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Generalizing Orthogonalization for Models with Non-Linearities
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Extends linear orthogonalization to non-linear models (GLMs, ReLU layers) and tensor-valued predictions so that protected attributes are removed from predictions and learned embeddings.
Problem
Classical orthogonalization (projecting features onto the orthogonal complement of protected features) only removes linear effects of protected attributes, so it fails when predictions pass through non-linear activations such as sigmoid, exp, or ReLU, leaving biases such as race or sex in prediction
Approach
Generalizes orthogonalization to non-linear predictors by (i) deriving a GLM correction that orthogonalizes the predictions via a constrained fit solved with a modified differential multiplier method, (ii) proving that projecting pre-activations with the orthogonal projection yields zero ReLU-evaluation coefficients, and (iii) extending the projection to tensor-valued outputs via a mode-1 product
Key results
- On MIMIC-CXR embeddings, generalized orthogonalization reduces sex, age, and race effects to non-significant (e.g., Black race coefficient -0.001, p=0.753) while classical orthogonalization leaves significant effects, with AUC increasing by 0.02.
- On adult income, the proposed correction brings race and sex coefficients to near zero (e.g., sex male -0.000, p=0.998) versus 1.066 without correction, with accuracy dropping from 0.854 to 0.833.
- On UTKFace, correcting the ResNet-50 embedding for age and race lowers AUC from 0.831 to 0.826 while making race and age effects non-significant.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply generalized orthogonalization | Fairness benchmark GLMs · adult | accuracy: 0.854 -> 0.833 | Hurt |
| Apply generalized orthogonalization | Fairness benchmark GLMs · bank | accuracy: 0.899 -> 0.887 | Hurt |
| Apply generalized orthogonalization | Fairness benchmark GLMs · compas | accuracy: 0.739 -> 0.724 | Hurt |
| Apply generalized orthogonalization | Fairness benchmark GLMs · health | root mean squared error: 0.750 -> 0.789 | Hurt |
| Apply generalized rather than classical orthogonalization | MIMIC-CXR Pleural Effusion GLM · MIMIC-CXR Pleural Effusion | absolute protected-feature coefficient: 2.237 -> -0.011; -0.483 -> -0.001 | Helped |
| Correct Adult Income predictions for sex and race | Adult Income GLM · Adult Income | absolute protected-feature coefficient: 1.066 -> -0.000; 0.522 -> 0.005 | Helped |
| Correct learned representations for age and race | ResNet50 sex classifier · UTKFace sex classification | AUC: 0.831 -> 0.826 | Hurt |
| Correct movie-description embeddings for protagonist and director gender | LSTM movie review-count model · Movie Reviews | root mean squared error: 1% | Hurt |
| Orthogonalize the first convolutional layer with respect to red-channel information during training | CNN · MNISTred counterfactual test set | accuracy | Helped |
Surprising findings
- Classical linear orthogonalization of embeddings still leaves significant race and age effects on the probability scale for GLM predictions, showing linear orthogonalization is insufficient for non-linear models.
- Fairness methods (Komiyama, Berk, Zafar) still show significant protected-feature effects, so they cannot be used as substitutes for orthogonality.
Limitations
- Requires n larger than number of features p and more prediction features q than protected features p.
- Requires knowledge of features Z related to the protected information; cannot detect unknown biases.
- Orthogonality does not guarantee stochastic independence or absolute protection of sensitive information; the evaluation model is linear in protected features.
Experimental setup
Datasets: Adult income Bank Compas Health Retirement MNIST (colorized) MIMIC-CXR embeddings UTKFace Movie Reviews
Benchmarks: Adult income Fairness benchmark (adult, bank, compas, health.retirement) MIMIC-CXR pleural effusion UTKFace sex classification Movie Reviews Colorized MNIST
Metrics: coefficient estimates and p-values of protected features accuracy AUC RMSE
Baselines: classical orthogonalization Komiyama1 Komiyama2 Berk Zafar
Builds on: classical orthogonalization (metadata normalization) modified differential multiplier method fair regression (Komiyama et al.) Zafar fairness constraints
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