Equity-Conformalized Quantile Regression

• The paper introduces a method that combines split conformal prediction with quantile regression and optimal transport to achieve finite-sample coverage and fairness constraints.
• ECQR is a nonparametric approach that adapts to local heteroscedasticity and enforces demographic parity through a fair post-processing step based on Wasserstein barycenters.
• Empirical evaluations demonstrate competitive interval lengths and a precise fairness-accuracy trade-off in settings with imbalanced representation and groupwise noise.

Equity-Conformalized Quantile Regression (ECQR) designates a class of nonparametric predictive interval procedures that unite three critical statistical demands: (1) finite-sample, distribution-free marginal coverage, (2) adaptivity to local distributional features such as heteroscedasticity, and (3) certified fairness—specifically, demographic parity—across sensitive subpopulations. This methodology synthesizes split conformal prediction, quantile regression, and post-hoc optimal transport synchronization, providing a distribution-free pipeline for learning quantile functions that respect fairness constraints and yield reliable, fair prediction intervals for continuous outcomes (Romano et al., 2019, Liu et al., 2022).

1. Formal Framework and Motivations

Let $(X, S, Y)$ denote the covariates, sensitive group indicator, and continuous response. The foundational goal is to construct, for each $(x, s)$, a predictive interval $C(x, s)$ such that the marginal coverage (over the data-generating distribution) satisfies $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$ for a chosen miscoverage rate $\alpha \in (0, 1)$, and, crucially, both endpoints of $C(x, s)$ are Demographic Parity (DP)–fair under the empirical group-wise distribution of predicted quantiles. Demographic Parity at quantile level $\tau$ demands that $\mathsf{Law}(Q_s(X;\tau))$ be invariant across all sensitive attribute values $s$.

These methods arose in response to the observation that vanilla conformal prediction, while affording valid marginal coverage, ignores subpopulation representation and can yield systematically unequal intervals—typically, wider for underrepresented or noisier groups—thus violating algorithmic fairness criteria (Liu et al., 2022).

2. Base Methodology: Conformalized Quantile Regression

The standard Conformalized Quantile Regression (CQR) algorithm operates in two main stages (Romano et al., 2019):

1. Quantile Regression Step: Fit lower and upper quantile regression models on the training set,

$$\hat q_{\alpha/2}(x) \approx Q_{Y|X}\left(\alpha/2 \mid x\right), \qquad \hat q_{1-\alpha/2}(x) \approx Q_{Y|X}\left(1 - \alpha/2 \mid x\right)$$

by check-loss minimization.

2. Split Conformal Calibration: Compute nonconformity scores $(x, s)$0 on a calibration set. The $(x, s)$1-quantile of the calibration set's scores, $(x, s)$2, is used to form the prediction interval

$(x, s)$3

This procedure yields marginal finite-sample coverage under the sole assumption of data exchangeability

$(x, s)$4

CQR intervals exhibit local adaptivity to heteroscedasticity owing to data-adaptive quantile regression levels, yet the conformal correction is global and does not address group fairness (Romano et al., 2019).

3. Fairness Constraints and Demographic Parity for Quantiles

Equity-Conformalized Quantile Regression extends CQR by enforcing Demographic Parity for the induced distributions of quantile predictions. For categorical $(x, s)$5, DP at quantile level $(x, s)$6 is defined as

$(x, s)$7

In practice, let $(x, s)$8 be a base quantile regressor. One forms the group-$(x, s)$9 marginal CDF $C(x, s)$0 and its generalized inverse (quantile function) $C(x, s)$1. Demographic Parity then becomes

$C(x, s)$2

Thus, a quantile prediction rule is DP-fair if, across all groups, the marginal distribution of predicted quantiles coincides (Liu et al., 2022).

4. Pipeline: Fair Post-processing and Split Conformalization

The ECQR pipeline proceeds as follows (Liu et al., 2022):

1. Data Splitting: Partition $C(x, s)$3 data points $C(x, s)$4 into
   • Proper training set $C(x, s)$5
   • Calibration set $C(x, s)$6
2. Base Quantile Regression: Fit quantile regression models (possibly including $C(x, s)$7 as input)

$C(x, s)$8

1. Functional Synchronization (Fair Quantile Post-processing):
   • For each group and each $C(x, s)$9, estimate the groupwise CDF and invert/smooth to obtain an empirical quantile function.
   • Form the Wasserstein-2 (W$\Pr\{Y \in C(X,S)\} \geq 1-\alpha$0) barycenter quantile:

   $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$1

   where $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$2 is the empirical weight for group $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$3. - For each prediction, map the base quantile $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$4 through its group-specific CDF and into the barycenter quantile (the monotone transport step):

   $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$5

• This guarantees that the distribution of $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$6 is identical across groups (exact DP).

1. Split-Conformal Calibration:
   • Using the calibration set, compute nonconformity scores

   $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$7

• Set $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$8 as the empirical $\Pr\{Y \in C(X,S)\} \geq 1-\alpha$9-th smallest among $\alpha \in (0, 1)$0.
• The final prediction interval is

  $\alpha \in (0, 1)$1

This procedure is model-agnostic in the quantile regression step and post-processes the learned quantile predictions to enforce fairness with negligible computational overhead (Liu et al., 2022).

5. Theoretical Guarantees

Coverage

Under the assumption of exchangeable samples, split-conformalization ensures that

$\alpha \in (0, 1)$2

with at most $\alpha \in (0, 1)$3 overcoverage when the conformity scores are non-tied. The construction is distribution-free and holds for arbitrary finite samples (Liu et al., 2022).

Exact Demographic Parity

The functional synchronization step guarantees

$\alpha \in (0, 1)$4

where $\alpha \in (0, 1)$5 denotes the barycenter-transported predictor at level $\alpha \in (0, 1)$6 (Liu et al., 2022).

Optimality under Fairness Constraint

Given any base quantile predictor $\alpha \in (0, 1)$7, the barycenter-transported version $\alpha \in (0, 1)$8 uniquely minimizes the squared L$\alpha \in (0, 1)$9 distance to $C(x, s)$0 among all Demographic Parity–fair functions:

$C(x, s)$1

The minimized excess risk equals the barycenter objective $C(x, s)$2, indicating that the W$C(x, s)$3 barycenter synchronizes groupwise distributions with minimal distortion (Liu et al., 2022).

6. Practical Considerations and Empirical Performance

Empirical evaluations confirm that ECQR methods deliver marginal coverage near the nominal level (e.g., $C(x, s)$4 within $C(x, s)$5) and achieve group-level parity in the distribution of interval bounds (Liu et al., 2022). The average interval length is generally competitive with or shorter than classical conformal approaches, especially in regimes with groupwise heteroscedasticity or imbalanced representation. The fairness-accuracy trade-off is precisely characterized: the barycenter transport achieves the smallest loss to the original quantiles among all fair solutions.

All standard quantile regression learners are supported, including linear methods, random forests, and neural networks. The additional computational burden is minimal—consisting primarily of one sort and basic post-processing per group—and the procedure supports fully nonparametric, high-dimensional settings.

7. Limitations and Directions for Extension

ECQR provides only marginal, not conditional, coverage:

$C(x, s)$6

may differ across $C(x, s)$7 and $C(x, s)$8 except under restrictive assumptions. The method attains exact DP at the interval endpoints but does not address more stringent fairness notions, such as equalized coverage conditional on covariates or landscape-level risk parity. Extensions could incorporate groupwise or intersectional splits (stratified conformalization) and leverage Mondrian conformal prediction to further localize coverage control. The methodology is applicable to any setting where equitable treatment across sensitive groups and robust, interpretable uncertainty quantification are simultaneous requirements (Liu et al., 2022, Romano et al., 2019).