Asymmetric Heterogeneous Fairness Constraint Aggregation

• The paper introduces a framework that aggregates heterogeneous, asymmetric fairness constraints into a unified optimization pipeline to address group- and individual-level fairness.
• It proposes novel algorithms in clustering, federated, and multi-task learning using constant-factor approximations and convex-concave formulations to balance competing fairness notions.
• Empirical evaluations show enhanced representation, improved recommendation utility, and Pareto-optimal trade-offs, while also highlighting challenges in scalability and the integration with other fairness definitions.

Asymmetric Heterogeneous Fairness Constraint Aggregation is a foundational paradigm for multi-dimensional algorithmic fairness, wherein disparate, often task- or group-specific, fairness constraints—possibly subject to asymmetric requirements—are simultaneously enforced or harmonized within a single machine learning or optimization pipeline. This approach generalizes beyond parity or symmetric fairness, supporting the aggregation of constraints that differ in type, scope, or strength (e.g., group-level, individual-level, structural, or utility-based fairness metrics) and addressing heterogeneous agents, tasks, or subpopulations. Modern instantiations span fair clustering, federated learning, recommendation, multi-task learning, social choice, and resource allocation. Fundamental frameworks rigorously formalize the aggregation methods, establish approximation guarantees, analyze asymmetries in constraint satisfaction, and develop optimization strategies tailored to the interactions between multiple fairness notions.

1. Formal Definitions and Heterogeneity of Fairness Constraints

Formal aggregation entails that distinct fairness desiderata—often defined at different granularities or reflecting disparate stakeholder objectives—are mapped onto precise mathematical constraints. For example, in fair clustering, the two dominant notions are:

• Group Fairness (GF): Ensures clusters reflect global group proportions up to prescribed bounds; for center $i$ and group $h$, $B_h|C_i| \le |C_i \cap C_h| \le A_h|C_i|$, admitting additive violation $p$.
• Diversity in Center Selection (DS): The center set $S$ itself must contain prescribed numbers $\kappa_h$ of members from each group $h$.

Constraints may be defined as upper/lower bounds, gap metrics (as in AEOD, CAEOD, AEP (Hu et al., 29 Nov 2025)), proportional targets, or stochastic dominance relationships. In federated optimization, clients impose heterogeneous, local fairness metrics (such as Demographic Parity regularizers (Lei et al., 19 Mar 2025)), while recommendation and social choice frameworks introduce agent-specific metrics (e.g., Group Proportional Fairness, Group MRR (Aird et al., 6 Oct 2024)).

These constraints are typically encoded as feasibility sets, per-group target bounds, or per-task vectors $\Psi = (\psi_1, ..., \psi_T)$, each reflecting a nonnegative fairness violation in disparate spaces and with different normalization or aggregation semantics.

2. Aggregation Mechanisms: Algorithms and Optimization Frameworks

Aggregation of heterogeneous fairness constraints demands sophisticated algorithmic routines designed to meet multiple objectives, sometimes with conflicting requirements or differing cost implications.

Fair Clustering Aggregation (Dickerson et al., 2023):

• Starting from an approximation for GF or DS, one can construct a constant-factor solution for the intersection (GF+DS). For GF→GF+DS, pick additional centers per group as needed and "divide" clusters such that both constraints are satisfied with bounded cost inflation. For DS→GF+DS, post-processing via max-flow and local reassignments can ensure GF, but may be computationally heavier and not always possible without unbounded cost.
• Subroutine DIVIDE reassigns cluster points to maintain active centers and small additive violation.

Multi-Task Learning Aggregation (Hu et al., 29 Nov 2025):

• Each task $t$ is assessed with its own metric (AEOD for binary detection, CAEOD for multi-class, AEP for regression). Violations are aggregated by $\Phi(\Psi)$, which is a strongly convex-concave maximization over task weights $w \in \Delta_T$:

$$\Phi(\Psi) = \max_{w \in \Delta_T} \left( w^\top \Psi - \frac{\mu}{2} \|w\|_2^2 \right)$$

This supports asymmetric attention to tasks/groups with the highest violations.

• Jointly optimized with task utility via a primal-dual saddle-point formulation and a head-aware multi-objective update proxy.

Rank Aggregation (Chakraborty et al., 15 May 2025):

• Algorithms for fair rank aggregation under arbitrary, heterogeneous group constraints are constructed via two-stage combinatorial reductions, greedy bi-partitioning, and use of a fairness-ranking oracle for diverse constraints. The generic approach yields an approximation factor of 2.881 independent of the fairness definition, including asymmetrically bounded top-$k$ group representations.

Federated Learning (Lei et al., 19 Mar 2025, Chu et al., 2022, Zhang et al., 29 Nov 2024):

• Personalized federated learning (pFedFair) allows each client to inject a private fairness gradient into model updates, and global aggregation is performed by simple averaging, implicitly integrating heterogeneous constraints via weighted ensemble.
• Agent clustering (FOCUS) means high-quality agents are not forced to bear the cost of low-quality ones for "fairness".
• Orthogonal aggregation (PPOA) ensures each group's update is encoded into a subspace, making groupwise averages exactly recoverable without explicit regularization or group suppression.

3. Asymmetry and Interactions Between Constraints

Aggregation may exhibit fundamental asymmetry: the feasibility of transitioning from one fairness notion to another is often one-directional.

• In clustering, any GF-feasible solution can be extended to DS at bounded cost, but DS cannot in general be extended to GF without unbounded cost (Dickerson et al., 2023):
  • GF→DS: Always possible (bounded cost via DIVIDE).
  • DS→GF: Not always possible (counterexamples show unbounded cost blow-up).
• In federated settings, "fairness via agent-awareness" (FAA) optimizes excess risk relative to each agent’s Bayes error, allowing high-quality agents not to sacrifice accuracy for others, contrasting with symmetric accuracy-parity which enforces strict equality (Chu et al., 2022).
• In multi-task learning, AHFDA ensures that attention is only paid to groups or tasks lagging behind the best, so fairness does not require degrading well-served groups (Hu et al., 29 Nov 2025).
• Recommendation via social choice allows allocation weights $\lambda_i$ that asymmetrically prioritize under-served agents according to realized fairness scores and user compatibility (Aird et al., 6 Oct 2024).

4. Theoretical Guarantees and Complexity

Many aggregation frameworks establish approximation guarantees and convergence properties specific to the aggregated fairness constraints and optimization formulations.

Problem Domain	Guarantee/Rate	Reference
Clustering (GF+DS)	Constant-factor (2α_GF) for bounded violation	(Dickerson et al., 2023)
Rank Aggregation	(2+ε)-approx for colorful bi-partition, 2.881 for generic	(Chakraborty et al., 15 May 2025)
Multi-Task Learning (AHFDA)	$O(1/k)$ convergence in dual gap for Frank–Wolfe inner loop	(Hu et al., 29 Nov 2025)
Personalized FL (pFedFair)	Pareto-optimal trade-off across heterogeneous fairness	(Lei et al., 19 Mar 2025)
FAA Aggregation	Provable improvement over FedAvg for fairness and convergence	(Chu et al., 2022)

Complexity is dictated by the reduction steps (e.g., bi-partitioning, LP/max-flow, scenario decomposition), number of constraints, and scenario/task dimensions. Strongly convex-concave structures or linear aggregation mappings (orthogonality, projection) are often exploited to yield efficient solvers.

5. Empirical Demonstrations and Practical Impact

Empirical results across domains consistently demonstrate substantial gains in both fairness measures and utility when asymmetric heterogeneous constraint aggregation is applied.

• Clustering algorithms enforcing GF+DS simultaneously achieve both demographic representation and center selection diversity at moderate PoF, while single-constraint approaches fail to deliver on the other metric (Dickerson et al., 2023).
• pFedFair yields strictly better accuracy–fairness Pareto fronts compared to non-personalized or global-constraint baselines in federated image and tabular classification (Lei et al., 19 Mar 2025).
• PPOA achieves optimal gender fairness and improvements in group-wise NDCG/Hit-Rate metrics over classical aggregated recommendations (Zhang et al., 29 Nov 2024).
• FOCUS achieves lower FAA (more uniform excess risk) and higher accuracy than accuracy-parity or agnostic baselines (Chu et al., 2022).
• Social choice aggregation (SCRUF-D) leads to near-perfect fairness across heterogeneous agent metrics and maintains high recommendation utility with weighted or rescored choice functions (Aird et al., 6 Oct 2024).
• In rank aggregation, the flexible approximation algorithms outperform strict parity methods and handle arbitrary group constraints efficiently (Chakraborty et al., 15 May 2025).
• Resource aggregation in energy markets via acceptability constraints supports agent-specific benefit thresholds and fair cost/utility sharing even under scenario uncertainty (Fornier et al., 1 Feb 2024).

6. Limitations and Open Challenges

• Many methods require careful tuning or specification of the number of tasks, clusters, or agents. Clustering-based approaches are sensitive to mis-specification and rely on data separability (Chu et al., 2022).
• Extending strong theoretical guarantees to nonconvex/deep RL or DNN architectures is a significant open direction (Chu et al., 2022).
• Scalability may be an issue where the number of constraints or required binary/continuous auxiliary variables becomes large (e.g., stochastic dominance scheduling in resource allocation (Fornier et al., 1 Feb 2024)).
• Automatic selection, robustness against adversarial agents, and privacy-preserving aggregation of asymmetric constraints remain areas for further development.

7. Compatibility and Incompatibility with Other Fairness Notions

Simultaneous enforcement of multiple fairness notions, especially those with a distance-based or coalition focus, may be infeasible or yield an empty intersection of feasible solutions.

• GF and DS in clustering are incompatible with distance-based notions like "Fairness in Your Neighborhood," "Socially Fair," or "Proportionally Fair" in worst-case community-structured instances (Dickerson et al., 2023).
• Asymmetric agent and group constraints may preclude certain symmetry guarantees or require trade-offs that cannot be efficiently realized.

This suggests that selection and aggregation of constraints must be guided by the specific application domain and stakeholder requirements, and that universal enforcement of all fairness desiderata is not always possible or desirable.