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Understanding Fairness and Prediction Error through Subspace Decomposition and Influence Analysis
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Proposes Sequential Fair Projection (SFP), which uses sufficient dimension reduction to split the feature space into target-relevant, sensitive, and shared directions, then adds shared directions incrementally to trade fairness against accuracy, with influence-function asymptotics.
Problem
Fairness methods usually add constraints at the prediction level and ignore biases encoded in data representations, so the utility-fairness trade-off is not controlled at the representation level.
Approach
Decompose the central subspaces of the target Y and sensitive attribute Z into Z-orthogonal directions and shared directions via SDR (including an interaction-based eigen-decomposition of M_Y Σ M_Z), then build partially fair projections by sequentially adding shared directions, and analyze the resulting error, dCov-based dependence, and estimator asymptotics via influence functions.
Key results
- On Adult, SFP reports the highest accuracy (76.88%) and lowest DP gap (3.78), but is not best on MCDP (10.52 vs RLACE 7.45).
- On Bank, SFP reports 89.66% accuracy and DP gap 0.51, while its TPR gap (2.33) and MCDP (10.57) are not best.
- In simulations, adding shared dimensions lowers test RMSE and parameter distance to the original OLS estimator but raises the RMSE gap between sensitive groups.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SFP instead of AdvDebias | Adult classification · Adult | accuracy: 76.49 (0.65) -> 76.88 (0.47) | Helped |
| Use SFP instead of FairMixup | Adult classification · Adult | DP: 3.93 (1.34) -> 3.78 (0.15) | Helped |
| Use SFP instead of SUP | Adult classification · Adult | TPR gap: 4.37 (0.29) -> 4.43 (0.26) | Hurt |
| Use SFP instead of RLACE | Adult classification · Adult | MCDP: 7.45 (0.86) -> 10.52 (0.72) | Hurt |
| Use SFP instead of Logistic Regression | Bank classification · Bank | accuracy: 91.17 -> 89.66 (0.27) | Hurt |
| Use SFP instead of INLP | Bank classification · Bank | DP: 0.93 (0.21) -> 0.51 (0.26) | Helped |
| Use SFP instead of DRAlign | Bank classification · Bank | TPR gap: 1.96 (0.21) -> 2.33 (0.27) | Hurt |
| Use SFP instead of RLACE | Bank classification · Bank | MCDP: 8.98 (0.38) -> 10.57 (0.34) | Hurt |
| Add shared target-sensitive directions to the projection | SFP · Synthetic simulation | MSE | Helped |
| Add shared target-sensitive directions to the projection | SFP · Synthetic simulation | MSE gap | Hurt |
| Add shared target-sensitive directions to the projection | SFP · Synthetic simulation | distribution discrepancy | Hurt |
| Include shared directions in the representation | SFP · Theoretical prediction setting | approximation error | Helped |
Surprising findings
- On Bank, the fair SFP model reaches 89.66% accuracy, close to unconstrained logistic regression (91.17%), while other debiasing baselines fall to about 60-72%.
- Under model misspecification the estimated shared dimension was 28 for p=30, so the low-rank structure was not recovered.
Limitations
- Relies on linear SDR assumptions; nonlinear dependence is not captured.
- Asymptotic results rely on Hadamard differentiability and consistency claims that are largely omitted or stated without proof for the SDR estimators.
- Real-data gains are mixed: SFP is not best on TPR gap or MCDP for several settings.
Experimental setup
Datasets: Adult Bank
Benchmarks: Adult Bank synthetic simulation
Metrics: accuracy DP gap TPR gap MCDP RMSE RMSE gap Wasserstein distance
Baselines: Logistic Regression AdvDebias FairMixup DRAlign DiffMCDP INLP RLACE SUP
Builds on: sufficient dimension reduction RLACE INLP distance covariance
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