Enforcing Fair Predicted Scores on Intervals of Percentiles by Difference-of-Convex Constraints
Pith reviewed 2026-05-22 14:11 UTC · model grok-4.3
The pith
Machine learning models can enforce fairness only inside a chosen percentile interval of predicted scores by reformulating the requirement as difference-of-convex constraints.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The central claim is that fairness metrics restricted to a single percentile interval of predicted scores can be expressed as difference-of-convex functions, turning the training task into a constrained optimization problem solvable by an inexact difference-of-convex algorithm that finds a nearly KKT point with explicit complexity bounds; this formulation delivers high predictive accuracy while meeting the interval-specific fairness requirement.
What carries the argument
Difference-of-convex constraints that encode statistical fairness metrics applied only to a selected percentile interval of scores, handled by the inexact difference-of-convex algorithm (IDCA).
If this is right
- Predictive performance remains higher than under full-range fairness constraints.
- Fairness can be applied selectively to intervals that correspond to high-stakes decisions or vulnerable groups.
- The same difference-of-convex approach can be reused for multiple standard fairness definitions after suitable reformulation.
- The IDCA solver supplies a convergence guarantee to a stationary point of the constrained problem.
- Real-world datasets confirm that the partial-fairness models meet the interval target without global accuracy collapse.
Where Pith is reading between the lines
- Regulators could specify fairness rules that apply only above or below certain risk thresholds rather than across all cases.
- The method could be extended to let the interval itself be chosen adaptively from data characteristics during training.
- Multiple overlapping or disjoint intervals could be handled by adding several difference-of-convex constraints in the same optimization.
- The framework suggests testing whether interval fairness improves downstream outcomes such as loan approval equity or medical triage accuracy.
Load-bearing premise
That enforcing a fairness metric inside one chosen percentile interval is sufficient for stakeholder needs and that the difference-of-convex reformulation does not create unintended unfairness or performance loss outside that interval.
What would settle it
Train the method on a standard dataset with a chosen interval, then measure the fairness metric on the complementary score ranges and compare it to the same metric from an unconstrained model; a substantial rise in unfairness outside the interval would show the constraint does not isolate its effect.
read the original abstract
Fairness in machine learning has become a critical concern. Existing approaches often focus on achieving full fairness across all score ranges generated by predictive models, ensuring fairness in both high- and low-percentile populations. However, this stringent requirement can compromise predictive performance and may not align with the practical fairness concerns of stakeholders. In this work, we propose a novel framework for building partially fair machine learning models that enforce fairness only within a specific percentile interval of interest while maintaining flexibility in other regions. We introduce statistical metrics to evaluate partial fairness within a given percentile interval. To achieve partial fairness, we propose an in-processing method by formulating the model training problem as constrained optimization with difference-of-convex constraints, which can be solved by an inexact difference-of-convex algorithm (IDCA). We provide the complexity analysis of IDCA for finding a nearly KKT point. Through numerical experiments on real-world datasets, we demonstrate that our framework achieves high predictive performance while enforcing partial fairness where it matters most.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a framework for partial fairness in predictive models by enforcing statistical fairness metrics only within a user-specified percentile interval of the score distribution. Fairness requirements are reformulated as difference-of-convex (DC) constraints, the resulting non-convex program is solved via an inexact DC algorithm (IDCA), and a complexity bound is given for reaching a nearly KKT point. Experiments on real-world datasets are used to illustrate that predictive performance remains competitive while partial fairness is achieved inside the chosen interval.
Significance. If the DC reformulation is faithful to the partial-fairness metrics and the IDCA procedure reliably returns points that satisfy the interval-specific constraints to the stated tolerance, the approach would provide a principled way to trade off full fairness against accuracy when stakeholder concerns are localized to particular score ranges. The explicit complexity analysis for IDCA and the empirical demonstration on benchmark datasets constitute concrete strengths that would support adoption if the feasibility gap is closed.
major comments (2)
- The manuscript states that fairness metrics are converted into DC constraints but does not exhibit the explicit functional form of these constraints (e.g., how the indicator for the percentile interval is combined with the chosen fairness metric). Without the concrete expressions it is impossible to verify that the DC representation is exact inside the interval and does not inadvertently relax or distort the metric outside it.
- The complexity result for IDCA guarantees convergence to a nearly KKT point of the DC program, yet supplies no a-priori bound on the violation of the DC fairness inequality itself. Because the feasible set defined by a DC constraint is generally non-convex, stationarity alone does not imply small primal infeasibility; the current analysis therefore leaves open the possibility that the returned model violates the target partial-fairness tolerance by an arbitrary amount.
minor comments (2)
- Notation for the percentile interval bounds and the indicator function used to restrict the fairness metric should be introduced once and used consistently throughout the optimization formulation.
- The experimental section would benefit from an explicit statement of the chosen percentile intervals, the numerical tolerance used for the DC constraints, and a direct comparison of constraint violation before and after optimization.
Simulated Author's Rebuttal
We thank the referee for the constructive feedback on our manuscript. The comments highlight important aspects of clarity and theoretical rigor that we will address in the revision. We respond to each major comment below.
read point-by-point responses
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Referee: The manuscript states that fairness metrics are converted into DC constraints but does not exhibit the explicit functional form of these constraints (e.g., how the indicator for the percentile interval is combined with the chosen fairness metric). Without the concrete expressions it is impossible to verify that the DC representation is exact inside the interval and does not inadvertently relax or distort the metric outside it.
Authors: We agree that the explicit functional forms are necessary for full verification. While the high-level reformulation is described in Section 3, the detailed expressions that incorporate the percentile indicator function with the fairness metric (e.g., demographic parity or equalized odds restricted to the interval) were not expanded in the main text. In the revised manuscript we will add a dedicated subsection that derives and displays these concrete DC constraint expressions, allowing readers to confirm exactness inside the target interval and the lack of unintended relaxation outside it. revision: yes
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Referee: The complexity result for IDCA guarantees convergence to a nearly KKT point of the DC program, yet supplies no a-priori bound on the violation of the DC fairness inequality itself. Because the feasible set defined by a DC constraint is generally non-convex, stationarity alone does not imply small primal infeasibility; the current analysis therefore leaves open the possibility that the returned model violates the target partial-fairness tolerance by an arbitrary amount.
Authors: We acknowledge the distinction between stationarity and primal feasibility for non-convex DC constraints. The provided complexity bound establishes convergence to a nearly KKT point but does not directly control the violation of the fairness inequalities. Our experiments indicate that the returned solutions satisfy the interval-specific constraints to within a small tolerance in practice. In the revision we will add a discussion of this limitation, report quantitative feasibility gaps observed across the benchmark datasets, and note that post-hoc projection or penalty augmentation can be used to further reduce any residual violation if needed. revision: partial
Circularity Check
No significant circularity in the derivation chain
full rationale
The paper introduces a new in-processing framework that formulates partial fairness enforcement inside a chosen percentile interval as difference-of-convex constraints and solves the resulting problem with an inexact difference-of-convex algorithm (IDCA) whose complexity analysis is supplied for nearly-KKT points. These elements are presented as independent algorithmic constructions rather than reductions of previously fitted parameters, self-cited closed-form expressions, or load-bearing uniqueness theorems from the same authors. The central claims rest on the DC reformulation itself and the solver, which remain self-contained against external benchmarks such as standard fairness-constrained optimization methods. No load-bearing step reduces by construction to the paper's own inputs or to a self-citation chain that would force the reported outcomes.
Axiom & Free-Parameter Ledger
free parameters (1)
- percentile interval bounds
axioms (1)
- domain assumption Fairness metrics within a percentile interval admit a difference-of-convex representation suitable for the IDCA solver.
discussion (0)
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