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A Comprehensive View of Fairness through Distributional Stability

Gayane Taturyan, Charlotte Laclau, Stephan Clémencon

arXiv:2609.37061v1cs.LGcs.AI

TL;DR

The paper addresses fairness guarantees that may not carry over when protected-group composition changes between training and deployment. It defines fairness through prediction stability under such distributional perturbations and develops a constrained learning approach. The framework links classical fairness notions to stability and provides guarantees across demographic compositions, while experiments illustrate its practical relevance.

  • Problem

    Fairness is commonly assessed under a fixed data distribution, although deployment may involve a different composition of protected groups.

    Source: §1 · p. 1

  • Method

    The paper defines fairness as stability of a predictor under distributional perturbations and uses convex mixtures of reweighted classifiers in a second-order cone program.

    Source: §1 · p. 2

  • Results

    The framework establishes equivalences between individual fairness, demographic parity, equal opportunity, and stability under specific perturbations, and reports competitive benchmark results with tradeoffs that vary by setting.

    Source: §1 · p. 2, §5 · p. 9

  • Takeaways & Limitations

    The stability perspective links individual and group fairness and provides guarantees across demographic compositions at deployment without requiring prior knowledge of the deployment distribution.

    Source: §1 · p. 2, §6 · p. 9

    The method addresses a tractable surrogate rather than the original problem in full generality.

    Source: §6 · p. 10

Abstract

from arXiv · show

We view fairness as a property of distributional stability. Rather than assessing a predictor under a fixed data distribution, we study how its predictions change under perturbations that modify the composition of protected groups. A predictor is fair if it remains stable under such shifts. Under this perspective, several classical notions of fairness arise as stability with respect to specific perturbations, with the associated unfairness gap given by a Lipschitz constant of a prediction-rate functional. This formulation also yields guarantees that hold uniformly over a range of demographic compositions at test time, without requiring knowledge of the deployment distribution. It leads to a learning procedure based on convex combinations of reweighted predictors, formulated as a second-order cone program, for which we establish generalization bounds. Experiments on standard benchmarks illustrate the approach.

1 Introduction

The paper recasts fairness as distributional stability under changes in protected-group composition, rather than evaluation under a fixed distribution. This framework connects individual and group fairness and aims to provide deployment guarantees across demographic compositions without knowing the deployment distribution in advance.

1 Introduction

  • Existing fairness approaches assess predictors under fixed distributions, while prior work on shifts may assume a known deployment distribution or leave the sensitive-attribute distribution unchanged.The paper identifies changes in the marginal distribution of protected groups as a central deployment concern.Source: §1 · pp. 1–2
  • The framework defines fairness as stability of a fixed prediction function under distributional perturbations, linking individual and group fairness to different perturbation classes.It aims to provide guarantees uniformly across demographic compositions at deployment, without prior knowledge of the deployment distribution.Source: §1 · p. 2
  • The paper establishes equivalences between individual fairness, demographic parity, equal opportunity, and stability under specific perturbation classes, with unfairness gaps given by Lipschitz constants.For group fairness, it formulates a constraint that measures deviation from a target level across deployment contexts.Source: §1 · p. 2
  • Restricting learning to convex mixtures of reweighted classifiers yields a second-order cone program with generalization bounds.The paper also reports numerical experiments on the practical applicability of the approach.Source: §1 · p. 2
  • Unlike distributionally robust optimization, the framework uses prescribed perturbation rules with an importance-weight representation, which the paper says yields non-vacuous deployment guarantees.The introduction notes that the distinction is discussed in Appendix A.2.Source: §1 · p. 2

2 Defining fairness as a property of stability

The section presents stability under distributional changes as a way to unify fairness notions. It reviews individual and group fairness definitions, then defines fairness through changes in a predictor’s output distribution under specified perturbations.

2 Defining fairness as a property of stability

  • The paper proposes distributional stability as a fairness perspective that can subsume existing fairness notions within a unified formal framework.It concerns changes in the data distribution, rather than the learning-theoretic stability of an algorithm to replaced training examples.Source: §2 · p. 2, §2.2 · p. 3

2.1 Background and preliminaries

  • Individual fairness requires a Lipschitz condition so similar individuals receive similar predictions, while demographic parity and equal opportunity require, respectively, group-independent predictions and equal true positive rates.Equalized odds strengthens equal opportunity by requiring equality conditional on both label values.Source: §2.1 · p. 3

2.2 A new definition of fairness for a predictive rule

  • Fairness as distributional stability evaluates how a fixed predictor’s output distribution changes under a class of perturbation operators, with the perturbation class and statistical distance chosen for the application.For group fairness, the paper identifies perturbations to the distribution of the sensitive attribute as a natural choice.Source: §2.2 · p. 3

3 From various existing concepts of fairness to a unified approach

This section shows how individual and group fairness notions can be expressed as distributional stability under appropriately chosen perturbations and statistical distances.

3.1 Individual fairness and local perturbations

  • Individual fairness is equivalent to stability under local feature transformations, with stability parameter Lρ proportional to the perturbation radius ρ.The transformations move each feature vector by at most ρ in the feature-space metric.Source: §3.1 · p. 4

3.2 Group fairness and resampling perturbations

  • Group perturbations vary the protected-group proportion globally or within the positive-label class, while holding specified conditional distributions fixed.For global resampling, P(X, Y | G) is preserved; for positive-class resampling, P(X | Y, G) and the distribution of Y are preserved.Source: §3.2 · p. 4
  • The prediction-rate functional is affine in the perturbation parameter, and its Lipschitz constant equals the associated unfairness gap.This links group fairness to stability of prediction rates as demographic composition changes.Source: §3.2 · pp. 4–6
  • For demographic parity and equal opportunity, the relevant prediction rate is constant across the corresponding perturbation parameter exactly when the fairness criterion holds.The stability deviation coincides with the demographic-parity gap; the equal-opportunity condition is constancy of the true-positive rate under positive-class resampling.Source: §3.2 · p. 5
  • Variance control measures average sensitivity over a distribution of perturbations, yielding smoother and more tractable constraints than worst-case deviation.The perturbation variance equals γ(h)^2 Var(π), and the stated bound on the unfairness gap is equivalent to a corresponding variance bound.Source: §3.2 · p. 5

4 Methodology: variance and mean control with mixture models

The method controls both variation in prediction rates across perturbed distributions and their mean distance from a target, then learns a convex mixture under this joint constraint. It constructs perturbations with reweighting and gives convergence and deployment guarantees under stated assumptions.

4.1 Problem formulation

  • The perturbation t-variance functional controls both prediction-rate variance across shifts and the mean prediction-rate deviation from target t.Constraining it also bounds each component by δ.Source: §4.1 · p. 6
  • The method mixes predictors trained on perturbed distributions and optimizes a linearized risk surrogate through a second-order cone program.The convex-loss surrogate upper-bounds the true risk of the mixture.Source: §4.1 · pp. 6–7

4.2 Constructing perturbations

  • Perturbations are sampled by stratifying the allowed π range, then represented through bounded importance weights rather than resampling.The stated assumptions bound group probabilities and restrict Π to [π̄, 1 − π̄].Source: §4.2 · p. 7

4.3 Convergence rates

  • Regression estimation contributes rate ρ_N ≍ N^-β/(2β+d), fixed-π empirical estimation and risk linearization contribute n^-1/2, and stratified perturbation sampling contributes m^-3/2.These errors combine in the informal convergence theorem under Assumptions 1–5.Source: §4.3 · p. 8
  • Unlike classical fairness constraints at the training marginal, the SOCP constraint controls prediction-rate slope and level, yielding a stronger deployment guarantee across π′.The passage states that the deployment gap is largest at π′ ∈ {0, 1}, and broader Π variance strengthens the guarantee.Source: §4.3 · p. 8

5 Numerical experiments

Across Adult Income and COMPAS, STABLE is competitive but its accuracy-fairness tradeoff depends on dataset and constraint. In the ACS geographic evaluation, it remains closest to the target prediction rate across held-out states.

Experimental setup

  • The experiments compare STABLE with ERM under constraints, PostProc, and Shifty on Adult Income and COMPAS, and test geographic generalization from California to 11 held-out ACS states.The ACS evaluation uses state racial-composition differences as a real geographic demographic shift.Source: §5 · pp. 8–9

Results

  • On Adult, STABLE outperforms all baselines under EO but loses accuracy under DP; on COMPAS, it outperforms all baselines under DP and has best accuracy and deployment gap under EO, with higher unfairness.The paper attributes occasional accuracy loss to the linearized-risk gap of its surrogate.Source: §5 · p. 9
  • On ACS Income, STABLE stays closest to target t across held-out states, while other baselines, including Shifty, drift below t in states with more White residents.The left panel plots prediction rate against each state's White-resident proportion; the right compares accuracy and DP unfairness across states.Source: §5 · p. 9, Figure 2 · p. 9

6 Conclusion and perspectives

The paper frames individual and group fairness as stability under appropriate perturbations and proposes a stronger joint constraint with a tractable SOCP formulation. Its guarantees and experiments concern a tractable surrogate, not the original problem in full generality.

Conclusion

  • Individual and group fairness arise as special cases of stability under appropriate perturbation classes.Source: §6 · p. 9
  • A stronger constraint jointly controls group fairness and prediction-rate stability across perturbations, and yields a tractable SOCP when using linearized risk.Source: §6 · p. 9

Perspectives

  • The method addresses a specific tractable surrogate rather than the original problem in full generality; the authors identify direct solution and end-to-end guarantees as future work.Source: §6 · p. 10

Organization of the Supplementary Material

The supplement collects proofs, framework extensions, and additional experiments. Its discussion situates the paper's mathematical focus alongside broader social questions and distinguishes predictor distributional stability from learning-algorithm stability.

Organization of the Supplementary Material

A Discussions on related works

  • The discussion focuses on fairness's mathematical aspects and does not aim to resolve the broader philosophical and social question of what fairness ought to be.Source: A Discussions on related works · p. 13

A.1 Stability and learning theory

  • In learning theory, stability often concerns how replacing training examples changes a learned function; this paper instead studies a fixed predictor under distribution changes.Source: §A.1 · p. 13

A.2 Comparison with Distributionally Robust Optimization

The section contrasts demographic-shift stability with distributionally robust optimization (DRO), whose uncertainty sets can include changes unrelated to fairness. It also states that individual fairness is equivalent to stability under specified perturbations.

A.2 Comparison with Distributionally Robust Optimization

  • DRO uncertainty sets covering demographic shifts can also allow arbitrary corruption of P(X | G) and P(Y | G), making worst-case risk uninformative about performance under those shifts.The required radius is positive uniformly in p1 and is Ω(1) when |π̄ − p1| is bounded below by a positive constant.Source: §A.2 · p. 14
  • STABLE restricts perturbations to changing group proportions while preserving P(X, Y | G), enabling simultaneous control of prediction-rate slope and absolute level.The passage states that Theorem 1 guarantees converge to zero as n, m →∞, unlike DRO bounds with radius ε = O(1).Source: §A.2 · p. 14
  • For binary demographic parity, Group DRO shares the demographic perturbation family but minimizes worst-case risk, whereas STABLE controls prediction-rate variance around a fairness target.The section also notes that Group DRO has no natural counterpart for the positive-class resampling used to capture equal opportunity.Source: §A.2 · p. 14

B.1 Individual fairness

  • Individual fairness is equivalent to stability under the specified perturbations of point-mass distributions, with prediction changes measured by Wasserstein distance.The proposition states equivalence between a finite (dY, dX)-Lipschitz constant and Lρ-fairness for all ρ > 0 under the stated perturbation class.Source: §B.1 · p. 15

B.2 Group fairness

The section connects demographic parity and equal opportunity to invariance of prediction rates under group-composition perturbations. It also develops importance-weight representations and error-rate results for the estimation procedure.

B.2 Group fairness

  • Demographic parity is equivalent to an invariant global positive prediction rate under resampling of the sensitive group.The rate is expressed as a mixture of group-specific positive rates weighted by the perturbed group proportion.Source: §B.2 · pp. 15–16
  • Equal opportunity is equivalent to an invariant true-positive prediction rate when the sensitive group is resampled within the positive-label class.The stated equivalence is between equal true-positive rates across groups and constancy of the rate over π ∈ [0, 1].Source: §B.2 · p. 16

C.1 Importance weights representation

  • The appendix represents the two demographic perturbations using importance weights derived from the corresponding factorizations of the data distribution.For Perturbation 1 the factorization is P(G)P(X, Y | G); for Perturbation 2 it is P(Y)P(G | Y)P(X | Y, G).Source: §C.1 · pp. 16–17

C.2 Convergence rates

  • The convergence analysis recalls Hoeffding and Bernstein inequalities and examines errors in estimating population-level quantities.The regression-error lemma is stated for mixtures and π in the deployment interval under Assumption 2 and Claims 2–3.Source: §C.2 · p. 17, §C.2.1 · p. 17

C.2.2 Perturbation error rates

The appendix develops error bounds for perturbation sampling, sample estimation, and the final fairness constraints. It also states uniform deployment and risk-rate results under the specified conditions.

C.2.2 Perturbation error rates

  • Stratified sampling is analyzed for an L-Lipschitz function over a partition of the perturbation interval, yielding a probability bound for the sampling error.The lemma samples independently within each interval; the bound’s expression is not included in the supplied passage.Source: §C.2.2 · p. 17

C.2.3 Sample generalization rates

  • The sample-generalization lemma gives a high-probability bound conditionally on the training sample, with a second probability statement also specified.The supplied statement does not include the bound’s expression.Source: §C.2.3 · p. 18

C.2.5 Risk rates

  • The appendix states an unfairness-rate corollary under Theorem 2’s conditions and a risk-rate theorem for a PSOCP solution under Assumptions 2–5.The risk-rate theorem requires n, m, and N to be large enough; the bound itself is not included in the supplied passages.Source: §C.2.4 · p. 19, §C.2.5 · p. 23

C.2.4 Final constraint rates

  • Theorem 2 states constraint-rate bounds, and Theorem 3 gives a uniform deployment guarantee simultaneously over π′ in the specified interval for a feasible PSOCP solution.The deployment guarantee is conditional on the stated conditions and probability over the sample and perturbation draws.Source: §C.2.4 · pp. 18, 20

D Extension to Equalized Odds

The appendix extends the stability equivalence to equalized odds by varying group proportions within each label class. It also specifies the experimental reporting and validation protocol.

D Extension to Equalized Odds

  • Equalized odds is connected to stability by perturbing group proportions separately within each label class and requiring prediction rates to remain invariant.The perturbation preserves the conditional distribution of X given (Y, G) and the distribution of Y.Source: D Extension to Equalized Odds · pp. 23–24

E Experiments: additional details and results

E.1 Evaluation protocol

  • STABLE’s δ and π̄ are selected by validation-grid search for deploy gap, then evaluated on 20 independent seeds or ACS test states.The protocol specifies fixed validation splits and reports that the selected hyperparameters are evaluated independently of selection.Source: §E.1 · p. 25

E.2 ACS geographic experiment.

On ACS Income, STABLE stays closest to the target across held-out states under equal opportunity. On Adult, the target sweep shows constraint-specific changes in accuracy and fairness measures.

E.2 ACS geographic experiment.

  • The 2018 ACS Income task covers over 1.5 million working-age adults; STABLE trains in California, tunes on three states, and tests on 11 unseen states spanning 62%–93% White residents.The test states were not seen during hyperparameter selection.Source: §E.2 · p. 25
  • Figure 3 maps accuracy, U(h), and deployment gap across δ and πbar; Table 4 reports selected hyperparameters, with K = 30 and m = 60 fixed.The Figure 3 experiment uses Adult under the DP constraint.Source: Figure 3 · p. 26, Table 4 · p. 26

E.3 Results on ACS with EO constraint

  • On ACS Income under EO, STABLE stays closest to t across held-out states, while STABLE and ERM+EO form the low-unfairness, high-accuracy frontier.Shifty drifts below t despite accounting for demographic shift, whereas PostProc drifts above it.Source: §E.3 · p. 26, Figure 4 · p. 26

E.4 The effect of t

  • The t sweep uses 25 targets in (0.05, 0.95), with tERM estimated from ERM and hyperparameters tuned there.STABLE is evaluated under DP and EO over 20 random seeds.Source: §E.4 · pp. 26–27
  • For DP, accuracy peaks slightly around t = 0.5 then drops dramatically while unfairness and deploy gap stay stable; for EO, accuracy drops away from tERM, indicating non-convergence.Under EO, unfairness and deployment gap rise near tERM, peak slightly, then decrease gradually.Source: §E.4 · p. 27, Figure 5 · p. 27
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