Problem

Machine Learning algorithms are increasingly used in crucial decision-making sectors like justice, healthcare, and finance. However, their widespread adoption has highlighted significant concerns regarding fairness, as these algorithms can perpetuate or even amplify existing biases against certain demographic groups or individuals. There is a critical need for methods that can build accurate models while actively mitigating these harmful biases. Current theoretical understanding of how models achieve fairness, particularly the intricate relationship between individual and group fairness, remains underdeveloped, limiting the development of truly equitable AI systems.

Method

The authors propose Fair Sheaf Diffusion (FSD), a novel theoretical framework based on Sheaf Diffusion, which utilizes tools from dynamical systems and homology to model fairness. The core idea is to project input data into a "bias-free" space by encoding fairness constraints as algebraic equations within the structure of cellular sheaves. The method introduces various types of sheaves, such as the identity sheaf for pre- or post-processing, and a vector sheaf for in-processing, which creates a bottleneck in information exchange to specifically target logits. A key aspect of FSD involves designing network topologies that inherently encourage fairness metrics in the kernel of their Laplacian operators. These topologies include global structures like the subset topology (to address group fairness and disconnected communities) and local structures like k-Nearest Neighbor (kNN) graphs and unit ball graphs (to promote individual fairness like consistency and small Lipschitz constants). By combining these different network topologies through linear combinations of their sheaf Laplacians, the framework allows for a unified method to tackle both individual and group bias simultaneously. A notable advantage is the method's interpretability, providing closed-form expressions for SHAP values.

Results

The simulation study demonstrated that FSD models generally improve individual fairness metrics, with kNN configurations achieving a 33% median decrease in consistency, though often at a cost to accuracy (e.g., 3-13% decrease). Group fairness results were less consistent, with the global topology underperforming, while local topologies sometimes matched logistic regression. SHAP value analysis revealed that graph models modulate variable importance, sometimes dampening influence but at other times increasing the importance of sensitive attributes for debiasing. A sensitivity analysis of hyper-parameters showed that increasing diffusion strength and the number of layers in discrete implementations generally improves fairness at the expense of performance, while continuous models exhibited less predictable behavior. When tested on standard fairness benchmarks (German, Compas, Adult datasets), FSD models consistently reduced individual fairness metrics like consistency and generalized entropy, with the kNN topology showing significant improvements in both individual and group fairness with minimal accuracy reduction. Pareto frontiers were used to quantify the trade-offs between accuracy, independence, and consistency, illustrating that small compromises in accuracy could lead to substantial fairness gains (e.g., 2% accuracy reduction yielding 33-50% improvement in fairness for the German dataset).

Implications

This work introduces Fair Sheaf Diffusion as a versatile and interpretable approach for tackling algorithmic bias. Its ability to model fairness constraints through customizable graph configurations, even for non-graph data, and to combine different topologies to simultaneously address individual and group fairness represents a significant contribution. The framework offers practitioners fine-grained control over fairness-accuracy trade-offs via hyper-parameters and provides readily available explanations through SHAP values, enhancing responsible AI development. The flexibility of FSD to function as a pre-processor, in-processor, or post-processor, or to be concatenated into hybrid multi-stage pipelines, expands its applicability. Future research directions include adapting FSD to more complex data structures like directed graphs and hypergraphs, exploring non-linear sheaf diffusion for non-linear constraints, investigating the wave equation for alternative debiasing methods, and applying local topologies to achieve counterfactual fairness. The theoretical foundation in algebraic topology could also open new avenues for understanding and defining fairness.