PaperlayerView exact revision on arXiv

Source-linked AI summary

Conditional Learning of Fair Representations

Han Zhao, Amanda Coston, Tameem Adel, Geoffrey J. Gordon

arXiv:1910.07162v3cs.LGcs.AIstat.ML

TL;DR

Fairness and utility can conflict, and whether equalized odds and accuracy parity can be guaranteed together has remained an open question. The paper combines conditional alignment of representations with balanced error rate to address it. The authors report theoretical support for the joint fairness goals and better utility-fairness trade-offs on balanced datasets than existing fair representation methods.

  • Problem

    Fairness can conflict with utility, and whether equalized odds and accuracy parity can be achieved together remains an open question.

    Source: §1 · pp. 1–2

  • Method

    The algorithm aligns representation distributions conditional on the target and uses balanced error rate over the target and protected attributes.

    Source: §1 · p. 2

  • Results

    On balanced datasets, the method reports better utility than existing fair representation methods, while on an imbalanced dataset it is the only method reported to achieve accuracy parity, at decreased utility.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The authors show that conditional alignment and balanced error rate can support accuracy parity and equalized odds without impacting demographic parity.

    Source: §1 · p. 2, §6 · p. 10

    Whether calibration harms utility and the fundamental utility limit of calibrated tools remain unclear.

    Source: §6 · p. 10

Abstract

from arXiv · show

We propose a novel algorithm for learning fair representations that can simultaneously mitigate two notions of disparity among different demographic subgroups in the classification setting. Two key components underpinning the design of our algorithm are balanced error rate and conditional alignment of representations. We show how these two components contribute to ensuring accuracy parity and equalized false-positive and false-negative rates across groups without impacting demographic parity. Furthermore, we also demonstrate both in theory and on two real-world experiments that the proposed algorithm leads to a better utility-fairness trade-off on balanced datasets compared with existing algorithms on learning fair representations for classification.

1 INTRODUCTION

Fair representation learning seeks to reduce group disparity, but fairness and utility can conflict, and it remains open whether equalized odds and accuracy parity can be guaranteed together. The paper proposes conditional alignment with balanced error rate (BER) to address that question, with theoretical and empirical support for the joint fairness goals.

1 INTRODUCTION

  • Fairness is often at odds with utility, and equalized odds is incompatible with demographic parity except in degenerate cases where group membership is independent of the target.The introduction frames accuracy parity and equalized odds as alternatives to demographic parity when utility is a concern.Source: §1 · p. 1
  • Whether equalized odds and accuracy parity can be guaranteed simultaneously remains an open question in the cited prior work.The paper presents its algorithm as an affirmative answer to this question.Source: §1 · p. 2
  • The paper proposes conditional alignment of representations with BER to simultaneously ensure accuracy parity and equalized odds across groups.The authors report theoretical and real-world experimental support for this approach.Source: §1 · p. 2
  • The authors report that BER helps ensure accuracy parity and small joint error, and under equalized odds it upper-bounds each subgroup’s error.Together with conditional alignment, BER supports simultaneous equalized odds and accuracy parity.Source: §1 · p. 2
  • The proposed method is reported to achieve equalized odds without impacting demographic parity, preserving the demographic-parity gap while satisfying equalized odds.The authors also report better utility among existing fair representation methods on balanced datasets; on an imbalanced dataset, CFAIR alone achieves accuracy parity but with decreased utility.Source: §1 · p. 2
  • The authors position the algorithm as an alternative for real-world settings where accuracy parity and equalized odds are desired.They describe the work as contributing to understanding the relationship between these two fairness notions.Source: §1 · p. 2

2 PRELIMINARY

The preliminaries set up group fairness with a binary sensitive attribute and review the fairness criteria and error measures used in the paper. They also describe a trade-off result for demographic-parity representations when group base rates differ.

2 PRELIMINARY

  • The notation defines D_a and D_y as distributions conditional on A = a and Y = y, respectively, and uses total variation to measure distribution discrepancy.For binary labels, the total variation between marginal label distributions equals the difference in their base rates.Source: §2 · pp. 2–3
  • For binary prediction, balanced error rate is the sum of the false-negative and false-positive rates, and the paper defines it for predicting both Y and A.The error of a predictor is defined as expected absolute difference between its prediction and the label.Source: §2 · p. 3
  • The setup assumes the sensitive attribute A is available during training but not inference, so post-processing methods to ensure fairness are infeasible in this setting.The paper uses A to denote group membership and considers binary classification with labels Y in {0, 1}.Source: §2 · pp. 2–3
  • Demographic parity requires the prediction to be independent of group membership; for deterministic binary classifiers, both groups receive positive outcomes at the same rate.The introduction notes that demographic parity may reduce utility when group base rates differ.Source: §2 · p. 3
  • The cited bound relates the sum of subgroup errors to the base-rate difference minus the total-variation distance between the represented group distributions [Zhao & Gordon, 2019].When representations are independent of the sensitive attribute, the total-variation term is zero.Source: §2 · p. 3
  • If dTV(g♯D0, g♯D1) = 0, then for any hypothesis h, at least one subgroup has error at least ∆BR(D0, D1)/2.This applies to fair representations in the demographic-parity sense; the bound could be large in settings like criminal justice where ∆BR(D0, D1) is large.Source: §2 · p. 3
  • Accuracy parity requires equal error rates across groups, while equalized odds requires the prediction to be independent of group membership conditioned on Y.Equalized odds corresponds to equal true-positive and false-positive rates between groups.Source: §2 · pp. 3–4
  • Predictive rate parity requires the groups to have equal probabilities of Y = 1 conditional on each prediction value.For deterministic classifiers with unequal base rates, prior work rules out simultaneous equalized odds and predictive rate parity when the classifier is imperfect.Source: §2 · p. 4

3 ALGORITHM AND ANALYSIS

The section develops a fair-representation objective using balanced error rate (BER) and conditional alignment, then analyzes their implications for equalized odds, demographic-parity gaps, and subgroup errors. It concludes with a cost-sensitive cross-entropy relaxation for practical optimization.

3.1 CONDITIONAL LEARNING OF FAIR REPRESENTATIONS

  • The objective uses BER for prediction and two adversaries, one per target-label condition, to encourage conditional alignment across demographic groups.The BER objective emphasizes conditional errors rather than marginal error.Source: §3.1 · p. 4
  • Perfect conditional alignment implies that any classifier on the representations satisfies equalized odds, but alignment alone is only a necessary condition for small joint error.A small BER is also needed for a sufficient joint-error guarantee.Source: §3.1 · p. 5

3.2 THE PRESERVATION OF DEMOGRAPHIC PARITY GAP AND SMALL JOINT ERROR

  • Conditional alignment bounds the classifier’s demographic-parity gap by the gap of the true labels, so it does not add demographic-parity disparity.Theorem 3.1 states ΔDP(bY) ≤ ΔBR(D0, D1) = ΔDP(Y).Source: §3.2 · p. 5
  • Theorem 3.2 bounds the sum of subgroup errors by twice the BER: ErrD0(bY) + ErrD1(bY) ≤ 2BERD(bY || Y).The bound motivates minimizing BER while aligning conditional representations.Source: §3.2 · pp. 5–6

3.3 CONDITIONAL ALIGNMENT AND BALANCED ERROR RATES LEAD TO SMALL ERROR

  • Theorem 3.3 bounds the error gap by ΔBR·BER + 2ΔEO; under equalized odds, minimizing BER reduces the gap, while Corollary 3.1 bounds each subgroup’s error.The error gap is the accuracy-parity gap between subgroups.Source: §3.3 · p. 6
  • When base rates differ, equalized odds requires equal global FPR and FNR for equalized accuracy, a weaker condition than zero BER.This follows from the error-gap relationship under equalized odds.Source: §3.3 · p. 6

3.4 PRACTICAL IMPLEMENTATION

  • Because binary 0/1 loss is NP-hard to optimize over many hypothesis classes, the practical formulation uses cost-sensitive cross-entropy with inverse class-probability weights.This relaxation avoids explicitly computing conditional distributions.Source: §3.4 · p. 7

4 EMPIRICAL STUDIES

Experiments on Adult and COMPAS assess whether CFAIR can reduce error and fairness gaps while controlling the utility trade-off. CFAIR performs best on balanced COMPAS, while Adult’s target-label imbalance limits its utility.

4 EMPIRICAL STUDIES

  • Across the two real-world datasets, the experiments examine how fairness parity affects target utility and how dataset imbalance shapes the fairness-utility trade-off.The section reports that imbalanced target labels can compromise utility, while balanced datasets show a better trade-off for the proposed method.Source: §4 · p. 7

4.1 EXPERIMENTAL SETUP

  • The experiments use Adult income prediction and COMPAS recidivism prediction, with gender and race, respectively, as sensitive attributes; Adult is imbalanced, whereas COMPAS is balanced in target and sensitive attributes.Adult has different group base rates and about 24.6% positive labels; COMPAS has different group base rates despite its stated balance.Source: §4.1 · p. 7, Table 1 · p. 7
  • CFAIR minimizes conditional errors for both target and adversary losses; comparisons include CFAIR-EO, LAFTR, FAIR, and an unconstrained NODEBIAS baseline under a shared network architecture.CFAIR-EO uses BER only for adversary loss; LAFTR uses global cross-entropy for the target and conditional L1 for the adversary.Source: §4.1 · p. 8

4.2 RESULTS AND ANALYSIS

  • On Adult, CFAIR’s error gap approaches 0 at λ=1e3 while other algorithms’ gaps are around 0.12, but target-label imbalance can substantially hurt utility when BER is used for the target loss.CFAIR, CFAIR-EO, and LAFTR achieve small equalized-odds gaps; FAIR is most effective at reducing demographic-parity gaps.Source: §4.2 · pp. 8–9, Figure 1 · p. 8
  • On balanced COMPAS, CFAIR has the smallest utility-fairness trade-off and reduces error, equalized-odds, and demographic-parity gaps; it slightly outperforms CFAIR-EO on parity and joint error.The authors identify CFAIR as particularly suited to balanced datasets where accuracy parity and equalized odds are both desired.Source: Figure 2 · p. 8, §4.2 · p. 9

5 RELATED WORK

Prior work studies group and individual fairness, theoretical compatibility among fairness criteria, and fair representation learning. This paper distinguishes its approach by combining balanced error rate with conditional representation alignment.

5 RELATED WORK

  • Group-fairness methods include post-processing for equalized odds, which requires the sensitive attribute at inference, and causal-fairness approaches that rely on untestable assumptions.In settings with treatment effects, fairness adjustment based on Y may lead to adverse outcomes; applying this method there would require matching conditional counterfactual distributions.Source: §5 · p. 9
  • Prior theory establishes incompatibilities among fairness notions and relates accuracy parity and equalized odds to base-rate gaps and balanced error rate.The paper also states that, for predictors satisfying equalized odds, balanced error rate bounds joint error across demographic subgroups.Source: §5 · p. 9
  • Fair representation methods use adversaries or distribution distances; unlike those approaches, this paper optimizes BER on both target and adversary losses to pursue accuracy parity and equalized odds.The authors report a better utility-fairness trade-off on balanced datasets.Source: §5 · p. 10

6 CONCLUSION AND FUTURE WORK

The paper proposes fair representation learning that combines conditional alignment and balanced error rate to target accuracy parity and equalized odds. It reports improved fairness-utility trade-offs on balanced datasets, while leaving calibration’s utility effects unresolved.

6 CONCLUSION AND FUTURE WORK

  • CFAIR aligns conditional representation distributions and uses balanced error rate to pursue accuracy parity and equalized odds without impacting demographic parity.The authors also state theoretical guarantees concerning joint error across demographic subgroups and report experiments on two real-world datasets.Source: §6 · p. 10
  • On balanced datasets, the experiments show CFAIR has a better utility-fairness trade-off than existing fair representation methods.The conclusion describes the proposed algorithm as effectively leading to the desired fairness notions across two real-world experiments.Source: §6 · p. 10
  • Whether statistical calibration harms utility, and the fundamental utility limit for calibrated tools, remain theoretically unclear.The paper identifies calibration as relevant in some settings, such as COMPAS.Source: §6 · p. 10
  • The relationship between privacy and group fairness is proposed as a direction for future work, including possible use of privacy techniques for more efficient fairness-aware learning.The paper also notes potential reciprocal relevance to privacy-preserving machine learning.Source: §6 · p. 10

A.1 PROOF OF PROPOSITION 3.1

The proof uses total-variation contraction under a classifier to transfer conditional representation alignment into equalized odds. It applies the bound separately for each target label.

A.1 PROOF OF PROPOSITION 3.1

  • When the conditional pushforward distributions of the representations agree for both labels, any classifier composed with the representation map satisfies equalized odds.The proof concludes that the binary output is independent of group membership given the target label.Source: §A.1 · p. 13
  • For any two distributions over the representation space and any binary hypothesis, the proof establishes that applying the hypothesis cannot increase their total-variation distance.This contraction is the intermediate step used in the proposition’s proof.Source: §A.1 · p. 13
  • The proof applies the contraction separately for each target label to relate the conditional distributions after classification.The preceding inequality is applied twice, once for each y in {0, 1}.Source: §A.1 · p. 13

A.2 PROOF OF PROPOSITION 3.2

Under the stated representation condition, Proposition 3.2 bounds the classifier-output disparity between groups by the balanced-rate disparity. The proof applies equalized-odds relations to the binary predictions.

A.2 PROOF OF PROPOSITION 3.2

  • The classifier-output total-variation disparity is bounded by ΔBR(D0, D1) under the stated representation condition.The proof applies the bound to both possible binary predictions.Source: §A.2 · p. 13
  • For each binary prediction value y, the displayed inequality starts by bounding the groups’ absolute prediction-probability difference by their absolute Y=0 prevalence difference multiplied by an unspecified factor.A second displayed expression is the sum of the absolute group differences for bY=0 and bY=1.Source: §A.2 · p. 13

A.3 PROOF OF LEMMA 3.1

Lemma 3.1 bounds the difference in group prediction probabilities using the difference in their negative-label prevalences. Its proof invokes equalized odds after expanding probabilities by the law of total probability.

A.3 PROOF OF LEMMA 3.1

  • Lemma 3.1 bounds each group difference in the probability of a predicted label by |γ0 − γ1| times a further factor.The supplied lemma statement does not show the factor following the multiplication sign.Source: §A.3 · p. 13
  • The lemma’s prevalence terms are γa := Da(Y = 0) for each group a.These terms denote each group’s probability of a negative label.Source: §A.3 · p. 13
  • The proof expands the prediction probabilities by the law of total probability before applying equalized-odds relations.It equates the relevant conditional prediction probabilities across groups for each label.Source: §A.3 · p. 14

A.4 PROOF OF THEOREM 3.1

The appendix proves that the representation conditions yield a demographic-parity bound and a bound on summed group errors. The latter is established by expressing errors through conditional prediction probabilities and applying equalized odds.

A.4 PROOF OF THEOREM 3.1

  • Theorem 3.1 bounds the classifier’s demographic-parity gap by ΔBR(D0, D1), which equals the demographic-parity gap in Y.The proof applies Lemma 3.1 to both possible predictions and sums the resulting bounds.Source: §A.4 · p. 14

A.5 PROOF OF THEOREM 3.2

  • Theorem 3.2 bounds the sum of the two groups’ classification errors by 2BER_D(bY || Y).Its proof expands the error probabilities and uses the classifier’s equalized-odds property.Source: §A.5 · pp. 14–15

A.6 PROOF OF THEOREM 3.3

Theorem 3.3 relates the accuracy gap to balanced error, balanced-rate disparity, and equalized-odds disparity. A corollary bounds the worst group error when the classifier satisfies equalized odds.

A.6 PROOF OF THEOREM 3.3

  • The accuracy gap is bounded by ΔBR(D0, D1) · BER_D(bY || Y) + 2ΔEO(bY).The proof bounds group error differences through conditional prediction probabilities and equalized-odds deviations.Source: §A.6 · p. 15
  • The proof decomposes each group’s error into false-negative and false-positive terms before bounding the group error difference.It invokes two lemmas to bound terms involving conditional prediction probabilities and the equalized-odds gap.Source: §A.6 · p. 16
  • When the classifier satisfies equalized odds, the maximum of the two group errors is bounded by ΔBR(D0, D1) · BER_D(bY || Y)/2 + BER_D(bY || Y).The corollary combines the error-gap bound with the bound on summed group errors.Source: §A.6 · pp. 15–16

B.1 THE ADULT EXPERIMENT

The Adult and COMPAS experiments specify baseline classifier and adversary architectures. In both experiments, CFAIR and CFAIR-EO use two adversaries with the same architecture as the FAIR and LAFTR adversaries.

B.1 THE ADULT EXPERIMENT

  • The Adult baseline is a three-layer ReLU network with 114 input units, 60 hidden units, and a one-unit logistic-regression output for the target probability.The experiment's hyperparameters are listed in Table 2.Source: §B.1 · p. 17, Table 2 · p. 17
  • Adult FAIR and LAFTR adversaries have 60 input units and 50 ReLU hidden units; CFAIR and CFAIR-EO use two adversaries with the same architecture.Each adversary has a one-unit output representing its inference probability.Source: §B.1 · p. 17

B.2 THE COMPAS EXPERIMENT

  • The COMPAS baseline is a three-layer ReLU network with 11 input units, 10 hidden units, and a one-unit logistic-regression output for the target probability.The experiment's hyperparameters are listed in Table 3.Source: §B.2 · p. 17, Table 3 · p. 17
  • COMPAS FAIR and LAFTR adversaries have 60 input units and 10 ReLU hidden units; CFAIR and CFAIR-EO use two adversaries with the same architecture.Each adversary has a one-unit output representing its inference probability.Source: §B.2 · p. 17
Loading 1910.07162v3…