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Meta Optimality for Demographic Parity Constrained Regression via Post-Processing
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Gives meta-theorems showing fair minimax optimal regression under demographic parity can be obtained by post-processing a conventional minimax-optimal regressor with an estimated Wasserstein-barycenter transport map, with a new convergence-rate analysis for that map.
Problem
Fair minimax optimal regression under demographic parity has been analyzed only under specific data-generating models; a general theory linking fair and conventional minimax rates is lacking.
Approach
A meta-theorem bounds the fair minimax error by the conventional minimax error (per group) plus the estimation error of barycenter transport maps; the algorithm fits a conventional regressor on half the data, then estimates transport maps via congruent-potential minimization (sieved M-estimation) on the other half and composes them.
Key results
- Meta-theorem: fair minimax error is bounded above by L^2 * weighted sum of conventional per-group minimax errors plus (n~/ln n~)^{-alpha/(alpha+beta)}, and below by the conventional error.
- Illustrative Sobolev example gives fair minimax rate Theta(max_i n^{-beta_i*/(2beta_i*+t_i)} + n^{-2gamma/(2gamma+1)}) up to log factors, which can be faster than n^{-1/2}.
- Transport-map estimator analysed via sieved M-estimation achieves expected squared error O((n/ln n)^{-alpha/(alpha+beta)}).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Post-process a conventional minimax-optimal regressor with estimated Wasserstein barycenter transport maps | Fair regression under demographic parity · Fair regression (general P) | fair minimax error: \mathcal{E}_n(\dom{P}_1) \le \bar{\mathcal{E}}_n(\dom{P}) \le C\ab(L^2\sum_{s \in [M]}w_s\mathcal{E}_{n_s}(\dom{P}_s) + | Helped |
| Exploit smoothness structure of the optimal transport maps | Fair regression under demographic parity · Fair regression (general P) | convergence rate: faster than n^{-1/2} | Helped |
| Use the post-processing estimator based on empirical multiple-correlation minimization over a sieve | Wasserstein barycenter transport-map estimation · Wasserstein barycenter transport-map estimation | expected squared estimation error: C\ab(\frac{\tilde{n}}{\ln(\tilde{n})})^{-\alpha/(\alpha + \beta)} | Helped |
Surprising findings
- When the conventional regression problem is harder than transport-map estimation (e.g., high-dimensional X), the fair optimal error is dominated by the conventional minimax error, so fairness adds no extra cost.
Limitations
- Minimization over congruent potentials may be computationally hard; convergence under approximate congruency left open.
- Requires assumptions (Lipschitz/strictly increasing transport maps, Poincaré-type inequality, complexity condition) that must be verified per application.
- Purely theoretical; no experiments reported in the excerpt.
Experimental setup
Baselines: Chzhen et al. 2020 fair post-processing Chzhen et al. 2022 minimax fair regression Fukuchi and Sakuma 2023
Builds on: Chzhen et al. 2020 fair regression via optimal transport Korotin et al. 2020 continuous wasserstein barycenter estimation Chzhen et al. 2022 minimax fair regression Fukuchi and Sakuma 2023 demographic parity regression
Closest papers by meaning
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