Fair Risk Minimization under Causal Path-Specific Effect Constraints (2408.01630v1)

Abstract: This paper introduces a framework for estimating fair optimal predictions using machine learning where the notion of fairness can be quantified using path-specific causal effects. We use a recently developed approach based on Lagrange multipliers for infinite-dimensional functional estimation to derive closed-form solutions for constrained optimization based on mean squared error and cross-entropy risk criteria. The theoretical forms of the solutions are analyzed in detail and described as nuanced adjustments to the unconstrained minimizer. This analysis highlights important trade-offs between risk minimization and achieving fairnes. The theoretical solutions are also used as the basis for construction of flexible semiparametric estimation strategies for these nuisance components. We describe the robustness properties of our estimators in terms of achieving the optimal constrained risk, as well as in terms of controlling the value of the constraint. We study via simulation the impact of using robust estimators of pathway-specific effects to validate our theory. This work advances the discourse on algorithmic fairness by integrating complex causal considerations into model training, thus providing strategies for implementing fair models in real-world applications.

Fair Risk Minimization Under Causal Path-Specific Effect Constraints

The paper "Fair Risk Minimization under Causal Path-Specific Effect Constraints" presents an advanced framework for embedding fairness within machine learning models using path-specific causal effects. The authors, Razieh Nabi and David Benkeser, propose a structured approach to achieve fair optimal predictions while minimizing risk based on mean squared error (MSE) and cross-entropy criteria. This research is situated within the broader discourse on algorithmic fairness, addressing the significant challenge of eliminating or controlling biases in predictive models grounded in sensitive attributes like race or gender.

Summary of Key Contributions

This paper's contributions are multifaceted:

1. Path-Specific Effects Integration: The authors offer a detailed exposition of integrating path-specific causal effects into fairness constraints. They systematically derive closed-form solutions under MSE and cross-entropy risks, presenting a comprehensive analytical treatment of these solutions. This derivation facilitates the identification and quantification of risk reductions and the trade-offs inherent in achieving fairness constraints.
2. Mechanisms of Optimal Fair Prediction: By scrutinizing the derived closed-form solutions, the paper provides insight into how fairness adjustments modify the optimal unconstrained prediction functions. One of the pivotal findings is that these adjustments are contingent not only on the magnitude of the fairness constraint but also significantly on its gradient and the variance of this gradient.
3. Robust and Flexible Estimation: A significant portion of the paper is dedicated to developing and validating flexible semiparametric estimation strategies for nuisance components essential for fairness adjustments. The authors explore various estimation techniques, including plug-in and doubly robust estimators, ensuring robustness against partial model misspecification.

Theoretical Development and Analysis

Central to this work is the utilization of Lagrange multipliers for solving constrained optimization problems in infinite-dimensional functional spaces. By employing these techniques, the authors characterize the constrained functional parameter as the minimizer of a penalized risk criterion. For any given Lagrangian multiplier value, the parameter minimizing the unconstrained risk satisfies a differential equation involving gradients of the risk function and the constraint functional. This solution defines a path through the parameter space, where the optimal parameter is identified by selecting the appropriate Lagrange multiplier.

Key Theoretical Results:

• MSE Risk: For the MSE risk, the optimal fairness-constrained parameter $\psi^*_0$ can be expressed as $$\psi_0(z) - \Theta_{\Delta, P_0}(\psi_0) \frac{D_{\Theta_\Delta, P_0}(z)}{\sigma^2(D_{\Theta_\Delta, P_0})}$$. This closed-form solution provides an explicit adjustment to the unconstrained risk minimizer $\psi_0$. The mean squared difference between $\psi^*_0$ and $\psi_0$ is proportional to $$\frac{\Theta^2_{\Delta, P_0}(\psi_0)}{\sigma^2(D_{\Theta_\Delta, P_0})}$$, quantifying the trade-off between risk minimization and fairness constraints.
• Cross-Entropy Risk: Under the cross-entropy risk, the adjustment depends on the product $$\psi^*_0(z) = \psi_0(z) - \Theta_{\Delta, P_0}(\psi_0) \frac{D_{\Theta_\Delta, P_0}(z) \ \sigma^2_{\psi^*_0}(z)}{ E[ D^2_{\Theta_\Delta, P_0}(Z) \ \sigma^2_{\psi^*_0}(Z)]}$$. The paper also discusses a quadratic characterization of this path for practical estimation, facilitating numerical approximations.

Implications and Future Directions

The implications of this research are profound, both theoretically and practically. The closed-form solutions and the nuanced understanding of fairness adjustments offer a robust framework for integrating fairness into real-world predictive models. This work lays the groundwork for further exploration into more complex and varied fairness constraints, potentially expanding beyond the causal path-specific effects discussed.

The practical implications are equally significant. The proposed estimation strategies provide a flexible and robust approach to learning fair prediction functions, adaptable to various real-world contexts. The application of these methods in different domains, such as healthcare, finance, and criminal justice, can lead to more equitable decision-making systems.

Future Developments:

• Extending to Other Fairness Constraints: Future research could explore different types of fairness constraints, including those conditional on covariates or more complex causal structures, to broaden the scope of fair machine learning.
• Sensitivity Analyses: Developing sensitivity analyses to account for model misspecification or unmeasured confounding would enhance the robustness and applicability of the proposed methods.
• Empirical Validation: Applying these methodologies to large-scale datasets in diverse domains will be crucial for validating their practical utility and further refining the techniques based on empirical evidence.

In conclusion, this paper significantly advances fair machine learning by embedding causal considerations into model training. The authors' thorough theoretical and practical contributions provide a solid foundation for addressing fairness in predictive modeling, thereby advancing both the discourse and the practice of algorithmic fairness.