transport post-processing dominates global demographic-parity post-processing on the counterfactual-fairness frontier, while proxy quality for the latent variable $a,a'$14 remains critical (Lince et al., 27 May 2026). FairPFN, GCFN, EXOC, and the imperfect-SCM minimax model all report strong fairness–accuracy trade-offs on synthetic and real datasets, but they also identify recurrent limitations: prior mismatch and distribution shift, lack of formal guarantees, unobserved confounding, path-specific needs, support or overlap failures, sensitivity to the transport cost or auxiliary-node specification, and the possibility that counterfactual fairness may be undesirable or infeasible when the protected attribute has downstream effects that encode legitimate signal (Robertson et al., 2024, Ma et al., 2023, Tian et al., 2024, Duong et al., 2023).

In this sense, relaxed counterfactual fairness is not one criterion but a research program. Its unifying idea is to keep the counterfactual perspective while replacing exact, unit-level invariance by a bounded or computable surrogate: a distance on predictions, a transport coupling, a conditional Wasserstein barycenter, a soft adversarial invariance objective, a decision-dependent parity constraint on $a,a'$15, or an observable group metric justified by causal context.