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Controlling for Omitted Variable Bias in Deep Neural Networks
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Proposes controlling for omitted variable bias in DNNs by refitting the last layer with covariate additive effects under cross-fitting and ridge penalisation, with optional orthogonalisation and covariate marginalisation.
Problem
Shortcut learning in deep networks is a form of omitted variable bias when image-inferable covariates (e.g., sex, age) correlate with the outcome; correlation-based confound control and fairness methods do not remove this bias.
Approach
Refit the final layer of a pre-trained DNN as a partial (generalised) additive model with covariate effects, using cross-fitting so backbone and refit samples are disjoint, ridge penalisation for concurvity, optional orthogonalisation to get residual effects, and marginalisation over covariates for predictions.
Key results
- On the age-confounded training setting, the uncontrolled DNN's AUC on the balanced test set drops from ~0.70 to ~0.62, and age control with disjoint backbone/refit data recovers most of this gap.
- Cross-fitting (K=2) recovers the performance drop seen with simple sample splitting in the age-balanced setting.
- In simulations, the proposed control method is consistent with bias and variance decaying independently of the covariate effect strength, whereas uncontrolled DNNs and post-hoc orthogonalisation retain bias growing with beta_Z.
Surprising findings
- Model outputs can be biased even when predictions are uncorrelated with the omitted covariate, so decorrelation-based confound control does not remove omitted variable bias.
- Controlling for a confounder only helps when the backbone and last-layer refit use disjoint data; without sample splitting the control does not recover the gap.
Limitations
- Assumes confounding covariates Z are observed and correctly modelled; unmeasured confounding is out of scope.
- Estimated image effect depends on expressiveness of the DNN backbone's last-layer features.
- Noisy measurement of Z leads to attenuated and only partially corrected image effects.
Experimental setup
Datasets: UK Biobank
Benchmarks: UK Biobank resampled unconfounded test set simulated images
Metrics: AUC MSPE bias variance
Baselines: Neural Additive Model Semi-structured Network post-hoc orthogonalisation baseline DNN without covariates adversarial confound control regress-out
Builds on: Semi-structured Networks Neural Additive Models double/debiased machine learning Speckman estimator
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