Individual Fairness

The foundational mathematical constraint for individual fairness. A model M satisfies the Lipschitz condition if, for any two individuals x and y, the difference in their predictions is bounded by their distance according to a task-specific metric d.

• Formal definition: |M(x) - M(y)| ≤ d(x, y)
• Intuition: The model's output cannot change arbitrarily fast; similar inputs must map to similar outputs.
• Enforcement: Achieved via adversarial regularization or constrained optimization during training.

The core design challenge of individual fairness is defining the distance function d(x, y) that captures 'similarity' for a specific domain. This metric is not universal; it must be constructed by domain experts.

• Credit lending: Distance based on financial history, income, and debt-to-income ratio.
• College admissions: Distance based on grades, test scores, and extracurricular rigor.
• Non-attributes: Race, gender, and other protected characteristics are explicitly excluded from the metric.

Individual fairness is often operationalized through counterfactual fairness, which compares an individual to their counterfactual self rather than to another person. This avoids the need to define a pairwise similarity metric.

• Mechanism: A decision is fair if it is identical in the actual world and a counterfactual world where a sensitive attribute (e.g., race) was altered.
• Requirement: Requires a Structural Causal Model (SCM) to compute the intervention.
• Advantage: Bypasses the difficulty of defining a universal distance metric between different people.

Individual fairness provides granular guarantees that group-level metrics cannot. A model can satisfy demographic parity (a group metric) while still treating specific individuals arbitrarily unfairly.

• Group fairness: Averages outcomes across protected groups (e.g., equal acceptance rates).
• Individual fairness: Constrains the treatment of every single pair of similar individuals.
• Tension: Satisfying both simultaneously is often impossible, a consequence of the impossibility theorem in algorithmic fairness.

When a domain expert cannot manually define the similarity metric d, it can be learned from data using fair metric learning. This approach uses human judgments or implicit feedback to construct an embedding space where similar individuals are close.

• Input: Triplets of the form 'A is more similar to B than to C' from human annotators.
• Output: A Mahalanobis distance matrix or neural embedding that captures the ground-truth notion of task-relevant similarity.
• Constraint: The learned metric must be orthogonal to sensitive attributes to prevent encoding bias.

A practical method to enforce the Lipschitz condition during model training is to use an adversarial auditor. A second network is trained to find pairs of individuals who are similar by the metric d but receive maximally different predictions.

• Minimax game: The predictor minimizes loss while the auditor maximizes the disparity between similar individuals.
• Convergence: At equilibrium, the predictor satisfies the individual fairness constraint.
• Implementation: Used in frameworks like SenSR to produce individually fair classifiers without a pre-defined metric.