Subgroup-Subset Fairness in ML

• Subgroup-subset fairness is a fairness paradigm that targets selective unions of sensitive attributes to combat data sparsity and ensure statistically sound evaluations.
• It leverages the supremum Integral Probability Metric (supIPM) and surrogate discrepancy measures to capture nuanced distributional gaps beyond simple mean differences.
• The DRAF algorithm integrates adversarial optimization with efficient group selection to balance accuracy and fairness in high-dimensional, intersectional settings.

Subgroup-subset fairness is a class of fairness criteria and algorithmic strategies aimed at ensuring fair treatment and performance guarantees across collections of subgroups defined by multiple sensitive attributes. Its development responds to practical and statistical challenges that arise in realistic fairness-aware learning scenarios, especially when the number of sensitive attributes and their intersections grows, leading to a proliferation of fine-grained subgroups—many of which may be small or even data-sparse. This concept relaxes classic “subgroup fairness” by targeting an explicitly chosen or data-driven collection of subgroup–subsets, balancing statistical reliability, computational tractability, and coverage of interpretable or societally salient groups. The resulting algorithms and metrics (such as the supremum Integral Probability Metric, supIPM) enable robust fairness control in high-dimensional fairness settings and permit practical learning solutions that would otherwise be infeasible under intersectional fairness requirements. Central to these approaches are advances in adversarial and distributional fairness optimization, surrogate fairness gap measures, and efficiency improvements through carefully designed group selection and regularization principles.

1. Motivation and Definition

Subgroup-subset fairness generalizes beyond marginal fairness (which only considers individual sensitive attributes) and classic subgroup fairness (which targets all possible intersections of sensitive attributes). As the number of sensitive attributes increases, the intersectional lattice of possible subgroups grows exponentially (for q binary attributes, $2^q$ subgroups), quickly resulting in subgroups with insufficient sample sizes for reliable statistical estimation or for practical model optimization.

To reconcile statistical soundness with comprehensive fairness, subgroup-subset fairness is formalized by selecting a collection $\mathcal{W}$ of subgroup-subsets. Each $W \in \mathcal{W}$ is a union of intersectional subgroups (i.e., an arbitrary subset of attribute-value combinations). Fairness objectives and metrics are then defined with respect to these $\mathcal{W}$, enforcing that for each $W$, the distribution of predictions $f(X)$ conditional on $S \in W$ matches that on $S \notin W$ (the complement). By construction, this approach:

• Enables coverage of interpretable, regulatory, or stakeholder-specified subgroups while avoiding data sparsity risks from ultra-rare intersections.
• Includes the marginal groups (e.g., fairness for each attribute individually) as required, simply by including singleton $W$ corresponding to each value.
• Permits incorporation of unions of intersections or aggregated groups for enhanced statistical reliability.

This framing provides flexibility and rigor, allowing fairness-aware algorithms to adapt to the practical realities of high-dimensional sensitive attribute spaces.

2. Statistical and Computational Challenges

Traditional approaches to subgroup fairness (enforcing constraints on every intersectional subgroup) are often intractable for moderate or large $q$ due to both data sparsity and computational cost. Specifically:

• Statistical challenge: Many intersectional subgroups have very small support, precluding stable estimation of fairness metrics such as risk, calibration, or distributional distance.
• Computational challenge: Algorithms that require enforcing constraints or computing penalties for each group (e.g., group DRO, individualized Loss Regularization) have costs that scale with the number of subgroups, leading to infeasibility for large $q$.

Subgroup-subset fairness circumvents these problems by:

• Restricting the targeted collection $\mathcal{W}$ to include only “active” subgroups or subsets—those with sufficient empirical mass (e.g., support above a sample threshold $\gamma$).
• Leveraging distributional rather than mean-based fairness metrics, enabling practical surrogate optimization that simultaneously controls marginal and intersectional fairness.

3. Formalization and the supremum Integral Probability Metric

The central distributional fairness metric in this context is the supremum Integral Probability Metric (supIPM), defined as follows. For a learned predictor $f$, function class $\mathcal{G}$, and subgroup-collection $\mathcal{W}$, the supIPM is

$$\Delta_{\psi, \mathcal{W}}(f) = \sup_{W \in \mathcal{W}} \text{IPM}_{\mathcal{G}}(P_{f, W}, P_{f, W^c}),$$

where $P_{f, W}$ denotes the (empirical) distribution over predictions $f(X)$ for instances with $S \in W$, and IPM is computed as

$$\text{IPM}_{\mathcal{G}}(P, Q) = \sup_{g \in \mathcal{G}} | \mathbb{E}_{P}[g] - \mathbb{E}_{Q}[g] |.$$

This captures, for each group or union $W$, the maximal difference in expected value of any test function $g$, across $f(X)$ distributions for $W$ and its complement. Key properties:

• If supIPM is zero, the outcome distributions are indistinguishable for all chosen test functions.
• Distributional focus captures disparities beyond mean difference (unlike demographic parity), accommodating higher-order shape mismatches.
• supIPM can be estimated reliably and efficiently under sufficient group support (threshold $n_\mathcal{W}$ for minimum group size).

4. The DRAF Algorithm: Adversarial Optimization for Subgroup-Subset Fairness

The Doubly Regressing Adversarial learning for subgroup Fairness (DRAF) algorithm operationalizes subgroup-subset fairness through a single adversarial learning framework:

• Objective: Minimize prediction loss plus a regularization penalty proportional to the (surrogate) subgroup-subset fairness gap over $\mathcal{W}$.
• Surrogate gap: Instead of computing supIPM directly, which would require separate optimization for each $W$, a surrogate R²-based metric is introduced. For each $W$, a label $y_{W,i}$ is created for each example $i$ ($+1$ if $S_i \in W$, $-1$ otherwise), and a regression-based measure

$$R^2(f, W, g) = 1 - \frac{\sum_i (y_{W,i} - g(f(x_i, s_i)))^2}{\sum_i (y_{W,i} - \overline{y_W})^2},$$

is computed, possibly with a modification $\widetilde{R}^2$ (adding a mean correction term), which is proved to match the IPM. A Fisher $z$-transformation yields a stable scalar penalty.

• Adversarial optimization: The algorithm alternates steps:
  • *Classifier $f$ * is updated to minimize the empirical risk plus the weighted surrogate gap.
  • Adversary $g$, subgroup weights $v$ are updated to maximize the surrogate fairness violation over all subgroup-subsets—requiring only a single shared adversarial discriminator irrespective of $|\mathcal{W}|$.
• Efficiency: This approach lowers computational overhead from $O(|\mathcal{W}|)$ discriminators or constraints to a single adversarial term while robustly targeting fairness for all (sufficiently large) $W \in \mathcal{W}$.

5. Statistical Guarantees and Theoretical Foundations

The DRAF procedure and associated surrogate fairness gaps are supported by theoretical results:

• Generalization bounds: The uniform deviation between the empirical and population supIPM gap is upper bounded by $$O \left( \sqrt{\log |\mathcal{W}| / n_{\mathcal{W}}} \right)$$, where $n_{\mathcal{W}}$ is the minimum sample size among groups in $\mathcal{W}$. This allows selection of $\gamma$ (minimum support) so that estimation remains statistically sound as $|\mathcal{W}|$ grows.
• Upper bound property: The surrogate DR gap is shown to upper bound the true supIPM via direct equivalence to an Integral Probability Metric on regression residuals.
• Inclusion of marginal fairness: By including singleton attribute values into $\mathcal{W}$, DRAF guarantees marginal fairness alongside intersectional (subgroup-subset) fairness without explicitly separate constraints.

6. Empirical Performance and Applications

DRAF’s efficacy is demonstrated across benchmark datasets (Adult, Communities, Dutch, CivilComments), showing:

• Superior trade-off: DRAF achieves more favorable accuracy–fairness trade-offs than baselines focusing on marginal parity (REG), worst-case subgroup gap (GF), or sequential postprocessing (SEQ), especially as the number of sensitive attributes increases or when the data contains many sparse, small-size subgroups.
• Robustness to high-dimensional sensitive spaces: When subgroups proliferate but many are starved of samples, DRAF maintains strong fairness on all sizable and interpretable groupings while sidestepping overfitting risks and computational bottlenecks.
• Graceful degradation: As the fairness parameter is tightened (lowering $\gamma$ or widening $\mathcal{W}$), accuracy and subgroup-fairness gaps degrade smoothly, enabling systematic exploration of the fairness–accuracy frontier.

7. Comparison to Related Approaches and Broader Implications

Key distinctions and connections:

• Baseline methods: DRAF avoids the excessive computational cost of “full subgroup fairness” (one adversary per intersection) and outperforms marginal-only objectives when sensitive attribute structure is complex and high-dimensional.
• Design flexibility: The practitioner can tailor $\mathcal{W}$ to societal, regulatory, or stakeholder priorities (e.g., including specific at-risk intersectional groups or all groups above a support threshold).
• Theoretical soundness: Use of supIPM and its surrogate guarantees rigorous control over distributional disparities, advancing beyond mean gap penalties.
• Policy and practice: These techniques facilitate realistic, interpretable, and efficient fairness controls for ML models used in deployment settings with numerous or intersectional sensitive features.

Conclusion

Subgroup-subset fairness, as formalized and operationalized via metrics such as supIPM and optimized by algorithms like DRAF, addresses the dual challenges of data sparsity and computational cost in fairness-aware learning scenarios with multiple sensitive attributes. By focusing on collections of subgroup-subsets that balance interpretability, statistical reliability, and coverage, these approaches offer a principled and tractable pathway for reliably mitigating bias across a rich landscape of possible groups in modern ML applications. The DRAF algorithm exemplifies this paradigm by reducing the fairness gap for all meaningful groupings efficiently using a single adversarial architecture, providing provable guarantees and strong empirical performance even when traditional subgroup fairness strategies become impractical (Lee et al., 24 Oct 2025).