Fairness risk measures

Robert C. Williamson; Aditya Krishna Menon

Source-linked AI summary

Fairness risk measures

Robert C. Williamson, Aditya Krishna Menon

arXiv:1901.08665v1Jan 24, 2019cs.LGstat.ML

TL;DR

Existing fairness measures may not support generic sensitive features or convex objectives. The paper introduces fairness risk measures based on commensurate subgroup risks and derives a convex CVaR objective, which improves fairness as accuracy decreases and is competitive with fair-ERM in an illustration.

Problem
Existing fairness measures may require binary sensitive features, incur |S|^2 terms for categorical features, or produce non-convex objectives.
Method
The paper defines fairness risk measures that constrain subgroup risks to have low deviation and applies the framework to a convex CVaR-based objective.
Results
Increasing CVaR’s α improves fairness while reducing predictive accuracy, and its results are generally competitive with fair-ERM in the empirical illustration.
Takeaways & Limitations
Fairness risk measures provide a convex framework for handling generic sensitive features by aligning expected subgroup losses or risks.
Takeaways & Limitations
Existing general fairness measures typically produce non-convex objectives, while pairwise categorical measures require |S|^2 terms and are infeasible for real-valued sensitive features.

Abstract

from arXiv · show

Ensuring that classifiers are non-discriminatory or fair with respect to a sensitive feature (e.g., race or gender) is a topical problem. Progress in this task requires fixing a definition of fairness, and there have been several proposals in this regard over the past few years. Several of these, however, assume either binary sensitive features (thus precluding categorical or real-valued sensitive groups), or result in non-convex objectives (thus adversely affecting the optimisation landscape). In this paper, we propose a new definition of fairness that generalises some existing proposals, while allowing for generic sensitive features and resulting in a convex objective. The key idea is to enforce that the expected losses (or risks) across each subgroup induced by the sensitive feature are commensurate. We show how this relates to the rich literature on risk measures from mathematical finance. As a special case, this leads to a new convex fairness-aware objective based on minimising the conditional value at risk (CVaR).

1 Introduction

The paper introduces a fairness definition based on low deviation in subgroup risks, supporting generic convex losses and complex sensitive features while yielding a convex objective. It connects this framework to mathematical-finance risk measures and proposes a CVaR-based fairness-aware objective.

Framework scope: The proposal applies to generic convex base losses, including hinge loss, and complex sensitive features, including multi-valued groups, while producing a convex objective.
Definition of fairness: The framework requires low deviation among average losses or risks for subgroups induced by the sensitive feature.This definition builds on fairness approaches based on subgroup errors.
Risk-measure connection: The framework connects subgroup-risk fairness to risk measures from mathematical finance, enabling the use of established tools and analyses.
CVaR objective: The paper proposes a convex fairness-aware objective based on minimizing conditional value at risk (CVaR) and relates it to existing learning objectives.
Risk aggregation: The approach breaks standard risk into subgroup risks and combines them with an aggregator measuring their mean and deviation.

2 Background

Fairness-aware learning augments supervised prediction with constraints on discrimination across sensitive-feature subgroups, but existing approximate fairness measures face scalability or non-convexity limitations. This paper focuses on direct optimization and introduces a subgroup-risk notion connected to mathematical-finance risk measures, yielding a convex objective for generic sensitive features.

Fairness-aware learning: Supervised learning seeks a predictor f that estimates target labels while minimizing disagreement measured by a loss function.The setup uses feature and label spaces, a predictor class, and a nonnegative loss.
Fairness-aware learning: Fairness-aware learning adds a sensitive feature and requires a fairness measure to remain small alongside predictive-risk minimization.The sensitive feature may be included in the feature set, and omitting it does not guarantee fairness because it can correlate with other features.
Approximate fairness: Existing approximate fairness measures include subgroup-prediction deviations, independence-based measures, and covariance, but they can require |S|^2 terms, fail for real-valued S, or yield non-convex objectives.Independence-based measures handle multi-class, multi-label, and continuous sensitive features, whereas covariance provides a similarly general convex alternative.
Learning approaches: Fairness-aware algorithms can pre-process data, post-process predictions, or directly optimize the fairness-constrained learning objective; this paper focuses on direct optimization.Implications for pre-processing and post-processing methods are deferred to future work.
Present work: The paper introduces approximate fairness based on low deviation in subgroup risks, connects it to mathematical-finance risk measures, and obtains a convex objective applicable to generic sensitive features.Its stated scope includes a new fairness notion, a method optimizing it, and connections to existing learning paradigms.

3 Fairness as subgroup risk deviation

The section defines fairness through subgroup risks induced by the sensitive feature, requiring equal average losses for perfect fairness and small risk deviation for approximate fairness. It then formulates fairness-aware learning by aggregating subgroup risks to keep them both small and commensurate.

Subgroup risks: The sensitive feature partitions instances into subgroups, whose conditional losses define subgroup risks and whose distribution forms the random variable L(f).For finite S, L(f) takes subgroup-risk values {L_s(f)} with probabilities P(S = s), and overall risk is their average.
Perfect fairness: Perfect fairness requires every subgroup to attain the same average loss, making L(f) a constant random variable.With zero-one loss, this becomes equality of conditional misclassification probabilities across subgroups.
Risk-measure perspective: The proposed approximate-fairness notion provides a bridge to financial risk measures and motivates axiomatizing admissible deviation measures.The framework supports empirical optimization by replacing subgroup risks with empirical subgroup risks.
Approximate fairness: Approximate fairness requires a deviation measure D(L(f)) of subgroup risks to be smaller than a tolerance ϵ, and applies to generic sensitive features.For binary sensitive features with standard deviation deviation, the construction recovers a scaled difference between subgroup error rates.
Fairness-aware objective: The fairness-aware objective aggregates subgroup risks so that their average risk is small while the subgroup risks remain roughly commensurate.The aggregator augments expected-risk minimization with a penalty on subgroup-risk deviation, discouraging systematic misprediction bias toward one subgroup.

4 Fairness risk measures

Fairness risk measures are defined by seven axioms, with convexity preserving fair optimization and aversity penalizing deviations from perfect fairness. Every such measure is a law-invariant coherent and regular financial risk measure, enabling construction from regular deviation measures.

Axioms: Fairness risk measures satisfy seven axioms, including convexity, positive homogeneity, monotonicity, lower semicontinuity, translation invariance, aversity, and law invariance.They operate on real-valued random variables in L2(S) with finite second moment.
Axioms: Convexity ensures subgroup splitting cannot artificially lower risk and preserves convexity of the fairness objective when losses and predictors are convex.Thus, encouraging fairness does not impose an optimization burden for convex loss and prediction functions.
Axioms: Aversity penalizes deviation from perfect fairness, where perfect fairness corresponds to constant loss.The standard-deviation aggregator does not satisfy convexity and therefore is not a fairness risk measure, although it may still be sensible to use.
Relationship to financial risk measures: Every fairness risk measure is a law-invariant coherent and regular measure of risk.Coherent measures satisfy F1–F5, while regular measures add the relevant regularity and aversity conditions.
Relationship to financial risk measures: A one-to-one correspondence links regular risk measures R and regular deviation measures D, with fairness measures constructible as R(Z) = E(Z) + D(Z).Positive homogeneity corresponds between R and D, while monotonicity of R is equivalent to D(Z) ≤ sup Z − E(Z).

5 The CVaR-fairness risk measure

The CVaR-fairness risk measure aggregates subgroup risks by emphasizing their upper tail, yielding objectives that interpolate between average and maximal subgroup risk. Its variational representation gives a convex optimization objective under convexity of the predictor-to-subgroup-risk map, and connects the framework to robust learning and ν-SVM.

CVaR fairness risk: Using CVaR to measure subgroup deviation yields a simple fairness-aware objective and connects the framework to existing learning paradigms.CVaR-based risk and deviation measures are regular and coherent, with DCV,α(Z) = CVaRα(Z)−E(Z).
CVaR fairness risk: Increasing α focuses on the largest subgroup risks, while α →1 yields the maximin principle and α →0 yields the impartial-observer principle.For finite uniformly distributed groups, the objective minimizes a weighted average of the largest subgroup risks, tightening their range.
Optimization: The CVaR-fairness objective is convex when f 7→L(f) is convex, such as with a convex base loss and hypothesis class.A variational representation introduces an auxiliary threshold ρ and focuses optimization on subgroups whose risks are not smaller than ρ.
Connections to existing learning objectives: The framework subsumes top-k per-instance-loss objectives as the special case in which each instance forms its own group.This links CVaR fairness to robust binary classification objectives that average the largest per-instance losses.
Connections to existing learning objectives: The ν-SVM is likewise a special case with one group per instance and linear loss, replacing low-margin-error instances with low-average-loss subgroups.Both formulations use CVaR-like weighting to ignore or down-weight the least adverse elements.

6 Extensions and discussion

The framework extends by changing subgroup weights and by accommodating generic or multiple sensitive features through suitable measures. CVaR-based optimization exhibits a fairness–accuracy tradeoff as its parameter varies.

Alternative subgroup weighting: The fairness-aware objective can use an alternative weighting νS to privilege selected sensitive groups rather than the underlying sensitive-feature distribution.For finite S, the weighting changes the subgroup-risk aggregation and the associated deviation measure D(LS(f)).
Empirical discussion: As α varies, CVaR-based optimization decreases predictive accuracy and fairness violation while attaining a reasonable fairness–accuracy tradeoff against two baselines.These results are shown on the synth and adult datasets.
Alternative subgroup weighting: Uniform νS treats groups equally in risk and corresponds to Harsanyi’s principle of justice and balanced error.This provides an alternative to weighting groups according to the underlying sensitive-feature distribution.
Generic sensitive features: The risk-measure framework handles generic sensitive features, including continuous S = R≥0, by defining a measure νS over S.For a finite sample with distinct sensitive values and the empirical measure, each instance is treated as its own group, equivalent to the top-k objective for k = mα.
Multiple sensitive features: Multiple sensitive features can be handled by structuring S as their product and defining νS as the product of the individual feature measures.The construction applies to sensitive features {S1, . . . , Sl}.

7 Empirical illustration

Experiments show that tuning the CVaR-fairness objective produces a controllable fairness–accuracy tradeoff and results generally competitive with fair-ERM. Across synthetic and UCI Adult datasets, increasing α reduces predictive accuracy while improving equality-of-opportunity fairness.

Experimental setup: CVaR-based optimisation is evaluated by tuning α over {0.1, 0.2, . . . , 0.9} against a standard SVM and fair-ERM baseline.All methods use square-hinge loss and regularised linear scorers.
Experimental setup: The evaluation measures balanced error for predictive performance and equality-of-opportunity violation for fairness over 100 random 80–20% train-test splits.Experiments use a synthetic two-dimensional dataset and the UCI Adult dataset with gender as the binary sensitive feature.
Results: Increasing α decreases predictive accuracy but improves fairness by reducing violation of the equality-of-opportunity condition.The CVaR method explicitly encourages subgroup fairness for the square-hinge loss.
Results: CVaR produces fairness–accuracy tradeoffs generally competitive with Donini et al.’s fair-ERM approach.Figure 1 compares CVaR operating points across all α values with the fair-ERM baseline on one train-test split.

8 Conclusion and future work

The paper introduces a convex fairness definition for generic sensitive features by requiring commensurate subgroup risks, connecting fairness to mathematical-finance risk measures. This framework yields a CVaR-based fairness objective and motivates studying broader risk-measure families.

Conclusion: The proposed fairness definition supports generic sensitive features and produces a convex objective by making subgroup expected losses or risks commensurate.It generalises some existing fairness proposals while avoiding their restrictions to binary sensitive features or non-convex objectives.
Conclusion: The connection between fairness and risk measures leads to a new convex fairness-aware objective based on minimising conditional value at risk (CVaR).The approach relates fairness to the literature on risk measures from mathematical finance.
Future work: Future work should investigate risk measures beyond CVaR, including spectral measures, optimised certainty equivalents, and entropic value at risk.This direction is motivated by the paper’s formulation of fairness through risk measures.

Supplementary material for “Fairness risk measures” · A Justification of fairness risk measure axioms · B Additional experiments

The supplementary material justifies fairness-risk axioms by linking them to subgroup fairness, optimization tractability, and invariance properties. It also introduces additional experiments supplementing the paper’s main results.

A Justification of fairness risk measure axioms: Convexity prevents finer subgroup partitioning from artificially decreasing the fairness risk and preserves optimization tractability when combined with monotonicity.Without convexity, subadditivity can fail, making the risk of combined subgroup losses exceed the sum of subgroup risks.
A Justification of fairness risk measure axioms: Positive homogeneity preserves the optimal classifier when losses are multiplied by a positive constant, although the axiom is not essential.The same invariance would hold under k-homogeneity for any k ≥ 0.
A Justification of fairness risk measure axioms: Monotonicity ensures that convex loss mappings produce convex fairness-risk objectives and that lowering every subgroup risk cannot increase overall risk.The supplementary material also notes related monotonicity assumptions in prior work by Dwork et al.
A Justification of fairness risk measure axioms: Lower semicontinuity is presented as a technical condition that avoids problems involving limits.The supplementary discussion characterizes this axiom as technical rather than substantive.
A Justification of fairness risk measure axioms: Translation invariance recognizes that adding a constant to losses changes expected risk but not unfairness.Replacing ℓ with ℓ + C leaves the unfairness measure unchanged.
A Justification of fairness risk measure axioms: Aversity is required for the fairness measure to penalize deviations from perfect fairness.Without aversity, the measure would not capture deviation from the ideal fairness definition.
A Justification of fairness risk measure axioms: Law invariance makes the fairness measure depend on the distribution of subgroup risks rather than the identities of sensitive-feature values.The measure operates through the distribution P_Z and the induced functional F_P.
B Additional experiments: The supplementary material presents additional experimental results that complement those reported in the paper’s main body.No specific experimental findings are stated in the supplied passage.

B.1 Results with real-valued sensitive feature

The model is evaluated on the adult dataset using fnlwgt as a real-valued sensitive feature, with essentially each instance forming its own subgroup. Setting α = 0.9 encourages margin scores to become roughly commensurate across instances.

Experimental setup: The adult dataset uses fnlwgt, an estimate of individual representativeness, as the real-valued sensitive feature.Following [31], essentially all instances are placed into separate subgroups when forming the CVaR objective.
Results: α = 0.9 encourages all margin scores to be roughly commensurate.Figure 2 compares margin-score histograms for α = 0.1 and α = 0.9.
Results: With essentially singleton subgroups, α = 0.9 encourages commensurate model predictions and margin scores across all instances.The comparison uses the adult dataset with fnlwgt as the continuous sensitive feature.

B.2 Additional results on synth and adult

On synth and adult, varying α in CVaR-based optimisation generally decreases predictive accuracy and fairness violation. The analysis evaluates performance using 0-1 error, pairwise disagreement, and square hinge loss, while noting that class imbalance limits the suitability of 0-1 error.

Evaluation metrics: The evaluation also uses 0-1 error and square hinge loss, together with their subgroup differences induced by S.The subgroup difference in 0-1 error measures violation of demographic parity, while 0-1 error is not ideal because the datasets are slightly imbalanced.
Evaluation metrics: Pairwise disagreement measures predictive performance and its subgroup difference to better reflect class imbalance.Pairwise disagreement is defined as one minus the area under the ROC curve.
CVaR behaviour with α: Increasing α generally decreases predictive accuracy and fairness violation on the synth and adult datasets.This pattern is shown across the reported figures for CVaR-based optimisation.