Exact demographic parity requires the same prediction distribution across sensitive groups. In regression, this can be too restrictive and reduce predictive accuracy. We instead bound every pairwise Wasserstein distance between group-specific prediction distributions by a chosen tolerance, allowing the groups to differ while keeping fairness tunable. On the real line, the prediction and constrained transport problems have the same infimum. If the transport problem attains this value and suitable Monge maps exist, a deterministic transport-form predictor realizes it. Under a standard one-dimensional condition, the problem becomes a bounded quantile optimization that is convex when the conditional expected loss is convex in prediction. Under squared loss with two positively weighted groups, the W1, W2, and W∞ constraints yield soft-thresholding, shrinkage, and clipping. Under the stated conditions, we establish rates for the known-cost plug-in estimator. When the cost is unavailable, we describe an observed-loss computational proxy. Synthetic and California 2018 ACS studies illustrate the tunable fairness-accuracy trade-off and distinguish calibration feasibility from held-out fairness and accuracy.
Figures & tables
p
Optimal difference Δ
Effect
2
Δ=Δ∗ if D≤ϵ ; otherwise Δ=(ϵ/D)Δ∗ .
shrink
1
Δ=Sλ(Δ∗) .
soft-thresh
∞
Δ=clip[−ϵ,ϵ](Δ∗) .
clip
Table 1: Closed-form Δ updates (squared loss, K=2 ). Let Δ∗=Q1∗−Q2∗ and Δ=Q1−Q2 , with constraint ∥Δ∥p≤ϵ . Write D=∥Δ∗∥2 , Sλ(t)=sgn(t)(∣t∣−λ)+ , and clip[−ϵ,ϵ](t)=min{ϵ,max{−ϵ,t}} . For p=1 , choose λ so that ∥Sλ(Δ∗)∥1=ϵ when 0<ϵ<∥Δ∗∥1 . Use λ=0 when the constraint is inactive. Use Δ=0 when ϵ=0 .
Constraint
Fair difference Δ^
Interpretation
W2
Δ^=Δ^∗ if D^≤ϵ ; otherwise Δ^=(ϵ/D^)Δ^∗ .
global shrinkage
W1
Δ^=Sλ^(Δ^∗) .
soft-thresholding
W∞
Δ^(u)=min{ϵ,max{−ϵ,Δ^∗(u)}} .
clipping
Table 2: Plug-in fairness adjustment for squared loss and two groups. Let Δ^∗=Q^1∗−Q^2∗ and let Δ^=Q^1−Q^2 . For W2 , write D^=∥Δ^∗∥2 . For W1 , choose λ^ so that ∥Sλ^(Δ^∗)∥1=ϵ when 0<ϵ<∥Δ^∗∥1 . Set λ^=0 when the constraint is inactive. Use Δ^=0 when ϵ=0 .
Figure 1: Fairness–accuracy frontiers for DGP1 (top) and DGP2 (bottom) at n=2000 . The curves become steep near exact parity: the last reduction in the held-out gap costs more MSE than earlier reductions. Each panel uses its named Wasserstein norm, so horizontal scales should not be compared. Curves are Monte Carlo means; ribbons show plus or minus one standard error.
DGP1
DGP2
Constraint
MSE
Test gap
MSE
Test gap
L1
1.5550(0.0179)
1.0236(0.0351)
1.7159(0.0204)
1.2659(0.0394)
L2
1.5493(0.0178)
1.0700(0.0341)
1.7519(0.0219)
1.3802(0.0377)
L∞
1.3862(0.0159)
2.4619(0.0545)
1.4192(0.0147)
3.8044(0.0743)
Table 3: Simulation results at τ=0.5 (that is, ϵ=0.5Dmax ). Entries are means over R=50 replications, with standard errors (SE) in parentheses. “Test gap” is the achieved test-set Wasserstein distance in the norm named by that row.
Figure 2: Calibration quantiles for the linear mean-shift process (DGP1) at ϵ=0.5Dmax . Because the groups differ across nearly all ranks, each rule makes a broad adjustment. Dashed curves are original; solid curves are adjusted.
Figure 4: Held-out pairwise W2 gaps in the K=3 experiment. As the tolerance grows, the low-versus-high pair becomes the largest held-out gap, while the other two pairs remain closer. The dashed line is the largest gap between the fitted group curves on the calibration grid; the dotted line is the calibration tolerance.
Figure 5: Calibration quantiles before and after the K=3 adjustment at half the baseline gap. Panel (A) shows the unadjusted calibration quantile curves. Panel (B) overlays the unadjusted curves (dashed) and the adjusted curves (solid). The smallest group moves most, while the middle and largest groups move much less. Unequal weights and the active pairwise constraints jointly shape these changes. Flat stretches arise when the ordering projection pools neighboring ranks.
Figure 6: Calibration-grid W∞ gap versus held-out accuracy. Light points show the 30 observed values. The solid line is a fitted summary constrained not to decrease. The black star marks ϵ=0 , where the grid gap is zero, and the red square marks the unadjusted model at its gap before adjustment. The line does not imply that held-out accuracy is monotone.
Figure 7: California 2018 ACS income regression with K=2 groups. The curves show all 30 observed settings per norm in tolerance order. The marked near-half setting is τ=14/29 . At τ=1 , the identity convention preserves the unadjusted predictions, so the final point coincides with the unadjusted model. The final rise reflects both the tolerance change and the bypass of empirical rank reconstruction. The exact-calibration-parity setting can retain a nonzero held-out gap. Each panel uses its own Wasserstein norm and should be read separately.
Figure 8: Maximum pairwise W2 distance between the fitted group curves on the calibration grid versus held-out R2 . Light points show the 25 observed values. The solid line is a fitted summary constrained not to decrease. The black diamond marks ϵ=0 , where the grid gap is zero, and the red star marks the unadjusted model at its gap before adjustment. The line does not imply that held-out R2 is monotone.
Figure 9: Smoothed distributions of the California 2018 ACS calibration predictions before and after the K=3 adjustment at ϵ=0.5Dmax . The smoothed group densities remain distinct at this positive tolerance.
Algorithmic fairness studies how automated decisions distribute errors, opportunities, and harms across groups, and how to measure and reduce disparities.