Research map / Learning theory and algorithms / Fairness
Kernel-based Equalized Odds: A Quantification of Accuracy-Fairness Trade-off in Fair Representation Learning
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Proposes a kernel/MMD-based equalized odds statistic EO_k for fair representation learning, analyzing its relations to independence and calibration under unbiased and biased Y-S dependence, with a concentration bound; purely theoretical.
Problem
Fair representation learning must trade off incompatible fairness criteria (independence, separation/equalized odds, calibration) against predictive accuracy, and existing metrics do not characterize this trade-off when Y and S are dependent.
Approach
Defines EO_k as an MMD between Y-reweighted mixtures of group-conditional representations over an affine RKHS unit ball of predictors; shows it recovers DP and EO when Y is independent of S, and lower-bounds DP and calibration violations when Y and S are dependent, while preserving Bayes-optimal accuracy.
Key results
- When Y is independent of S, sup over the RKHS-ball predictor class of DP equals (2 nu^{1/2})^{-1} times EO_k, so minimizing EO_k enforces both independence and separation.
- When Y and S are dependent, DP (and hence calibration) is lower-bounded by a term involving |p_{0|0}-p_{0|1}| and EO_k, showing an inherent accuracy-fairness conflict.
- A concentration bound gives |EO_k^hat^2 - EO_k^2| of order sqrt((log d + log(1/delta))/(n0+n1)) for feed-forward encoders with high probability.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Minimize EO_k in the unbiased setting (Y independent of S) | Kernel-based Equalized Odds (EO_k) for fair representation learning · Y independent of S | Demographic Parity (DP) | Helped |
| Use EO_k rather than DP as the fairness constraint in the biased setting (Y dependent on S) | Fair representation learning · Y dependent on S | predictive accuracy | Helped |
| Enforce calibration for all downstream predictors | Fair representation learning in the biased setting · Y dependent on S | predictive accuracy | Hurt |
Surprising findings
- Calibration violation is linearly lower-bounded by DP violation, so enforcing calibration in biased settings is stricter than enforcing independence alone.
Limitations
- Purely theoretical; no empirical experiments validating the metric or the trade-off.
- Bounds depend on bounded kernel and Gaussian complexity assumptions; neural encoder bound relies on covering-number and weight-norm assumptions.
Experimental setup
Baselines: TVD-based fairness regularizers MMD fairness regularizers adversarial fair representation learning
Builds on: maximum mean discrepancy demographic parity equalized odds calibration fairness measures (DPC/DNC) total variation distance IPM penalties
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