Generalizing Orthogonalization for Models with Non-Linearities

TL;DR

This work generalizes orthogonalization to non-linear models, enabling removal of linear information from protected features in GLMs, neural networks, and tensor-valued predictions. By introducing a correction mechanism that cleanly orthogonalizes pre-activation signals (and, where appropriate, post-activation outputs) against protected attributes, the authors demonstrate zeroing of the linear influence $\hat{\bm{\beta}}^c = \bm{0}$ across diverse settings. Key contributions include a GLM-specific correction with a constrained optimization solved via MDMM, a ReLU-compatible theory ensuring $\hat{\bm{\beta}}^c = \bm{0}$, and tensor- and online-orthogonalization variants validated on tabular data, embeddings, and image/text representations. The findings suggest practical pathways to debias and safeguard information in complex models with manageable trade-offs in predictive performance.

Abstract

The complexity of black-box algorithms can lead to various challenges, including the introduction of biases. These biases present immediate risks in the algorithms' application. It was, for instance, shown that neural networks can deduce racial information solely from a patient's X-ray scan, a task beyond the capability of medical experts. If this fact is not known to the medical expert, automatic decision-making based on this algorithm could lead to prescribing a treatment (purely) based on racial information. While current methodologies allow for the "orthogonalization" or "normalization" of neural networks with respect to such information, existing approaches are grounded in linear models. Our paper advances the discourse by introducing corrections for non-linearities such as ReLU activations. Our approach also encompasses scalar and tensor-valued predictions, facilitating its integration into neural network architectures. Through extensive experiments, we validate our method's effectiveness in safeguarding sensitive data in generalized linear models, normalizing convolutional neural networks for metadata, and rectifying pre-existing embeddings for undesired attributes.

Figures (12)

• Figure 1: Exemplary optimization process for a fixed number of iterations (converging towards the black crosses) for logistic regression with features $z_1,z_2$, weights $\gamma_1,\gamma_2$, and one protected feature $x_1$ correlated with $z_1$. The upper row shows the loss surface and optimization path for the three different methods (columns) a small step size. The bottom row shows the correlation between the model's prediction and the protected feature along the optimization path, where only the generalized orthogonalization yields predictions uncorrelated with the sensitive information.
• Figure 2: Workflow of generating predictions via model $\mathcal{M}^p$ and checking the influence of $\bm{X}$ on the resulting predictions $\hat{\bm{y}}$ using $\mathcal{M}^e$. Green parts indicate the orthogonalization applied to $\mathcal{M}^p$.
• Figure 3: Comparison of orthogonalization properties of fairness methods in comparison with our approach. Ideally, methods should yield effects of zero (in the left plot) and large p-values (in the right plot). Missing boxes indicate that the method did not converge.
• Figure 4: Estimated influence of the color red on the convolutional layer's predictions (left plot) and train-, validation- and test-accuracy of the two models (with/out correction).
• Figure 5: Comparison of orthogonalization properties of fairness methods in comparison with our approach when extending the notion of orthogonality beyond linear feature effects. Ideally, methods should yield random forest feature importance (RFimp) of zero (in the left plot) and large p-values (in the right plot). Missing boxes indicate that the method did not converge.

Theorems & Definitions (10)

• Example 1
• Definition 1
• Definition 2
• Lemma 3
• Corollary 4
• Example 2
• Theorem 5
• Corollary 6
• Corollary 7
• Lemma 8