Tuning Derivatives for Causal Fairness in Machine Learning

Technical Summary: Tuning Derivatives for Causal Fairness in Machine Learning

Problem Statement
Artificial Intelligence systems frequently inherit societal biases regarding protected attributes such as race, gender, and age. Classical fairness metrics like Statistical Parity (SP), which demand that predictions be independent of protected attributes, are often overly restrictive. This is particularly true when protected attributes influence "business necessities" or "resolving variables" (e.g., prior offenses in recidivism risk assessment) that are legitimately required for prediction. Recent causal formulations address this by distinguishing between "not-allowed" paths (direct effects of protected attributes) and "allowed" paths (indirect effects through business necessities), often incorporating Predictive Parity (PP) to ensure legitimate influences are preserved. However, existing path-based definitions relying on causal contrasts are primarily designed for categorical attributes. When applied to continuous protected attributes (e.g., age), these methods become impractical due to the need to evaluate contrasts at infinite points.

Methodology
The authors propose a framework for causal fairness within Structural Causal Models (SCMs) tailored specifically for continuous protected attributes.

1. Derivative-Based Definitions:

   • Statistical Parity in Derivatives (SPD): Instead of using contrasts, the authors define SP along a path $\pi$ as the product of partial derivatives of the structural mechanisms along that path being zero. This ensures the predictor is constant with respect to the protected attribute along not-allowed paths.
   • Predictive Parity in Derivatives (PPD): PP is defined by equating the partial derivatives of the predictor with respect to variables along allowed paths to the partial derivatives of the true outcome variable in the Data-Generating Process (DGP). This ensures the predictor replicates the legitimate influence of business necessities.

2. Existence Conditions:
   The paper establishes a necessary and sufficient condition for the existence of a predictor that simultaneously satisfies SPD and PPD. For a direct not-allowed path ($X \to Y$) and an allowed path ($W \to Y$), a fair predictor exists if and only if the mixed partial derivative of the expected outcome is zero: $\frac{\partial^2}{\partial w \partial x} E[Y | P_A^Y = p] = 0$. If this condition does not hold, a trade-off between SPD and PPD is unavoidable.

3. Fair Tuning Algorithm:
   To construct such predictors, the authors propose a "fair tuning" algorithm for neural networks:

   • Step 1: Fit an unconstrained predictor ($\hat{Y}_{\theta_U}$) to minimize prediction loss (e.g., MSE).
   • Step 2: Initialize a fair predictor with the unconstrained parameters.
   • Step 3: Optimize a composite loss function ($L_{FT}$) consisting of:
     • Prediction loss (MSE between the tuned predictor and the unconstrained predictor).
     • SPD loss (L1 norm of partial derivatives along not-allowed paths).
     • PPD loss (L1 norm of the difference between the predictor's derivatives and the unconstrained predictor's derivatives along allowed paths).
   • Hyperparameters $\lambda_{SPD}$ and $\lambda_{PPD}$ allow users to prioritize one fairness criterion over the other when the compatibility condition is not met.

Key Contributions

• Formalization: The paper formalizes SP and PP using path-specific partial derivatives, making causal fairness definitions applicable to continuous protected attributes without the computational burden of evaluating infinite contrasts.
• Theoretical Characterization: It derives conditions under which these derivative-based definitions coincide with prior contrast-based causal definitions and establishes the compatibility criterion (Theorem 1) for the simultaneous existence of fair predictors.
• Algorithmic Solution: It introduces a practical tuning algorithm that constructs fair predictors when the compatibility criterion is met and offers a flexible trade-off mechanism when it is not.
• Empirical Validation: The method is evaluated on simulated data (linear and multiplicative DGPs) and the real-world COMPAS dataset, comparing performance against methods focusing solely on SP (SPT) and marginalization schemes.

Results

• Simulated Data: In linear settings where SPD and PPD are compatible, the fair tuning (FT) algorithm successfully minimizes both losses simultaneously. In non-linear (multiplicative) settings where they are incompatible, the algorithm demonstrates the expected trade-off: increasing constraints on one metric degrades the other. The FT method outperforms marginalization and SPT-only approaches in maintaining predictive performance while adhering to fairness constraints, particularly when PPD is considered.
• COMPAS Dataset: Applied to recidivism prediction, the FT algorithm (specifically with balanced weights) achieved a superior compromise between SPD and PPD compared to SPT and Marginalize methods. While all constrained models showed reduced predictive performance compared to the unconstrained baseline, FT 10 maintained higher accuracy and F1 scores while significantly reducing PPD loss.
• Derivative vs. Contrast: When applied to binary features in the COMPAS dataset, minimizing derivative-based losses (SPD/PPD) generally resulted in reduced contrast-based losses (CSP/CPP), though discrepancies were observed in some high-constraint scenarios.

Significance and Claims
The paper claims that its derivative-based framework provides a practical and theoretically grounded solution for causal fairness in continuous domains, a gap left by existing contrast-based methods. It argues that by explicitly modeling the trade-off between statistical and predictive parity through partial derivatives, practitioners can construct predictors that respect business necessities while mitigating discrimination. The authors emphasize that their method does not require assumptions about the structural equations of the DGP, only a causal diagram, making it applicable in real-world scenarios. The work concludes that while derivative fairness may not always perfectly align with contrast fairness for binary attributes, it offers a robust and flexible approach for continuous attributes and a viable pathway for handling the tension between statistical independence and predictive utility.