Group-based Counterfactual Explanations

• Group-based Counterfactual Explanations (GCEs) are methods that generate shared counterfactual recourse for groups rather than individual instances.
• They employ approaches like shared shift vectors, functional maps, and action sets to balance validity, proximity, and compactness in explanations.
• Optimization paradigms range from gradient-based methods to clustering and mathematical programming, addressing trade-offs in fairness, feasibility, and interpretability.

Group-based Counterfactual Explanations (GCEs) are counterfactual explanations constructed for sets of instances rather than isolated points, so that recourse is expressed through shared actions, shared shift vectors, shared mappings, or subgroup-level policies. Across recent work, they appear as “group-wise counterfactual explanations,” “group counterfactuals,” “collective counterfactual explanations,” and, in closely related settings, as global counterfactual summaries with a small number of actions. A recurrent view is that they occupy a middle ground between purely local counterfactuals and a single global rule, exposing multiple modes of recourse while retaining more structure and compactness than one counterfactual per instance (Furman et al., 2024).

1. Terminology and conceptual scope

The literature does not use a single uniform label. Some papers reserve Global Counterfactual Explanations for explanations shared across an entire affected population, often as one or a few common actions or directions, while others use group counterfactuals, group-wise counterfactual explanations, or collective counterfactual explanations for explanations attached to automatically discovered or externally given subgroups (Ley et al., 2023). In GLANCE, this distinction is made explicit through a difference between a counterfactual summary, which is a small set of actions intended for the whole adversely affected population, and a counterfactual segmentation, in which the feature space is partitioned into interpretable regions and each region receives a single action (Kavouras et al., 2024).

A second distinction concerns what counts as a “group.” In some formulations, the group is a fixed set of individuals of interest, and the problem is to jointly optimize one counterfactual per member under linking constraints. In this sense, a collective counterfactual explanation is a vector $$\underline{\mathbf{x}} = (\mathbf{x}_1,\dots,\mathbf{x}_I)$$ together with a subset $\mathcal{I}^*$ of perturbed instances, chosen so that the total perturbation cost is minimized subject to shared constraints and desired class flips (Carrizosa et al., 2023). In other formulations, the group is discovered automatically by clustering or by a latent assignment matrix; the explanation then consists of one shared recourse pattern per discovered subgroup rather than one per individual (Furman et al., 2024).

A third distinction concerns the role of generalization. Some methods explain only a fixed set of instances. Others learn a reusable function $g:\mathcal{X}\to\mathcal{X}$ that maps any group member, including unseen ones, to a counterfactual. In the optimal-transport formulation, a group counterfactual is precisely such a map, optimized to minimize average squared transport cost while satisfying classifier and geometry constraints (Valero-Leal et al., 28 Jan 2026).

This diversity of usage also creates an acronym ambiguity. “GCE” denotes Global Counterfactual Explanations in some work and is naturally read as Group-based Counterfactual Explanations in others. The underlying commonality is not the acronym but the structural idea: counterfactual recourse is shared across more than one instance.

2. Formal representations of group explanations

Across these papers, group explanations are formalized in several recurring ways.

The most explicit “middle-layer” formulation is the unified gradient-based model

$$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$

or, after relaxation of the discrete assignment matrix, $$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$, where $\mathbf{D}_{GW}$ contains $K$ base shift vectors, $\mathbf{P}$ gives sparse group assignments, and $\mathbf{K}$ provides per-instance magnitudes (Furman et al., 2024). This parameterization interpolates directly between three limiting cases: local counterfactuals, obtained when each instance has its own vector; global counterfactuals, obtained when $K=1$; and group-wise counterfactuals, obtained when $\mathcal{I}^*$0. The counterfactual for instance $\mathcal{I}^*$1 is

$\mathcal{I}^*$2

which reduces to a subgroup-specific shift when assignments become one-hot.

A closely related but simpler representation is the direction-plus-magnitude model of GLOBE-CE, in which a global explanation is a translation direction $\mathcal{I}^*$3 and each instance $\mathcal{I}^*$4 receives its own scalar $\mathcal{I}^*$5: $\mathcal{I}^*$6 When several directions $\mathcal{I}^*$7 are used, each direction can be interpreted as a segment-level transformation for a subgroup of instances with similar recourse patterns (Ley et al., 2023). For one-hot categorical features, the same paper shows that additive translations correspond to If/Then rules with one “Then” category per feature, so the shared direction can be read as a structured group rule rather than merely as a dense numeric vector.

A different representation is the functional map view. In optimal-transport group counterfactuals, the explanation is a single map $\mathcal{I}^*$8 solving

$\mathcal{I}^*$9

subject to classifier validity and bi-Lipschitz constraints. In parametric forms, this may be affine, $g:\mathcal{X}\to\mathcal{X}$0, diagonal-affine, Gaussian-induced, or piecewise-affine under a Gaussian-mixture model (Valero-Leal et al., 28 Jan 2026). The key difference from pointwise formulations is that the explanation is a policy-like transformation rather than a set of unrelated edited instances.

A further representation is the action-set view. In collective mathematical optimization, the central object is a set of counterfactual instances $g:\mathcal{X}\to\mathcal{X}$1 optimized jointly with a global sparsity term

$g:\mathcal{X}\to\mathcal{X}$2

which counts how many features are changed anywhere in the group (Carrizosa et al., 2023). Here the group explanation is not one vector but a coordinated set of edits constrained to reuse a small number of globally active features.

In recommender systems, the representation changes again. A group counterfactual explanation is a set $g:\mathcal{X}\to\mathcal{X}$3 of group interactions such that, after removal,

$g:\mathcal{X}\to\mathcal{X}$4

where $g:\mathcal{X}\to\mathcal{X}$5 is a target recommended item and $g:\mathcal{X}\to\mathcal{X}$6 is the black-box group recommender (Stratigi et al., 23 Jan 2026). The explanation is therefore a deletion set over shared group history rather than a feature perturbation in Euclidean space.

Representation	Example form	Representative papers
Shared shift vectors	$g:\mathcal{X}\to\mathcal{X}$7	(Furman et al., 2024, Ley et al., 2023)
Shared subgroup policy	one action per cluster or leaf	(Kavouras et al., 2024, Carrizosa et al., 2023)
Functional counterfactual map	$g:\mathcal{X}\to\mathcal{X}$8	(Valero-Leal et al., 28 Jan 2026)
Interaction-deletion set	$g:\mathcal{X}\to\mathcal{X}$9	(Stratigi et al., 23 Jan 2026)

These formalisms are different in mechanics, but they all operationalize the same idea: individual recourse is constrained by a shared group structure.

3. Optimization paradigms

The dominant optimization pattern is joint learning of recourse and grouping. In the unified gradient-based framework, the variables $$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$0, $$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$1, and $$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$2 are optimized together; Sparsemax relaxes discrete assignments, $$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$3 encourages sharp one-group membership, and $$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$4 regulates effective group usage (Furman et al., 2024). The same framework adds a plausibility penalty

$$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$5

based on conditional normalizing flows, and replaces cross-entropy with a margin-based validity loss so that confidence does not dominate distance and density terms.

A second paradigm is translation-based search. GLOBE-CE generates candidate translation directions at fixed cost, evaluates them on a grid of scalar magnitudes, and selects one or a few directions by coverage and cost. The design emphasizes coverage curves and minimum-cost distributions rather than a single scalar objective, and its categorical analysis turns translations on one-hot blocks into monotone If/Then rule progressions (Ley et al., 2023).

A third paradigm is summary learning through clustering and trees. GLANCE first generates local counterfactual actions, then either merges clusters in joint feature/action space or clusters in an augmented space that includes each instance’s minimum-cost action. Its tree-based variant, T-GLANCE, recursively splits on user-chosen features and attaches one action per node, yielding a subgroup policy optimized for effectiveness and average recourse cost under a small action budget (Kavouras et al., 2024).

A fourth paradigm is exact or near-exact mathematical optimization. In collective counterfactual explanations for score-based classification, linear models and additive tree models are transformed into convex quadratic mixed-integer formulations with binary variables for feature changes, globally used features, and selected individuals (Carrizosa et al., 2023). In optimal-transport group counterfactuals, linear classifiers admit QP, QCQP, and SDP formulations depending on the parametrization; the diagonal-affine case reduces bi-Lipschitz control to simple box constraints $$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$6, whereas PSD affine maps use Loewner-order LMIs (Valero-Leal et al., 28 Jan 2026).

These optimization regimes embody different priorities. Gradient-based methods emphasize end-to-end differentiability and plausibility modeling. Translation and clustering methods emphasize summary quality and interactive coverage–cost inspection. Mathematical optimization emphasizes exactness, convexity, and explicit global sparsity or geometry control.

4. Evaluation criteria and recurring trade-offs

The most common evaluation axes are validity, proximity/cost, and group compactness. In the unified local–group-wise–global framework, validity is the fraction of instances whose counterfactual is classified as the desired class; proximity is measured by $$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$7 and $$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$8 distance; plausibility is measured by Local Outlier Factor, Isolation Forest score, log density, and probabilistic plausibility; and group granularity is measured by the number of distinct shift vectors actually used and the fraction of counterfactuals assigned to exactly one group (Furman et al., 2024). The reported qualitative conclusion is a four-way trade-off: local counterfactuals achieve the best proximity but maximal $$\mathbf{X}_{GW}' = \mathbf{X}_0 + \mathbf{K}\mathbf{S}\mathbf{D}_{GW},$$9, global methods minimize $$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$0 but often incur larger distances and poorer plausibility, and group-wise methods aim to balance validity, proximity, plausibility, and compactness.

Coverage–cost trade-offs are central in translation-based and action-set methods. GLOBE-CE evaluates directions by the fraction of affected instances that can be flipped as the scalar magnitude increases, together with the corresponding average cost and per-instance minimum-cost histograms (Ley et al., 2023). GLANCE makes the same three-objective structure explicit: maximize effectiveness $$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$1, minimize average recourse cost $$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$2, and minimize the number of actions $$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$3, with strong dominance, size dominance, and weak dominance used to compare competing solutions (Kavouras et al., 2024).

Fairness enters the evaluation in several distinct ways. One line of work defines group fairness of counterfactual explanations as parity of counterfactual cost distributions across protected groups, relaxed operationally to approximate equality of groupwise expected costs (Artelt et al., 2022). A second line defines equal effectiveness and equal choice of recourse. For a set of actions $$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$4, micro effectiveness is

$$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$5

macro effectiveness is

$$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$6

and fairness requires small disparity between protected groups under these quantities (Ezzeddine et al., 28 Jan 2026). In group recommender systems, fairness instead measures how evenly the burden of removed interactions is distributed among members, using the inverse standard deviation of per-user contribution counts in the explanation set (Stratigi et al., 23 Jan 2026).

A recurring misconception is that group-based explanations are intrinsically fairness methods. The literature does not support that equation. Some methods use groups to summarize recourse patterns or improve interpretability, while others use groups to equalize burden, opportunity, or explanatory stability. These are related but not identical objectives.

5. Applications and empirical uses

The tabular recourse setting remains the canonical application domain. There, GCEs are used to summarize how adversely affected populations could obtain recourse through a small number of actions, to analyze subgroup differences, and to surface globally critical features. GLANCE, GLOBE-CE, collective mathematical optimization, and the unified local–group-wise–global framework all operate in this setting, but with different abstractions: dictionaries of actions, translation directions, linked per-instance edits, or jointly learned subgroup shifts (Kavouras et al., 2024).

A notable early user-facing use case is the Group-CF study on tabular classification. It constructs groups of similar instances, identifies key features from individual DiCE explanations, samples shared target values from the contrasting class, and selects the feature-value pair that maximizes group coverage. In a controlled user study with $$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$7, the results show that group counterfactuals elicit modest but definite improvements in people’s understanding of an AI system, together with ordered trends in confidence, explanation satisfaction, and trust (Warren et al., 2023).

Recent work broadens the application surface considerably. In multivariate rehabilitation time series, group-based counterfactuals are defined over semantic modality groups such as muscle × modality units. Learnable group gates are optimized jointly with perturbation masks to improve modality-group sparsity while maintaining or improving validity, temporal smoothness, and generation efficiency, thereby aligning explanations with clinical reasoning (Chukwu et al., 2 Jul 2026). In graph classification, RSGG-CE learns residual graph perturbations in latent space and samples them through a partially ordered edge-generation scheme, so that correlated sets of edge edits rather than isolated flips explain the decision (Prado-Romero et al., 2023). In group recommender systems, GCEs become sets of removed interactions whose absence would make a group recommendation disappear, with explicit utility, fairness, and cost measures and heuristics such as Pareto filtering, GreedyGrow, Grow&Prune, and ExpRebuild (Stratigi et al., 23 Jan 2026).

Adjacent literatures also adapt group-counterfactual ideas beyond standard recourse. The evolution of group counterfactuals across time is used to diagnose concept drift by tracking changes in group centroids and counterfactual action vectors before and after drift (Stępka et al., 11 Sep 2025). Group-aware shift explanations optimize worst-group PercentExplained under Wasserstein-2 distribution shift, so that mappings respect demographic or semantic group structure instead of sacrificing some groups for a better global fit (Stein et al., 2023). Procedural fairness work defines group counterfactual explanations through integrated gradients computed relative to multiple group-conditional baselines and penalizes cross-group variation in those attributions during training, thereby treating explanation invariance as a fairness criterion (Popoola et al., 11 Mar 2026). In low-dimensional representation analysis, Global Counterfactual Explanations and Transitive Global Translations explain how entire groups of points move between regions of an embedding through sparse, globally consistent translations (Plumb et al., 2020).

Taken together, these applications show that GCEs are not limited to one data modality or one explanatory purpose. They are used for recourse, model understanding, fairness auditing, shift analysis, and domain-aligned decision support.

6. Limitations and open problems

Several limitations recur across the literature. Many methods assume a differentiable predictor and continuous feature space. The unified gradient-based framework explicitly requires a differentiable classifier, considers logistic regression and a 3-layer MLP, and does not support categorical variables in its current implementation (Furman et al., 2024). Mathematical optimization and optimal-transport methods provide stronger guarantees for linear or linearizable models, but those guarantees weaken in black-box or highly nonlinear settings, where heuristic search or metaheuristics re-enter the picture (Valero-Leal et al., 28 Jan 2026).

A second limitation concerns the meaning of groups. In some methods, groups are externally given; in others, they are induced by clustering or by an entropy-regularized assignment matrix. The literature repeatedly notes that automatically discovered groups may or may not align with semantically meaningful subpopulations, demographic categories, or actionable policy segments (Furman et al., 2024). Group-aware shift explanations likewise assume a partition $$\mathbf{X}' = \mathbf{X}_0 + \mathbf{K}\mathbf{P}\mathbf{D}_{GW}$$8, with disjoint, known groups in the theory section, although unsupervised grouping can be used in practice (Stein et al., 2023).

A third limitation is the gap between probabilistic plausibility and real-world feasibility. Density penalties, adversarial discriminators, and OT geometry control can keep counterfactuals close to the data manifold, but they do not by themselves encode causal constraints, action feasibility, legal restrictions, or temporal dependencies. Several papers state this explicitly: plausibility is modeled probabilistically rather than causally, and incorporating causal knowledge, fairness constraints, or domain-specific feasibility remains an open direction (Furman et al., 2024).

Scalability also remains unresolved in the strongest formulations. Exact mixed-integer and SDP-based methods are effective on moderate-sized problems, but larger groups, higher dimensions, richer structured data, and online environments remain challenging (Carrizosa et al., 2023). Recommendation-specific GCEs add a different bottleneck: the search space is the power set of group interactions, so heuristic search under recommender-call budgets is essential, and only removal operations are currently allowed (Stratigi et al., 23 Jan 2026).

The field therefore remains methodologically plural. One plausible synthesis is that mature GCE systems will need to combine four ingredients that are currently scattered across different papers: a compact shared representation of recourse, explicit control of subgroup structure, a plausibility or geometry model, and an objective that reflects whether the goal is interpretability, fairness, coverage, or distributional diagnosis.