GLANCE: Global Actions in a Nutshell for Counterfactual Explainability

## Abstract

Abstract The widespread deployment of machine learning systems in critical real-world decision-making applications has highlighted the urgent need for counterfactual explainability methods that operate effectively. Global counterfactual explanations, expressed as actions to offer recourse, aim to provide succinct explanations and insights applicable to large population subgroups. Effectiveness is measured by the fraction of the population that is provided recourse, ensuring that the actions benefit as many individuals as possible. Keeping the cost of actions low ensures the proposed recourse actions remain practical and actionable. Limiting the number of actions that provide global counterfactuals is essential to maximize interpretability. The primary challenge, therefore, is balancing these trade-offs—maximizing effectiveness, minimizing cost, while maintaining a small number of actions. We introduce GLANCE, a versatile and adaptive framework, comprising two algorithms, that allows the careful balancing of the trade-offs among the three key objectives, with the size objective functioning as a tunable parameter to keep the actions few and easy to interpret. C-GLANCE employs a clustering approach that considers both the feature space and the space of counterfactual actions, thereby accounting for the distribution of points in a way that aligns with the structure of the model. T-GLANCE provides additional features to enhance flexibility. It employs a tree-based approach, that allows users to specify split features, to build a decision tree with a single counterfactual action at each node that can be used as a subgroup policy. Our extensive experimental evaluation demonstrates that our method consistently shows greater robustness and performance compared to existing methods across various datasets and models.

## 1 Introduction

Machine learning models are increasingly deployed in critical domains such as loan approvals, hiring, and healthcare. This widespread adoption has intensified the need for transparency and interpretability in model decisions, requiring users to understand how their input features influence the outcomes and how they might change them to achieve favorable outcomes, known as recourse []. Counterfactual explanations have gathered extensive attention for their suitability for achieving algorithmic recourse [], their interpretability [], actionability [], utility in fairness audits [, ], etc. A counterfactual action, or simply an *action*, defines the specific feature changes needed to convert an unfavorable decision into a favorable one.

Traditionally, counterfactual explanations refer to *local explainability*, being tied to a particular negatively affected instance. However, in many real-world scenarios, *global counterfactual explainability* is more useful, offering generalized explanations that apply across the affected population. While a collection of all local counterfactuals could technically cover all affected individuals, this approach sacrifices interpretability, which is central to global explainability.

Similar to past work [, ], we define Global Counterfactual Explanations (GCE) as a small set of *global actions* that provide effective recourse for the affected population.
Any global counterfactual solution must meet three objectives as identified from previous research:
(1) be composed of a small number of actions to ensure interpretability (size),

Figure: (a) Feature space
Refer to caption: x1.png

(2) minimize the cost of implementing those actions (cost), and
(3) offer recourse to as many affected individuals as possible (effectiveness).

As noted by [], the relationships between multiple optimization objectives are often complex, and aggregating them into a single objective, which is common in practice [], can be problematic since they are typically non-commensurable.
Framing GCEs as multi-objective optimization, allows us to explore the inherent *trade-offs* between effectiveness and cost, especially when the solution size is constrained.

To understand this optimization problem, assume a 2-d numerical feature space, and consider the two affected instances $x_{1},x_{2}$ depicted in Figure 1(a). Assume that recourse cost equals the distance to the decision boundary, drawn as a line in the figure. Observe that $a_{1}$ (resp. $a_{2}$) is the local action that provides recourse for $x_{1}$ (resp. $x_{2}$) at minimum cost.

Further, consider the *action space* depicted in Figure 1(b), where every action is represented as a point (or equivalently a vector relative to the center $o$ of the coordinate system). The blue $l_{1}$ and red $l_{2}$ lines represent the decision boundary seen from the perspectives of $x_{1}$ and $x_{2}$, respectively. The blue line $l_{1}$ separates the actions that provide recourse for $x_{1}$ (any action on the outside, away from $o$) from those that do not. Action $a_{1}$ lies on $l_{1}$, and is the closest point to $o$, and thus the min-cost local action for $x_{1}$.
Similarly, the red line $l_{2}$ concerns $x_{2}$ and contains its min-cost action $a_{2}$.

Consider now the problem of finding a single action GCE. To provide recourse for both $x_{1}$ and $x_{2}$, we look for an action that lies outside both lines in Figure 1(b). Among all such actions, $a_{3}$ has the minimum cost and thus is the optimal global action that maximizes effectiveness.
If we trade off effectiveness for cost, $a_{2}$ is the optimal global action that minimizes cost, but has 50% effectiveness (brings recourse to $x_{2}$ but not to $x_{1}$).

GCE gives rise to a different optimization problem than its local counterpart.
Even if optimal local actions are generated for each instance and a small subset is chosen as the global actions, this may still result in a suboptimal GCE.
In our example, the optimal global action $a_{3}$ is not an optimal local action for either $x_{1}$ or $x_{2}$; in fact, $a_{3}$ can be viewed as a compromise between $a_{1}$ and $a_{2}$, the locally optimal actions. Finding good GCEs requires a nuanced exploration of the action space.

In this work, we distinguish between two variants of GCE that differ in how they assign actions to individuals. In *implicit* GCE, each individual chooses the best (i.e., min-cost) global action in GCE to achieve (if possible) recourse [as in e.g., ]. Implicit GCE is helpful when one wishes to obtain a model-level understanding of recourse, or to select a global (horizontal) policy to apply to the entire affected population. In *explicit* GCE, individuals are partitioned according to their features and a single action applies to each partition [as in e.g., , ]. Explicit GCE provides a more interpretable summary as it associates actions to partitions of the feature space, and can also be used to implement targeted policies to bring recourse to subpopulations.

### 1.1 Key Contributions

Recognizing that both the effectiveness and the cost of actions depend on the individuals’ features and the model’s decision boundary, we propose a novel clustering approach for GCEs that considers both feature and action spaces.
We introduce GLANCE, a novel and highly adaptable framework, that consists of two algorithms based on hierarchical clustering techniques, C-GLANCE and T-GLANCE.

Figure: (a)
Refer to caption: x3.png

#### C-GLANCE.

We present C-GLANCE, a clustering-based approach for the implicit GCE problem that returns $s$ global actions. At a high level, C-GLANCE consists of two phases: candidate action generation, and action selection.

To generate candidates, C-GLANCE first performs a fine-grained clustering of the individuals in the feature space. For each cluster, C-GLANCE then discovers a few diverse actions that bring recourse to the cluster centroid. The intuition here is that highly similar individuals are expected to have a similar perspective of the decision boundary, and can thus achieve recourse through similar cost-efficient actions. Figure 2(a) shows an example with four fine-grained clusters, where three diverse actions are discovered from each centroid. Each action represents an alternate way to bring recourse to cluster members, giving C-GLANCE the flexibility to later select among them.

At the end of the first phase, C-GLANCE has associated each cluster of individuals with a set of actions. In the second phase, C-GLANCE iteratively merges such clusters until exactly $s$ clusters remain. Two clusters are merged if they either concern similar individuals or contain similar actions. The intuition behind the former is similar to that of the candidate generation phase. The intuition for the latter is that if similar actions bring recourse to them it should be easier to identify a common low-cost, highly effective action, even though they are dissimilar feature-wise. Figure 2(b) shows an example where the red individuals are grouped together either through affinity in the feature or in the action space.

As discussed in [], performing a coarse-grained clustering on feature space and then assigning a single action per cluster leads to inadequate global actions, that suffer either in cost or effectiveness. For example, in Figure 2(c), individuals are clustered based on feature similarity and a single action is selected per cluster. Because the blue cluster is associated with rather dissimilar actions, choosing a single actions leads to a compromise in either effectiveness (as depicted) or cost.

#### T-GLANCE.

Recognizing the need for explicit GCE, we present T-GLANCE, an algorithm employing a tree-based approach, similar to explainable clustering techniques [e.g., ]. This allows assigning actions to individuals while satisfying the transparency and consistency properties required by [], making it ideal for policy-making and auditing. T-GLANCE also allows users to select features that define specific groups, to produce targeted GCEs. For the user’s facilitation, we also provide suggestions based on the importance feature permutation score for selecting features to be used in splits. As shown in our experiments, T-GLANCE delivers competitive results, outperforming previous tree-based methods while maintaining efficiency.

#### Experimental Evaluation.

We perform a comprehensive evaluation across various datasets and models, demonstrating that GLANCE consistently outperforms or matches state-of-the-art methods in both cost and effectiveness while remaining computational efficiency.
To facilitate comparison in global counterfactual methods, we invoke Pareto dominance criteria from multiobjective optimization, further showcasing
GLANCE’s stable, superior performance across diverse datasets and models.

### 1.2 Related Work

#### Counterfactual Explanations.

There has been a plethora of work with an emphasis on counterfactual explanations []. An overview of methods on algorithmic recourse, which provides explanations and recommendations to individuals impacted by automated decision-making systems is presented by Karimi et al. []. These methods can be model-agnostic or model-specific [e.g., for trees ], focus on properties like diversity [] or feasibility []), global or local.
While local counterfactuals are well defined, global counterfactuals present challenges, as they must provide recourse for all individuals within a specific group in the same manner while maintaining explainability and staying true to the notion of local counterfactual, which focuses on minimal changes. Achieving this balance globally presents significant difficulties.

Prior work does not make the distinction between explicit and implicit GCE.
Regarding explicit GCE, Rawal and Lakkaraju [] introduced AReS, a framework for global counterfactual explanations, that jointly optimizes recourse correctness, coverage, and cost, providing an interpretable summary of recourses, expressed in a two-level rule set.
However, the AReS framework may fail to cover the entire population. Ley et al. [] later improved the computational efficiency with Fast AReS.
In another direction, Kanamori et al. [] introduced CET, which partitions the space and assigns an action to each part transparently and consistently. Although effective, its computational complexity limits scalability.

Regarding implicit GCE, Warren et al. [] developed Group-CF, that generates counterfactuals that seeks to maximize effectiveness, though it can result in higher costs.
Ley et al. [] proposed GLOBE-CE, where global counterfactual explainability is defined differently, as a small set of action *directions* along which individuals can “move” to achieve recourse.
Therefore, GLOBE-CE attacks a different optimization problem than the one we study here. For the toy example in Figure 1, the single direction that minimizes the total recourse cost for $x_{1}$ and $x_{2}$ is the direction $d$ depicted in Figure 1(b). This direction contains actions $a_{4}$ and $a_{5}$ that bring recourse to $x_{2}$ and $x_{1}$, respectively, with minimum cost *along* $d$. It is important to note that neither $a_{4}$ nor $a_{5}$ is optimal as a GCE (or for local explainability). Even the set $\{a_{4},a_{5}\}$ is a suboptimal GCE—the set $\{a_{1},a_{2}\}$ dominates it with equal size and effectiveness but lower total cost. In general, (1) directly translating the output of GLOBE-CE into a GCE leads to numerous micro-actions (two in our example, but potentially as many as the number of individuals), which reduces interpretability, and (2) choosing a few actions along the optimal directions may result in suboptimal GCEs (in our example, any single action along $d$ is dominated by $a$).
Other works include Carrizosa et al. [, ], who use mixed-integer quadratic models for group-level explanations, and Koo et al. [], who employ Lagrangian methods. Research has also extended to generating global counterfactuals for graphs [] and auditing subgroup fairness [].

#### Other Related work.

Many clustering algorithms lead to clusters that are hard to explain. There is a plethora of work on explainable or interpretable clustering, mainly using decision trees that under specific assumptions come with provable guarantees, including the works of [, , , , , ], etc.
Another area of related work is multiobjective optimization and Pareto optimality. We refer the reader to Branke [].

## 2 Problem Formulation

We consider a black box binary classifier $h:{\mathcal{X}}\rightarrow\{-1,1\}$, where the positive outcome is favorable and the negative is unfavorable. We will focus on the set $\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}\subseteq{\mathcal{X}}$ of adversely affected individuals, i.e., those who receive the unfavorable outcome.
We denote as ${\mathbb{A}}$ the set of all possible actions (which is potentially infinite), where an action $a\in{\mathbb{A}}$ is a set of changes to feature values, e.g., $a=\{\text{country}\to\text{US},\text{education-num}\to+2\}$, which, when applied to an instance $x\in\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}$, results in a counterfactual instance $x^{\prime}=a(x)$. Every action $a$ has a cost, denoted as $\operatorname{\mathsf{cost}}(a,x)$, and is effective for an instance $x$ if $h(a(x))=h(x^{\prime})=1$. Let ${\mathbb{C}}\subseteq{\mathbb{A}}$. The recourse cost $\operatorname{\mathsf{rc}}({\mathbb{C}},x)$ of an instance $x$ is the minimum cost incurred from an effective action in ${\mathbb{C}}$:

$$ $\operatorname{\mathsf{rc}}({\mathbb{C}},x)=\min\{\operatorname{\mathsf{cost}}( a,x)|a\in{\mathbb{C}}:h(a(x))=1\}$ $$

Let $X_{{\mathbb{C}}}=\{x\in\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}|h(a(x))=1
,a\in{\mathbb{C}}\}$ be the set of instances that flip their prediction using one of the actions in ${\mathbb{C}}$. Then the effectiveness, also known as coverage Ley et al. [],
of the actions for the affected instances $\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}$ is defined as the percentage of $\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}$ that managed to flip their prediction using one of the actions in ${\mathbb{C}}$:

$$ $\operatorname{\mathsf{eff}}({\mathbb{C}},\operatorname{{\mathcal{X}}_{\mathsf{ {aff}}}})=\frac{|X_{{\mathbb{C}}}|}{|\operatorname{{\mathcal{X}}_{\mathsf{{aff }}}}|},$ $$

and their cost as the average cost of the instances that manage to flip their predictions using one of the actions in ${\mathbb{C}}$:

$$ $\operatorname{\mathsf{avc}}({\mathbb{C}},\operatorname{{\mathcal{X}}_{\mathsf{ {aff}}}})=\frac{\sum\limits_{x\in X_{{\mathbb{C}}}}\operatorname{\mathsf{rc}}( {\mathbb{C}},x)}{|X_{\mathbb{C}}|}.$ $$

Finally, let

$$ $\operatorname{\mathsf{size}}({\mathbb{C}})=|{\mathbb{C}}|$ $$

denote the cardinality of a set ${\mathbb{C}}$.

As it is clear, multiple sets of actions can produce recourse for $\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}$ and the quality of such a set is a factor of the three notions we introduced: effectiveness, cost, and size. An ideal global counterfactual should maximize effectiveness while minimizing the cost and the size, based on the properties that state-of-the-art works [, , , ] have argued are essential.
Formally, we define the global counterfactual explanation problem as follows:

###### Problem 1 (Global Counterfactual Explanations (GCE)) .

Given a black box model $h$ that classifies the $\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}$ instances to the negative class, our goal is to find the set ${\mathbb{C}}\subseteq{\mathbb{A}},{\mathbb{C}}\neq\emptyset$
that represents a solution to the following multi-objective optimization:

$$ $\min\quad\{\operatorname{\mathsf{size}}({\mathbb{C}}),\;-\operatorname{\mathsf {eff}}({\mathbb{C}},\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}),\; \operatorname{\mathsf{avc}}({\mathbb{C}},\operatorname{{\mathcal{X}}_{\mathsf{ {aff}}}})\}$ $$

The requirement for a small set of actions is to enhance the interpretability of the explanation. This can also be expressed as a constraint to the set size we can afford. In this case, the third objective of Problem 1 is replaced by the constraint

$$ $\operatorname{\mathsf{size}}({\mathbb{C}})\leq s$ $$

,
where $s$ is a small positive integer. For the rest of the paper, we will use the following problem formulation:

###### Problem 2 ( $s$ -GCE) .

Given a black box model $h$ that classifies the $\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}$ instances to the negative class, our goal is to find the set ${\mathbb{C}}\subseteq{\mathbb{A}}$
that represents a solution to the following bi-objective optimization problem:

$$ $\displaystyle\min\quad$ $\displaystyle\{-\operatorname{\mathsf{eff}}({\mathbb{C}},\operatorname{{ \mathcal{X}}_{\mathsf{{aff}}}}),\;\operatorname{\mathsf{avc}}({\mathbb{C}}, \operatorname{{\mathcal{X}}_{\mathsf{{aff}}}})\}$ s.t. $\displaystyle\operatorname{\mathsf{size}}({\mathbb{C}})\leq s$ $$

If the set of actions ${\mathbb{A}}$ is finite and explicitly given, $s$-CGE is NP-hard. For more details, we refer the reader to Appendix A .

###### Problem 1 (Global Counterfactual Explanations (GCE)) .

###### Problem 2 ( s 𝑠 s italic_s -GCE) .

## 3 C-GLANCE

C-GLANCE (Algorithm 1) presents a clustering algorithm that follows an agglomerative approach.
The first step is to establish multiple small clusters and determine representative diverse actions for each. The initial clusters will undergo merging, exploiting the notion of proximity in both the original feature space and the action space, until no more than $s$ clusters exist. Finally, a single action will be extracted from each cluster with universal applicability across all instances.

Figure: Algorithm 1 C-GLANCE

#### Algorithm Description.

*(1) Initial clusters and actions generation.*
The feature space is partitioned into $k$ clusters, by a clustering algorithm (e.g., $k$-means, line 1, alg. 1). We compute the *centroid* of each cluster and generate $m$ *diverse* counterfactual actions for each centroid (line 3, alg. 1), employing any candidate counterfactual generation method (see Appendix E for the methods used in the experimental evaluation). We aim to efficiently and effectively explore the action space by generating actions from widely dispersed points within the feature space and guiding them in diverse directions that cross the decision boundary.

*(2) Merging.* Similar to agglomerative clustering, our approach merges clusters with the goal of maximizing intra-cluster similarity and minimizing inter-cluster similarity with respect to both the feature and action spaces. Specifically, similarity is computed by a metric $D$, which is the sum of two distances: $d_{1}:\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}\times\operatorname{{
\mathcal{X}}_{\mathsf{{aff}}}}\rightarrow\mathbb{R}$, the distance between the centroids of each cluster, and $d_{2}:{\mathbb{A}}\times{\mathbb{A}}\rightarrow\mathbb{R}$, the distance between the average counterfactual actions of each cluster; other formulations for $d_{2}$ are also possible.
Thus, clusters are combined based on both their feature proximity and the proximity of their respective actions. Until the desired number $s$ of clusters is reached, we merge (line 7, alg. 1) the two clusters that minimize the total distance $d_{1}+d_{2}$ (line 6, alg. 1).
Merging two clusters means that their action sets are combined, ensuring that the most effective actions for each cluster are retained (line 8, alg. 1).

*(3) GCE extraction and optimality.*
For each of the $s$ clusters, we extract a single candidate action, and return the set of these $s$ actions (line 10, alg. 1).
Assuming that the action generation and subsequent merging has succesfully grouped together individuals that can achieve recourse through similar cost-efficient actions, in the final step we target effectiveness, selecting at each cluster the optimal action, in terms of effectiveness, among those associated with the cluster.

#### Approach strengths.

C-GLANCE supports various clustering methods, cost metrics, and action generation techniques making it adaptable to different applications. In the App. Table 13, we present the results using various action generation methods, all of which
lead to near-optimal solutions in terms of the effectiveness-cost tradeoff. Additionally, C-GLANCE demonstrates speed, robustness, and near-optimal solution quality across datasets, as shown in the experimental evaluation.

## 4 T-GLANCE

T-GLANCE (Algorithm 2) employs a top-down approach, similar to the decision tree methods in classification and regression trees (CART) []. In CART, splits are determined by minimizing an impurity measure and then recursively applying this process to the resulting child nodes []. At every iteration, T-GLANCE chooses a different direction to split on at every cell of the current partition, similar to Breiman’s Algorithm. However, the splitting criterion differs, since T-GLANCE aims to optimize the trade-off between effectiveness and cost to find non-dominated solutions. In addition, T-GLANCE offers flexibility by allowing users to specify the split features when building the decision tree. At each node, the algorithm assigns optimal counterfactual actions, enhancing interpretability while maintaining transparency.
We begin with the necessary definitions to introduce Algorithm 2.

Figure: Algorithm 2 T-GLANCE

#### Definitions.

A partition ${\mathcal{P}}$ of ${\mathbb{R}}^{d}$ is a family of sets $\{A_{1},\dots,A_{l}\}$ such that $A_{j}\subseteq{\mathbb{R}}^{d}$, $A_{j}\cap A_{k}=\emptyset$ for all $j,k\in[l]$ and $\bigcup_{j=1}^{s}A_{j}={\mathbb{R}}^{d}$. Let ${\mathcal{P}}$ a partition of $\operatorname{{\mathcal{X}}_{\mathsf{{aff}}}}$. Every element $A$ of ${\mathcal{P}}$ is called a cell of ${\mathcal{P}}$ or just a cell. Each cell can be further divided into two subcells $A_{0}^{i},A_{1}^{i}$ with respect to any direction $i$, defined as $A_{0}^{i}=\{x\in A|x_{i}\in v_{i}(0)\}$ and $A_{1}^{i}=\{x\in A|x_{i}\in v_{i}(1)\}$, where $v_{i}(k)$ represent the values that correspond to the specific split (e.g., 0 or 1 if it is a binary feature, specific ranges for continuous features).
Let $\bm{ca}(A):=\texttt{actions}(c,m)$ be the set of $m$ generated actions for the centroid $c$ of the cell $A$.
For simplicity, we will use $\texttt{bca}(\bm{ca}(A),A)$ to refer to the action in $\bm{ca}(A)$
with the highest effectiveness. Formally,

$$ $\texttt{bca}(\bm{ca}(A),A)=\operatorname*{arg\,max}_{a\in\bm{ca}(A)}\{ \operatorname{\mathsf{eff}}(\{a\},A)\}$ (1) $$

We define a local version of the total effectiveness gain achieved by a specific portion split $i$ to a local cell $A$.

$$ $\displaystyle\begin{split}V(A,i)&\triangleq\operatorname{\mathsf{eff}}(\{ \texttt{bca}(\bm{ca}(A),A)\},A)-\overline{\operatorname{\mathsf{eff}}}(\{ \texttt{bca}(\bm{ca}(A^{i}_{k}),A^{i}_{k})\},A^{i}_{k})\\ &=\operatorname{\mathsf{eff}}(\{\texttt{bca}(\bm{ca}(A),A)\},A)-\frac{1}{|k|} \sum_{k=\{0,1\}}{\operatorname{\mathsf{eff}}(\{\texttt{bca}(\bm{ca}(A^{i}_{k}) ,A^{i}_{k})\},A^{i}_{k}})\end{split}$ (2) $$

#### Algorithm description.

GLANCE-T selects a different direction for splitting at each iteration at every cell in the current partition.
Specifically, the algorithm begins with a single cluster containing all affected instances and generates $m$ *diverse* actions for the centroid, retaining the best action according to the criterion in Equation 1 (lines 1–2, alg. 2). The root then is established as the first cell $A$ to be examined, and the possible splits are examined based on the actions that were generated for each split (lines 8–9, alg. 2).
For simplicity, we present the algorithm with binary splits (branching factor of 2), though it can be generalized to accommodate any branching factor.
We proceed with the split that maximizes Equation 2, provided it yields a nonnegative value, and add the new cells as nodes of the tree to be examined in the next repetitions (lines 10–13, alg. 2); otherwise the node is not split. Importantly, the set of directions allowed for splitting is a subset of the features, provided by the user, allowing for controlled and interpretable partitioning of the data.

#### Termination criteria.

This process continues until one of the following termination criteria occurs: (1) a node has no remaining features to consider for splitting, (2) the maximum score of the actions from all remaining features is negative, or (3) the specified number of leaf nodes has been reached (i.e., size constraints).

#### Approach strengths.

A key strength of T-GLANCE is its user-driven flexibility, allowing policy makers or auditors to select the features that determine the partitioning of the tree. By allowing users to guide the feature selection, T-GLANCE can adapt to specific policy needs or audit requirements, making it highly customizable. T-GLANCE assigns a single optimal counterfactual action to each node of the tree, ensuring transparency and consistency across decision-making processes. Additionally, the method remains efficient, robust, and effective across various datasets and scenarios, as shown in our evaluations, making it well-suited for real-world applications.

## 5 Experimental Evaluation

### 5.1 Experimental Setting

#### Baselines.

We compare GLANCE framework methods against state-of-the-art methods in Global Counterfactual Explanations, specifically: AReS (using the Fast AReS implementation), CET, GroupCF, and GLOBE-CE, all of which are constrained by a predefined action set size. Details on the implementation used for each method can be found in Appendix D.

#### Datasets.

We use four established benchmark datasets from previous research: COMPAS [], German Credit [], Default Credit [], and HELOC []. Additionally, we introduce the Adult dataset [] for further evaluation. Details on the datasets and their preprocessing can be found in Appendix  B.

#### Models.

We trained three different model types: XGBoost (XGB), Logistic Regression (LR), and Deep Neural Network (DNN). Hyperparameters and training accuracy statistics are provided in Appendix C. We used 5-fold cross-validation, to also evaluate the robustness of the results.

#### Recourse Cost.

For computing recourse costs, we adhered to the guidelines established by Ley et al. [].

#### Reproducibility.

All experiments were conducted on an in-house server with cloud infrastructure equipped with an Intel(R) Core(TM) i9-10900X CPU @ 3.70GHz, 128 GB of RAM. No GPU acceleration was utilized during these experiments. We provide the code for the reproducibility of our experiments as supplementary material.

#### Running Time.

In our experiments, GLOBE-CE and dGLOBE-CE were the fastest methods, consistently delivering solutions within 20 seconds across all datasets and models. C-GLANCE and T-GLANCE followed, typically completing in under 300 seconds, although T-GLANCE reached up to 850 seconds in four cases. Other methods were slower, with Fast AReS ranging typically between 150–400 seconds, and peaking at 1,400 seconds in some runs. The least computationally efficient were Group-CF and CET, with maximum runtimes of 3,500 and 17,000 seconds, respectively, and CET failed to solve the underlying optimization problem after 20 hours of runtime for the Adult dataset across all models.

**Table 1: Evaluation of the effectiveness and cost of C-GLANCE and T-GLANCE against Fast AReS, CET, Group-CF, GLOBE-CE, and dGLOBE-CE for $s$-GCE problem with $s=4$. $s$-GCE solutions with effectiveness below 80% (applicability threshold) are highlighted in purple. Non-robust GCEs, identified by either a standard deviation (std) in effectiveness greater than 5% across folds or a std in cost greater than half the average cost, are highlighted in red.**
| Models | Algorithms | Datasets |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  | Adult | COMPAS | Default Credit | German Credit | HELOC |  |  |  |  |  |
|  |  | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ |
| dnn | Fast AReS | 12.39 ± 1.06 | 1.0 ± 0.0 | 55.0 ± 0.86 | 1.21 ± 0.09 | 18.88 ± 2.16 | 1.0 ± 0.0 | 52.39 ± 1.63 | 1.0 ± 0.0 | 12.19 ± 0.58 | 1.03 ± 0.05 |
|  | CET | nan ± nan | nan ± nan | 63.62 ± 10.35 | 0.96 ± 0.24 | 98.87 ± 0.62 | 6.32 ± 2.28 | 97.3 ± 2.46 | 1.58 ± 0.54 | 86.78 ± 10.62 | 8.67 ± 3.25 |
|  | Group-CF | 100.0 ± 0.0 | 10.08 ± 0.03 | 100.0 ± 0.0 | 4.48 ± 2.53 | 79.6 ± 20.79 | 1.53 ± 0.62 | 97.8 ± 4.4 | 1.85 ± 0.13 | 80.4 ± 10.17 | 3.09 ± 0.91 |
|  | GLOBE-CE | 99.92 ± 0.0 | 4.24 ± 0.42 | 100.0 ± 0.0 | 4.54 ± 3.31 | 76.94 ± 37.55 | 5.14 ± 0.35 | 93.31 ± 3.48 | 2.0 ± 1.55 | 42.72 ± 46.97 | 11.77 ± 15.87 |
|  | dGLOBE-CE | 99.92 ± 0.0 | 10.89 ± 1.37 | 100.0 ± 0.0 | 7.96 ± 3.91 | 87.38 ± 18.69 | 5.96 ± 4.14 | 97.36 ± 0.82 | 2.49 ± 0.27 | 99.96 ± 0.05 | 11.07 ± 8.6 |
|  | C-GLANCE | 100.0 ± 0.0 | 4.6 ± 0.73 | 100.0 ± 0.0 | 2.34 ± 0.43 | 100.0 ± 0.0 | 1.2 ± 0.4 | 95.31 ± 3.15 | 1.25 ± 0.33 | 99.94 ± 0.05 | 11.24 ± 1.37 |
|  | T-GLANCE | 100.0 ± 0.0 | 4.43 ± 0.43 | 99.53 ± 0.22 | 2.62 ± 0.24 | 100.0 ± 0.0 | 1.48 ± 0.43 | 96.97 ± 3.48 | 1.59 ± 0.8 | 99.8 ± 0.24 | 10.9 ± 1.22 |
| lr | Fast AReS | 11.74 ± 2.4 | 1.0 ± 0.0 | 62.5 ± 1.82 | 1.24 ± 0.14 | 10.85 ± 5.45 | 1.07 ± 0.13 | 75.27 ± 2.96 | 1.0 ± 0.0 | 9.23 ± 1.24 | 1.12 ± 0.1 |
|  | CET | nan ± nan | nan ± nan | 73.18 ± 4.34 | 1.24 ± 0.15 | 100.0 ± 0.0 | 3.79 ± 1.31 | 96.5 ± 2.85 | 2.42 ± 0.24 | 100.0 ± 0.0 | 3.57 ± 1.48 |
|  | GroupCF | 100.0 ± 0.0 | 1.71 ± 0.39 | 100.0 ± 0.0 | 3.97 ± 2.38 | 95.4 ± 9.2 | 1.94 ± 1.2 | 97.6 ± 2.94 | 9.34 ± 3.85 | 90.6 ± 3.93 | 2.4 ± 1.38 |
|  | GLOBE-CE | 99.92 ± 0.0 | 2.68 ± 0.17 | 95.74 ± 8.52 | 5.14 ± 3.77 | 99.94 ± 0.07 | 3.42 ± 1.99 | 57.09 ± 20.03 | 0.75 ± 1.04 | 99.9 ± 0.0 | 0.6 ± 0.54 |
|  | dGLOBE-CE | 99.92 ± 0.0 | 5.91 ± 0.93 | 100.0 ± 0.0 | 6.71 ± 0.23 | 99.94 ± 0.07 | 10.38 ± 7.76 | 69.89 ± 15.35 | 2.47 ± 0.23 | 99.9 ± 0.0 | 1.63 ± 0.35 |
|  | C-GLANCE | 100.0 ± 0.0 | 1.04 ± 0.07 | 100.0 ± 0.0 | 2.33 ± 0.38 | 100.0 ± 0.0 | 1.05 ± 0.11 | 100.0 ± 0.0 | 1.21 ± 0.06 | 100.0 ± 0.0 | 1.55 ± 0.54 |
|  | T-GLANCE | 100.0 ± 0.0 | 0.69 ± 0.01 | 99.79 ± 0.22 | 2.19 ± 0.1 | 100.0 ± 0.0 | 1.43 ± 0.36 | 99.58 ± 0.83 | 1.54 ± 0.32 | 100.0 ± 0.0 | 1.58 ± 0.37 |
| xgb | Fast AReS | 6.13 ± 0.42 | 1.0 ± 0.0 | 59.83 ± 3.12 | 1.1 ± 0.05 | 31.86 ± 5.12 | 1.05 ± 0.04 | 51.27 ± 1.57 | 1.0 ± 0.0 | 8.49 ± 1.32 | 1.16 ± 0.13 |
|  | CET | nan ± nan | nan ± nan | 58.4 ± 9.3 | 1.06 ± 0.24 | 86.29 ± 9.94 | 4.5 ± 2.64 | 100.0 ± 0.0 | 2.73 ± 0.49 | 86.78 ± 6.7 | 12.51 ± 2.75 |
|  | GroupCF | 96.8 ± 1.72 | 1.41 ± 0.54 | 100.0 ± 0.0 | 4.06 ± 2.1 | 95.2 ± 1.6 | 1.41 ± 0.64 | 100.0 ± 0.0 | 5.78 ± 4.11 | 78.4 ± 5.82 | 5.63 ± 1.93 |
|  | GLOBE-CE | 82.87 ± 12.14 | 30.1 ± 10.39 | 87.13 ± 11.14 | 9.75 ± 7.2 | 82.7 ± 7.26 | 20.82 ± 1.73 | 77.05 ± 11.26 | 1.14 ± 1.24 | 27.66 ± 5.06 | 12.52 ± 32.48 |
|  | dGLOBE-CE | 93.76 ± 1.98 | 64.76 ± 1.29 | 99.84 ± 0.31 | 12.46 ± 3.42 | 97.47 ± 0.82 | 42.58 ± 3.57 | 86.96 ± 9.79 | 2.66 ± 0.77 | 77.64 ± 11.51 | 128.0 ± 0.0 |
|  | C-GLANCE | 99.85 ± 0.12 | 5.98 ± 4.22 | 99.51 ± 0.46 | 2.96 ± 0.82 | 98.13 ± 1.05 | 3.68 ± 1.64 | 100.0 ± 0.0 | 1.06 ± 0.03 | 98.94 ± 0.66 | 19.99 ± 1.91 |
|  | T-GLANCE | 99.86 ± 0.14 | 1.8 ± 0.51 | 99.02 ± 0.93 | 2.6 ± 0.46 | 94.6 ± 3.13 | 2.75 ± 1.42 | 99.46 ± 1.08 | 1.33 ± 0.42 | 96.42 ± 2.45 | 24.85 ± 8.4 |

### 5.2 Experimental Evaluation

Table 1 presents the summarized results of all competing methods.
We compare C-GLANCE and T-GLANCE against the five other competitors across five datasets and three models, resulting in 75 head-to-head comparisons for each method, totaling 150. However, since CET failed to solve the underlying optimization problem for the Adult dataset across all models, the final count is 72 comparisons per GLANCE method.
Recall that all reported results concern the $s$-GCE problem for $s=4$; Appendix F presents an extensive experimental evaluation, including: additional and more detailed results for $s=4$; results for $s=8$ for all methods and results under no size constraints for GLOBE-CE; results utilizing different counterfactual generation methods.

#### Pareto Dominance.

We summarize method performance by determining whether one solution dominates another based on effectiveness and cost.
Specifically, *a solution ${\mathbb{C}}$ of $s$-GCE *Pareto dominates* another solution ${\mathbb{C}}^{\prime}$ if it offers equal or better effectiveness and cost, and is strictly better in at least one of these objectives*.

**Table 2: Pareto domination evaluation of solutions, for $s$-GCE problem with $s=4$. The table reports the rate (number of times over available comparisons) at which GLANCE methods dominate competitors (listed in the dominates column) and the rate at which GLANCE methods are dominated by competitors (listed in the is dominated column).**
|  | C-GLANCE | T-GLANCE |  |  |
| --- | --- | --- | --- | --- |
|  | dominates | is dominated | dominates | is dominated |
| Fast AReS | 1/15 | 0/15 | 1/15 | 0/15 |
| CET | 6/12 | 0/12 | 5/12 | 1/12 |
| GroupCF | 9/15 | 0/15 | 6/15 | 1/15 |
| GLOBE-CE | 11/15 | 0/15 | 9/15 | 0/15 |
| dGLOBE-CE | 12/15 | 1/15 | 9/15 | 0/15 |
| competitors | 39/72 | 1/72 | 30/72 | 2/72 |

As shown in Table  2, C-GLANCE dominates other methods in 39 out of 72 cases (54%), while is dominated only once by dGLOBE-CE (in HELOC-DNN—cf. Table 1). T-GLANCE dominates other methods in 30 out of 72 cases (41%), while it is dominated once by Group-CF (in Default Credit-XGB) and once by CET (in German Credit-DNN). In the three cases in which our method is dominated, the performance in terms of effectiveness is comparable. Overall, the GLANCE methods dominate other solutions in almost half of the cases (48%) and are dominated in only 2%.

#### Solution practicality.

In the prior Pareto-dominance evaluation (Table 2) we compare solutions with optimal or near-optimal effectiveness to many solutions that exhibit unacceptable or unsatisfactory effectiveness. These lower-performing solutions are impractical for GCE, as the goal is to offer recourse to a large population segment. Solutions with low effectiveness fail to meet this goal, limiting their applicability in real-world scenarios. A solution that leaves a significant percentage of individuals without recourse undermines the very purpose of GCE, as noted by []. It is also important to note that achieving low recourse costs is easier for smaller subpopulations, especially those near the decision boundary. We consider a solution to be practical if it achieves effectiveness of at least 80%. In Table 1, we highlight the impractical solutions in purple. GLANCE methods never return impractical solutions. In contrast, all the solutions returned by Fast AReS are deemed impractical, while the other methods return 2–5 impractical solutions each.

**Table 3: Pareto domination evaluation of solutions, for $s$-GCE problem with $s=4$, after applying the $\operatorname{\mathsf{eff}}>80\%$ threshold. The table reports the rate (number of times over available comparisons) at which GLANCE methods dominate competitors (listed in the dominates column) and the rate at which GLANCE methods are dominated by competitors (listed in the is dominated column).**
|  | C-GLANCE | T-GLANCE |  |  |
| --- | --- | --- | --- | --- |
|  | Dominates | is Dominated | Dominates | is Dominated |
| Fast AReS | 0/0 | 0/0 | 0/0 | 0/0 |
| CET | 6/9 | 0/9 | 5/9 | 1/9 |
| GroupCF | 8/13 | 0/13 | 5/13 | 1/13 |
| GLOBE-CE | 8/10 | 0/10 | 7/10 | 0/10 |
| dGLOBE-CE | 10/13 | 1/13 | 7/13 | 0/13 |
| Competitors | 32/45 | 1/45 | 24/45 | 2/45 |

Table 3 summarizes the head-to-head comparisons only when considering practical solutions. GLANCE framework dominates other methods in a much larger percentage—over 62% of the cases. It remains dominated in the same three cases, maintaining the percentage of dominated cases at a low 3%.

#### Robustness.

We expect methods to be robust, *consistently* generating highly effective and low-cost GCEs across different data splits, which is crucial for real-world deployment.
Without robustness, recourse actions can vary significantly, undermining trust and leading to unfair outcomes, especially in critical areas like healthcare or finance. Evaluating the stability of effectiveness and cost metrics across different folds is key to determining the practical applicability of a counterfactual explanation method.
Standard deviation measures this stability. An effectiveness deviation above 5% indicates an inconsistency in providing recourse, while a cost deviation exceeding half the average suggests unpredictable, impractical actions. These fluctuations make a solution unreliable and we highlight them in red in Table 1.
Figure 3 visualizes Table’s 1 results, i.e., the effectiveness of each solution vs. the cost (scaled to 0–1 values), along with their standard deviation. This figure provides a clear comparison of the trade-offs between cost and effectiveness, underscoring the robustness and consistency of the methods across different scenarios.

## 6 Conclusion

This paper introduces GLANCE, a robust and flexible framework for generating global counterfactual explanations that effectively balances the trade-offs between effectiveness, cost, and interpretability. An extensive experimental evaluation demonstrates that GLANCE constructs counterfactual explanations that are more effective and cheaper than those produced by the state of the art under size constraints.

## Acknowledgments and Disclosure of Funding

This work has been funded by the European Union’s Horizon Europe research and innovation
programme under Grant Agreement No. 101070568 (AutoFair).

## Appendix A Hardness

To analyze the complexity of the $s$-GCE problem, we suppose that the set of all actions is finite by discretizing continuous features.

###### Theorem 3 (NP-hardness) .

The $s$-GCE problem is NP-hard.

###### Proof.

We establish the NP-hardness of the $s$-GCE problem by reducing the well-known Max $k$-Coverage problem to it.

The Max $k$-Coverage problem is defined as follows: Given a universe $\mathcal{U}$ of elements and a collection of subsets $\mathcal{S}=\{S_{1},S_{2},\dots,S_{n}\}$ where each $S_{i}\subseteq\mathcal{U}$, along with an integer $k$, the goal is to select at most $k$ subsets from $\mathcal{S}$ such that the total number of covered elements is maximized.

To perform the reduction, we proceed by mapping each subset $S_{i}$ to an action $a_{i}$ in the $s$-GCE problem:

- •
Actions and Subsets Correspondence: Each action $a_{i}$ corresponds to a subset $S_{i}$ of elements in $\mathcal{U}$.
- •
Coverage Mapping: The effectiveness of an action $a_{i}$ in the $s$-GCE problem is defined as the elements in $S_{i}$.
- •
Ignoring Costs: We ignore the cost associated with actions in the $s$-GCE problem, focusing solely on coverage.

Under this mapping, the objective of selecting at most $k$ actions in the $s$-GCE problem to maximize the coverage of elements aligns exactly with the objective of the Max $k$-Coverage problem.

∎

Given that even the relaxed version of the $s$-GCE problem that ignores costs is NP-hard, identifying the optimal solution for the $s$-GCE is computationally infeasible. Moreover, the action space, although finite, may be too large to process efficiently. Therefore, our approach involves generating a manageable subset ${\mathbb{A}}^{\prime}\subseteq{\mathbb{A}}$ of actions before extracting the final $s$ actions. In the next section, we will introduce C-GLANCE, an efficient heuristic designed to solve the $s$-GCE problem.

###### Theorem 3 (NP-hardness) .

###### Proof.

## Appendix B Datasets & Preprocessing

We use five publicly available datasets as benchmarks. Our choice is based on their established use in previous works.
Table 4 summarizes the datasets’ information, including the number of instances, the number of categorical and continuous features, the input dimensions (i.e., the number of continuous features plus the number of categorical after a preprocessing step using one-hot-encoding), and the number of instances used for training and testing, evaluated using a 5-fold cross-validation strategy. This approach illustrates the resilience of our method across various splits.

The dataset preprocessing follows the approach in Ley et al. [], using the publicly available repository at [https://github.com/danwley/GLOBE-CE/](https://github.com/danwley/GLOBE-CE/). For completeness, we also provide a short description of each dataset below.

#### Adult

The Adult dataset (also known as “Census Income” dataset) is designed for the task of predicting whether an individual’s income exceeds $ 50K/yr, based on census data. The data as well as more detailed information can be obtained at [https://archive.ics.uci.edu/dataset/2/adult](https://archive.ics.uci.edu/dataset/2/adult).

For the preprocessing of this dataset, we first drop the “education-num” feature due to redundancy with the “education” feature (which takes on string values). Next, we remove missing values and map the class labels ‘$<$=50K’ and ‘$>$50K’ to 0 and 1, respectively. Additional minor transformations, mostly involving data types, are applied and can be reviewed in our source code.

#### COMPAS

The COMPAS dataset (Correctional Offender Management Profiling for Alternative Sanctions) [] is available at
[https://github.com/propublica/compas-analysis/blob/master/compas-scores-two-years.csv](https://github.com/propublica/compas-analysis/blob/master/compas-scores-two-years.csv). Detailed description and information on the dataset can be found at
[https://www.propublica.org/article/how-we-analyzed-the-compas-recidivism-algorithm](https://www.propublica.org/article/how-we-analyzed-the-compas-recidivism-algorithm). It categorizes recidivism risk based on several factors, including race.

For the preprocessing of this dataset, we drop the “days_b_screening_arrest” feature, as it contains missing values. We also turn jail-in and jail-out dates to durations and turn negative durations to 0. Some additional filters are taken from the [COMPAS analysis by ProPublica](https://github.com/propublica/compas-analysis). Finally, the target variable’s values are transformed into the canonical 0 for the negative class and 1 for the positive class.

#### German Credit

The German Credit dataset [] classifies people described by a set of attributes as good or bad credit risks. A detailed description and the dataset can be found in [https://archive.ics.uci.edu/ml/datasets/statlog+(german+credit+data)](https://archive.ics.uci.edu/ml/datasets/statlog+(german+credit+data)).

The only preprocessing step we performed for this dataset was the transformation of the target variable’s values into 0 - 1.

#### Default Credit

The Default Credit dataset [] is designed to classify the risk of default on customer payments, aiming to support the development and assessment of models for predicting creditworthiness and the likelihood of loan default. It can be obtained at [https://archive.ics.uci.edu/ml/datasets/default+of+credit+card+clients](https://archive.ics.uci.edu/ml/datasets/default+of+credit+card+clients).

To properly work with this dataset, we needed to drop the “ID” feature, since it holds no useful information, and transform the target labels into the canonical 0 - 1 values.

#### HELOC

The HELOC (Home Equity Line of Credit) dataset [] contains anonymized information about home equity line of credit applications made by real homeowners, classifying credit risk. It is available at [https://community.fico.com/s/explainable-machine-learning-challenge](https://community.fico.com/s/explainable-machine-learning-challenge). All the features on this dataset are numeric.

A substantial percentage of these features’ values are missing, so the main preprocessing step we performed here was to remove rows where all values are missing, and then replace all remaining missing values with the median of the respective feature. Other than that, we only needed to transform target labels to 0-1.

**Table 4: Summary of the datasets used in our experiments. Specifically, we list the number of instances, input dimensions (i.e., the number of continuous features plus the number of categorical after the one-hot-encoding preprocessing step performed internally by the models), the number of categorical and continuous features, and the number of instances used for training and testing, evaluated using a 5-fold cross-validation strategy.**
| Dataset | No. Instances | Input Dim. | Categorical | Continuous | Train | Test |
| --- | --- | --- | --- | --- | --- | --- |
| Adult | 30161 | 102 | 8 | 5 | 24128 | 6033 |
| COMPAS | 6172 | 15 | 4 | 2 | 4937 | 1235 |
| Default Credit | 30000 | 91 | 9 | 14 | 24000 | 6000 |
| German Credit | 1000 | 71 | 17 | 3 | 800 | 200 |
| HELOC | 9871 | 23 | 0 | 23 | 7896 | 1975 |

## Appendix C Models and hyperparameters

In our experimental evaluation, we utilize three distinct models: XGBoost (XGB), Logistic Regression (LR), and Deep Neural Networks (DNNs). Following Ley et al. [], we maintain an 80:20 train-test split, but, instead of splitting once, we perform 5-fold cross-validation, using each fold as a test set while training on the remaining four. The distinctive hyperparameters for each model are described in detail in this section. Additionally, we showcase the performance metric (accuracy), with all reported accuracies presented as the mean and standard deviation across the folds, providing a standardized foundation for comparative analysis of our methodologies.

#### XGBoost (XGB)

Implementation from the common xgboost(^1^11[https://xgboost.readthedocs.io/en/stable/](https://xgboost.readthedocs.io/en/stable/)) library. Hyperparameter values for each dataset and the model’s accuracy on the test set are shown in table 5.

**Table 5: XGBoost Hyperparameter Configurations.**
| Dataset | Depth | Estimators | $\gamma,\alpha,\lambda$ | Test Accuracy |
| --- | --- | --- | --- | --- |
| Adult | 6 | 100 | 0,0,1 | 86.62% $\pm$ 0.18% |
| COMPAS | 4 | 100 | 1,0,1 | 67.95% $\pm$ 1.73% |
| Default Credit | 10 | 200 | 2,4,1 | 81.35% $\pm$ 0.18% |
| German Credit | 6 | 500 | 0,0,1 | 76.9% $\pm$ 0.86% |
| HELOC | 6 | 100 | 4,4,1 | 72.97% $\pm$ 1.14% |

#### Logistic Regression (LR)

Implementation from the common sklearn(^2^22[https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LogisticRegression.html](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LogisticRegression.html)) library. Hyperparameter values for each dataset and the model’s accuracy on the test set are shown in table 6.

**Table 6: Logistic Regression Hyperparameter Configurations.**
| Dataset | Max Iter. | Class Weights(0:1) | Test Accuracy |
| --- | --- | --- | --- |
| Adult | 100 | 1:1 | 79.09% $\pm$ 0.12% |
| COMPAS | 1000 | 1:1 | 66.69% $\pm$ 0.97% |
| Default Credit | 1000 | 1:1 | 82.08% $\pm$ 0.19% |
| German Credit | 1000 | 1:1 | 74.80% $\pm$ 1.91% |
| HELOC | 3000 | 1:1 | 73.06% $\pm$ 0.96% |

#### Deep Neural Network (DNN)

Implementation using pytorch(^3^33[https://pytorch.org/docs/stable/index.html](https://pytorch.org/docs/stable/index.html)) library. Hyperparameter values for each dataset and the model’s accuracy on the test set are shown in table 7.

**Table 7: Deep Neural Network Hyperparameter Configurations.**
| Dataset | Hidden Layers Width | Test Accuracy |
| --- | --- | --- |
| Adult | [64, 32, 16] | 84.13% $\pm$ 0.30% |
| COMPAS | [64, 32, 16] | 66.98% $\pm$ 1.27% |
| Default Credit | [64, 32, 16, 8] | 81.37% $\pm$ 0.13% |
| German Credit | [8, 4] | 71.3% $\pm$ 2.27% |
| HELOC | [64, 32, 16, 8] | 70.97% $\pm$ 0.54% |

## Appendix D Competing Methods

GLOBE-CE. The GLOBE-CE method, introduced by Ley et al. [] was executed based on the implementation available on their GitHub ([https://github.com/danwley/GLOBE-CE](https://github.com/danwley/GLOBE-CE)).

**Table 8: GLOBE-CE results without constraining the action set size**
| Models | Datasets |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | Adult | COMPAS | Default Credit | German Credit | HELOC |  |  |  |  |  |  |  |  |  |  |
|  | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ |
| dnn | 100.0 ± 0.0 | 1.24 ± 0.06 | 322.4 | 100.0 ± 0.0 | 1.11 ± 0.6 | 186.2 | 98.9 ± 0.73 | 1.1 ± 0.04 | 18.4 | 95.69 ± 2.42 | 1.12 ± 0.09 | 4.2 | 84.32 ± 19.32 | 2.63 ± 1.03 | 224.8 |
| lr | 100.0 ± 0.0 | 1.2± 0.03 | 545.2 | 98.92 ± 8.52 | 1.41 ± 0.19 | 97.8 | 99.94 ± 0.07 | 1.08 ± 0.01 | 4.6 | 71.29 ± 14.27 | 1.05 ± 0.09 | 1.5 | 100.00 ± 0.0 | 0.44 ± 0.06 | 512.8 |
| xgb | 93.76 ± 0.91 | 4.28 ± 4.46 | 46.8 | 92.34 ± 8.87 | 1.22 ± 0.27 | 42.6 | 94.62 ± 2.85 | 1.05 ± 0.28 | 40.6 | 84.97 ± 10.46 | 1.17 ± 0.15 | 2.2 | 30.32 ± 4.07 | 2.11 ± 0.47 | 17 |

Additional results of GLOBE-CE can be found in Table 8. Here we do not constrain the set of actions and we observe better performance, especially in terms of cost. However, these solutions do not represent solutions for $s$-GCE problem, since the solution size can be up to an average of 545.2 actions using 5-fold cross-validation (in Adult-LR case).

dGLOBE-CE. The dGLOBE-CE method, introduced by [], was executed based on the implementation available on their GitHub repository ([https://github.com/danwley/GLOBE-CE](https://github.com/danwley/GLOBE-CE)).

AReS. The AReS method, first introduced by [], was executed using the enhanced Fast AReS version implemented by []. The code for this version was also obtained from the same GitHub repository ([https://github.com/danwley/GLOBE-CE](https://github.com/danwley/GLOBE-CE)).

GroupCF. The GroupCF method, introduced by [], was implemented using custom code due to some ambiguity in the instructions provided in the original GitHub repository ([https://github.com/e-delaney/group_cfe](https://github.com/e-delaney/group_cfe)), which made the original implementation challenging to follow.

CET. The CET method, introduced by [], was executed using the original implementation available on GitHub ([https://github.com/kelicht/cet](https://github.com/kelicht/cet)). We had to replace the solver they used for the MIP solved at each node of their tree. IBM CPLEX solver is proprietary and we replaced it with the GUROBI solver, for which we had an available license.

## Appendix E Candidate Counterfactual Action Generators

In this section, we outline the methods used to generate candidate counterfactual explanations and present comparative results employing different generation methods. To demonstrate the modularity of our framework, we developed and utilized various methods, showing that our approach is not tied to a specific counterfactual generation technique. The comparative results of these experiments are shown in Tables 13 and 14.
For the results in Table 1 of the main paper and Table 10, DiCE [] was used to generate candidate counterfactuals.

### E.1 DiCE

Detailed information on its methodology can be found in the corresponding paper by Mothilal et al. [], and we encourage interested readers to refer to it for a deeper understanding.

For the scope of this paper, it suffices to say that DiCE offers a number of alternative algorithms for finding local counterfactuals. In our experiments, we use the ‘random’ algorithm, which selects features to change at random and tests whether the change results in a successful counterfactual.

The other algorithms were either unsuitable for a black-box setting (e.g., requiring gradients), computationally inefficient (e.g., genetic algorithm), or failed to generate cost-efficient counterfactual (e.g., KDTree) during our initial experimentation.

### E.2 Random Sampling

We developed a method called ‘Random Sampling’.
To find counterfactuals for an affected instance, this method iteratively modifies its features one at a time. The process begins by randomly altering one feature at a time, generating multiple new candidate instances. Each modified instance is then evaluated by querying the black-box model to determine whether it qualifies as a valid counterfactual. The method proceeds to modify additional features, building a set of potential counterfactuals.

Key differences between our method and DiCE’s random approach include:

- 1.
We focus on modifying only the top $k_{f}$ most important features (using permutation feature importance), set to 3 in all experiments.
- 2.
For categorical features, only the top $k_{c}$ most frequent categories among unaffected individuals are considered replacement candidates (set to 10 for all experiments).
- 3.
We also introduce vectorization in certain operations, improving computational efficiency over DiCE’s implementation.

### E.3 Nearest Neighbors

This method is implemented by storing all unaffected individuals in memory. When queried to provide $k$ counterfactuals for an affected individual, it retrieves the $k$ nearest neighbors from the set of unaffected instances based on their proximity to the affected individual. This approach ensures that the generated counterfactuals are valid and closely aligned with the original instance, improving the relevance of the recommendations.

### E.4 Nearest Neighbors Scaled

This algorithm closely resembles the ‘Nearest Neighbors’ one but introduces a key enhancement. Rather than returning the $k$ nearest neighbors, it performs a localized search along the multidimensional line connecting the affected individual and any neighbor. We sample points along this line segment, and if any are classified as positive by the model, they are returned instead, as they are closer to the affected instance, potentially offering a more cost-efficient recourse.

## Appendix F Experimental Results

### F.1 Experimental Procedure

We evaluate our algorithms by comparing them with the following methods: Fast AReS by Ley et al. [], CET by Kanamori et al. [], GroupCF by Warren et al. [], GLOBE-CE and dGLOBE-CE by Ley et al. [].

All methods are constrained to solve the $s$-GCE problem, with $s=4$ and $s=8$.
These constraints are applied differently depending on the method. In C-GLANCE and GroupCF, we configure the resulting number clusters to match the target sizes.
GLOBE-CE is constrained by a maximum of 4 or 8 scalars, while dGLOBE-CE is limited to 4 or 8 directions, with up to two scalars per direction. Lastly, T-GLANCE and CET are constrained with a maximum of 4 or 8 leaves.

In the main paper, we present summarized results for $s=4$ in Table 1, with complementary results for $s=8$ in Table 10 of the following section. The summarized results include the effectiveness and cost of each solution for all the dataset and model combinations. In Tables 15, 19, 16, 18 and 17, we provide detailed results for each model, adding runtime and the number of actions per dataset and method alongside effectiveness and cost.

### F.2 Additional Results for s = 4

Figure: Figure 3: Comparison of effectiveness and average cost, normalized with the maximum cost achieved in each dataset/model combination) for the solution of $s$-GCE with $s=4$. Standard deviations are represented by error bars. The red horizontal lines represent the $\operatorname{\mathsf{eff}}>80\%$ threshold for evaluating the practicality of the solutions.
Refer to caption: x6.png

We conducted additional experiments to evaluate whether C-GLANCE could dominate the lower-effectiveness solutions provided by other algorithms when its effectiveness is intentionally reduced (provided as threshold). The experimental results confirm, that our C-GLANCE can outperform competitors under varying effectiveness thresholds. These can be found in Table 9.

**Table 9: Experiments when C-GLANCE effectiveness is intentionally reduced (provided as threshold) to find dominating solutions toward the lower-effectiveness solutions of other methods.**
| Models | Csompetitors | Datasets |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  | Adult | Default Credit | HELOC |  |  |  |  |  |  |  |  |  |
|  |  | competitor $\operatorname{\mathsf{eff}}$ | competitor $\operatorname{\mathsf{cost}}$ | C-GLANCE $\operatorname{\mathsf{eff}}$ | C-GLANCE $\operatorname{\mathsf{cost}}$ | competitor $\operatorname{\mathsf{eff}}$ | competitor $\operatorname{\mathsf{cost}}$ | C-GLANCE $\operatorname{\mathsf{eff}}$ | C-GLANCE $\operatorname{\mathsf{cost}}$ | competitor $\operatorname{\mathsf{eff}}$ | competitor $\operatorname{\mathsf{cost}}$ | C-GLANCE $\operatorname{\mathsf{eff}}$ | C-GLANCE $\operatorname{\mathsf{cost}}$ |
| dnn | Fast AReS | 12.39 ± 1.06 | 1.0 ± 0.0 | 26.59 ± 10.5 | 0.66 ± 0.23 | 18.88 ± 2.16 | 1.0 ± 0.0 | 21.54 ± 4.11 | 0.3 ± 0.1 | 12.19 ± 0.58 | 1.03 ± 0.05 | 21.32 ± 5.13 | 0.11 ± 0.11 |
|  | CET | NaN | NaN | NaN | NaN | 98.87 ± 0.62 | 6.32 ± 2.28 | 99.83 ± 0.1 | 0.82 ± 0.04 | 86.78 ± 10.62 | 8.67 ± 3.25 | 92.67 ± 4.69 | 3.52 ± 1.09 |
|  | GroupCF | 100.0 ± 0.0 | 10.08 ± 0.03 | 100.0 ± 0.0 | 1.73 ± 0.06 | 79.6 ± 20.79 | 1.53 ± 0.62 | 81.44 ± 2.18 | 0.8 ± 0.08 | 80.4 ± 10.17 | 3.09 ± 0.91 | 82.31 ± 1.81 | 3.09 ± 0.9 |
|  | GLOBE-CE | 99.92 ± 0.0 | 4.24 ± 0.42 | 99.96 ± 0.01 | 1.71 ± 0.05 | 76.94 ± 37.55 | 5.14 ± 0.35 | 81.44 ± 2.18 | 0.8 ± 0.08 | 42.72 ± 46.97 | 11.77 ± 15.87 | 45.49 ± 2.1 | 1.19 ± 0.71 |
|  | dGLOBE-CE | 99.92 ± 0.0 | 10.89 ± 1.37 | 99.96 ± 0.01 | 1.71 ± 0.05 | 87.38 ± 18.69 | 5.96 ± 4.14 | 91.22 ± 3.87 | 0.81 ± 0.07 | 99.96 ± 0.05 | 11.07 ± 8.6 | 100.0 ± 0.0 | 4.22 ± 1.10 |
| lr | Fast AReS | 11.74 ± 2.4 | 1.0 ± 0.0 | 57.91 ± 24.77 | 0.63 ± 0.26 | 10.85 ± 5.45 | 1.07 ± 0.13 | 65.14 ± 15.62 | 0.55 ± 0.11 | 9.23 ± 1.24 | 1.12 ± 0.1 | 71.55 ± 3.72 | 0.43 ± 0.04 |
|  | CET | NaN | NaN | NaN | NaN | 100.0 ± 0.0 | 3.79 ± 1.31 | 100.0 ± 0.0 | 0.73 ± 0.07 | 100.0 ± 0.0 | 3.57 ± 1.48 | 100.0 ± 0.0 | 0.72 ± 0.11 |
|  | GroupCF | 100.0 ± 0.0 | 1.71 ± 0.39 | 100.0 ± 0.0 | 0.78 ± 0.06 | 95.4 ± 9.2 | 1.94 ± 1.2 | 99.05 ± 0.38 | 0.71 ± 0.07 | 90.6 ± 3.93 | 2.4 ± 1.38 | 95.18 ± 1.75 | 0.6 ± 0.08 |
|  | GLOBE-CE | 99.92 ± 0.0 | 2.68 ± 0.17 | 99.96 ± 0.02 | 0.78 ± 0.06 | 99.94 ± 0.07 | 3.42 ± 1.99 | 100.0 ± 0.0 | 0.73 ± 0.07 | 99.9 ± 0.0 | 0.6 ± 0.54 | 99.92 ± 0.04 | 0.7 ± 0.11 |
|  | dGLOBE-CE | 99.92 ± 0.0 | 5.91 ± 0.93 | 99.96 ± 0.02 | 0.78 ± 0.06 | 99.94 ± 0.07 | 10.38 ± 7.76 | 100.0 ± 0.0 | 0.73 ± 0.07 | 99.9 ± 0.0 | 1.63 ± 0.35 | 99.92 ± 0.04 | 0.7 ± 0.11 |
| xgb | Fast AReS | 6.13 ± 0.42 | 1.0 ± 0.0 | 31.16 ± 8.02 | 0.66 ± 0.13 | 31.86 ± 5.12 | 1.05 ± 0.04 | 40.03 ± 7.82 | 0.13 ± 0.04 | 8.49 ± 1.32 | 1.16 ± 0.13 | 9.66 ± 1.12 | 0.4 ± 0.05 |
|  | CET | NaN | NaN | NaN | NaN | 86.29 ± 9.94 | 4.5 ± 2.64 | 89.31 ± 2.77 | 0.73 ± 0.06 | 86.78 ± 6.7 | 12.51 ± 2.75 | 90.72 ± 3.35 | 11.53 ± 2.94 |
|  | GroupCF | 96.8 ± 1.72 | 1.41 ± 0.54 | 97.16 ± 0.41 | 0.81 ± 0.02 | 95.2 ± 1.6 | 1.41 ± 0.64 | 95.73 ± 0.18 | 1.01 ± 0.05 | 78.4 ± 5.82 | 5.63 ± 1.93 | 82.46 ± 6.55 | 8.45 ± 3.1 |
|  | GLOBE-CE | 82.87 ± 12.14 | 30.1 ± 10.39 | 94.04 ± 0.46 | 0.77 ± 0.01 | 82.7 ± 7.26 | 20.82 ± 1.73 | 83.77 ± 0.84 | 0.63 ± 0.08 | 27.66 ± 5.06 | 12.52 ± 32.48 | 28.64 ± 0.99 | 0.85 ± 0.19 |
|  | dGLOBE-CE | 93.76 ± 1.98 | 64.76 ± 1.29 | 94.04 ± 0.46 | 0.77 ± 0.01 | 97.47 ± 0.82 | 42.58 ± 3.57 | 97.64 ± 0.12 | 1.37 ± 0.25 | 77.64 ± 11.51 | 128.0 ± 0.0 | 82.46 ± 6.55 | 8.45 ± 3.1 |

### F.3 Results for s = 8

Table 10 presents the comparison of our methods, C-GLANCE and T-GLANCE, with other state-of-the-art methods for a maximum of 8 actions. The results indicate that C-GLANCE and T-GLANCE demonstrate superior performance in most cases, achieving nearly 100% effectiveness with minimal and stable costs across all datasets and models. Exceptions are noted in the HELOC DNN and XGB experiments, where all methods report higher costs due to the dataset’s numeric-only features. In contrast, Fast AReS struggles significantly, particularly on the Adult dataset, showing effectiveness as low as 13.8% for DNN, 12.51% for LR, and 7.19% for XGB, explaining its lower costs.
Other methods, such as GroupCF, CET, GLOBE-CE, and dGLOBE-CE, generally perform well in terms of effectiveness but in almost all cases incur higher costs compared to C-GLANCE and T-GLANCE.

**Table 10: Evaluating the effectiveness and cost of C-GLANCE and T-GLANCE against Fast AReS, CET, Group-CF, GLOBE-CE, and dGLOBE-CE when $\operatorname{\mathsf{size}}({\mathbb{C}})\leq 8$. GCE solutions with effectiveness below 80% are highlighted in purple. Non-robust GCEs, identified by either a standard deviation (std) in effectiveness greater than 5% across folds or a std in cost greater than half the average cost, are highlighted in red.**
| Models | Algorithms | Datasets |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  | Adult | COMPAS | Default Credit | German Credit | HELOC |  |  |  |  |  |
|  |  | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ |
| dnn | Fast AReS | 13.8 ± 0.92 | 1.0 ± 0.0 | 63.02 ± 1.58 | 1.11 ± 0.04 | 23.75 ± 1.81 | 1.02 ± 0.03 | 65.47 ± 2.2 | 1.0 ± 0.0 | 15.44 ± 1.91 | 1.05 ± 0.06 |
|  | CET | nan ± nan | nan ± nan | 74.66 ± 5.33 | 1.02 ± 0.07 | 98.53 ± 2.11 | 3.68 ± 1.46 | 100.0 ± 0.0 | 1.27 ± 0.37 | 99.36 ± 0.92 | 8.86 ± 2.99 |
|  | GroupCF | 100.0 ± 0.0 | 11.54 ± 2.98 | 100.0 ± 0.0 | 5.65 ± 3.14 | 84.8 ± 13.23 | 2.15 ± 0.98 | 99.6 ± 0.8 | 2.49 ± 0.82 | 84.8 ± 12.32 | 1.98 ± 0.5 |
|  | GLOBE-CE | 100.0 ± 0.0 | 2.54 ± 0.14 | 100.0 ± 0.0 | 4.31 ± 2.84 | 93.25 ± 11.26 | 2.73 ± 0.85 | 95.37 ± 2.9 | 1.89 ± 0.38 | 45.49 ± 44.72 | 14.19 ± 14.14 |
|  | dGLOBE-CE | 100.0 ± 0.0 | 11.85 ± 1.53 | 100.0 ± 0.0 | 7.28 ± 1.25 | 99.95 ± 0.04 | 6.77 ± 3.67 | 98.09 ± 1.72 | 2.34 ± 0.17 | 100.0 ± 0.0 | 6.65 ± 0.15 |
|  | C-GLANCE | 100.0 ± 0.0 | 3.96 ± 0.56 | 100.0 ± 0.0 | 1.77 ± 0.16 | 100.0 ± 0.0 | 1.2 ± 0.4 | 96.43 ± 3.6 | 0.86 ± 0.18 | 99.98 ± 0.04 | 9.42 ± 2.22 |
|  | T-GLANCE | 100.0 ± 0.0 | 3.77 ± 0.27 | 99.65 ± 0.15 | 2.18 ± 0.15 | 100.0 ± 0.0 | 1.25 ± 0.38 | 96.65 ± 4.08 | 0.94 ± 0.3 | 99.78 ± 0.21 | 9.92 ± 1.83 |
| lr | Fast AReS | 12.51 ± 2.44 | 1.0 ± 0.0 | 68.19 ± 0.6 | 1.03 ± 0.01 | 11.67 ± 6.18 | 1.04 ± 0.08 | 71.9 ± 2.65 | 1.0 ± 0.0 | 10.95 ± 1.73 | 1.11 ± 0.07 |
|  | CET | nan ± nan | nan ± nan | 82.51 ± 2.97 | 1.51 ± 0.23 | 100.0 ± 0.0 | 2.0 ± 1.36 | 98.75 ± 1.67 | 2.02 ± 0.33 | 100.0 ± 0.0 | 2.99 ± 1.3 |
|  | GroupCF | 100.0 ± 0.0 | 2.33 ± 1.91 | 100.0 ± 0.0 | 4.67 ± 2.29 | 100.0 ± 0.0 | 3.98 ± 2.86 | 98.9 ± 1.65 | 10.56 ± 2.43 | 96.6 ± 1.96 | 1.98 ± 0.5 |
|  | GLOBE-CE | 100.0 ± 0.0 | 1.67 ± 0.11 | 97.74 ± 8.52 | 5.22 ± 3.3 | 99.94 ± 0.07 | 2.24 ± 0.98 | 58.42 ± 18.45 | 2.25 ± 0.33 | 99.9 ± 0.0 | 0.58 ± 0.5 |
|  | dGLOBE-CE | 100.0 ± 0.0 | 6.12 ± 0.64 | 100.0 ± 0.0 | 6.71 ± 0.23 | 99.94 ± 0.07 | 10.38 ± 7.76 | 69.89 ± 15.35 | 2.47 ± 0.23 | 99.9 ± 0.0 | 1.63 ± 0.35 |
|  | C-GLANCE | 100.0 ± 0.0 | 1.03 ± 0.07 | 100.0 ± 0.0 | 1.69 ± 0.06 | 100.0 ± 0.0 | 1.05 ± 0.11 | 100.0 ± 0.0 | 1.18 ± 0.05 | 100.0 ± 0.0 | 1.2 ± 0.12 |
|  | T-GLANCE | 100.0 ± 0.0 | 0.79 ± 0.16 | 99.71 ± 0.16 | 1.94 ± 0.11 | 100.0 ± 0.0 | 1.13 ± 0.18 | 99.55 ± 0.91 | 1.34 ± 0.22 | 100.0 ± 0.0 | 1.52 ± 0.32 |
| xgb | Fast AReS | 7.19 ± 0.3 | 1.0 ± 0.0 | 66.07 ± 2.5 | 1.14 ± 0.1 | 36.88 ± 5.46 | 1.07 ± 0.06 | 68.88 ± 1.27 | 1.0 ± 0.0 | 11.22 ± 2.05 | 1.09 ± 0.09 |
|  | CET | nan ± nan | nan ± nan | 68.62 ± 15.67 | 1.3 ± 0.52 | 90.01 ± 10.44 | 2.61 ± 1.06 | 100.0 ± 0.0 | 2.18 ± 0.37 | 90.25 ± 2.51 | 12.99 ± 3.49 |
|  | GroupCF | 99.4 ± 1.2 | 1.88 ± 1.36 | 100.0 ± 0.0 | 4.08 ± 2.15 | 97.8 ± 0.75 | 1.92 ± 1.33 | 100.0 ± 0.0 | 4.05 ± 2.55 | 81.4 ± 4.22 | 5.14 ± 1.46 |
|  | GLOBE-CE | 85.76 ± 9.11 | 13.48 ± 4.71 | 89.17 ± 11.09 | 9.78 ± 6.35 | 84.94 ± 6.59 | 11.21 ± 1.29 | 80.25 ± 9.62 | 2.51 ± 0.33 | 28.88 ± 4.71 | 12.51 ± 30.88 |
|  | dGLOBE-CE | 94.38 ± 2.01 | 62.37 ± 3.07 | 99.96 ± 0.08 | 9.17 ± 2.01 | 99.08 ± 0.66 | 28.73 ± 9.8 | 88.17 ± 8.2 | 2.64 ± 0.78 | 81.95 ± 10.38 | 128.0 ± 0.0 |
|  | C-GLANCE | 99.87 ± 0.09 | 3.85 ± 2.33 | 99.83 ± 0.24 | 2.06 ± 0.46 | 98.87 ± 0.59 | 2.14 ± 0.18 | 100.0 ± 0.0 | 1.02 ± 0.04 | 99.0 ± 0.55 | 16.65 ± 2.44 |
|  | T-GLANCE | 99.94 ± 0.04 | 1.65 ± 0.39 | 99.39 ± 0.26 | 2.41 ± 0.4 | 94.1 ± 3.57 | 1.85 ± 0.16 | 99.46 ± 1.08 | 1.07 ± 0.16 | 98.25 ± 1.85 | 18.75 ± 4.92 |

Figure: Figure 4: Comparison of effectiveness and average cost, normalized with the maximum cost achieved in each dataset/model combination) for the solution of $s$-GCE with $s=8$. Standard deviations are represented by error bars. The red horizontal lines represent the $\operatorname{\mathsf{eff}}>80\%$ threshold for evaluating the practicality of the solution.
Refer to caption: x7.png

**Table 11: Pareto domination evaluation of solutions, for $s$-GCE problem with $s=8$. This table reports the rate (number of times over available comparisons) at which GLANCE methods dominate competitors (listed in the dominates column) and the rate at which GLANCE methods are dominated by competitors (listed in the is dominated column).**
|  | C-CLANCE | T-CLANCE |  |  |
| --- | --- | --- | --- | --- |
|  | Dominates | is Dominated | Dominates | is Dominated |
| Fast AReS | 1/15 | 0/15 | 2/15 | 0/15 |
| CET | 6/12 | 0/12 | 5/12 | 0/12 |
| GroupCF | 9/15 | 0/15 | 7/15 | 0/15 |
| GLOBE-CE | 12/15 | 1/15 | 11/15 | 1/15 |
| dGLOBE-CE | 11/15 | 1/15 | 9/15 | 1/15 |
| competitors | 39/72 | 2/72 | 34/72 | 2/72 |

**Table 12: Pareto domination evaluation of solutions, for $s$-GCE problem with $s=8$, after applying the $\operatorname{\mathsf{eff}}>80\%$ threshold. This table reports the rate (number of times over available comparisons) at which GLANCE methods dominate competitors (listed in the dominates column) and the rate at which GLANCE methods are dominated by competitors (listed in the is dominated column).**
|  | C-CLANCE | T-CLANCE |  |  |
| --- | --- | --- | --- | --- |
|  | Dominates | is Dominated | Dominates | is Dominated |
| Fast AReS | 0/0 | 0/0 | 0/0 | 0/0 |
| CET | 6/10 | 0/10 | 5/10 | 0/10 |
| GroupCF | 9/15 | 0/15 | 7/15 | 0/15 |
| GLOBE-CE | 10/12 | 1/12 | 9/12 | 1/12 |
| dGLOBE-CE | 10/14 | 1/14 | 8/14 | 1/14 |
| competitors | 35/51 | 2/51 | 29/51 | 2/51 |

We repeat the experimental evaluation utilizing Pareto-dominance, now for $s=8$.
As shown in Table 11 C-GLANCE dominates other methods in 39 out of 72 cases (54%) while is dominated once by GLOBE-CE (in Adult-DNN—cf. Table 10) and once by dGLOBE-CE (in Heloc-DNN). T-GLANCE dominates other methods in 34 out of 72 comparisons, and is dominated one by Group-CF (in Default Credit-XGB) and once by CET (in German Credit-DNN). Overall, the GLANCE methods dominate other solutions in almost half of the cases (48%) and are dominated in only 3%.

Table 12 summarizes the head-to-head comparisons only when considering practical solutions. GLANCE framework dominates other methods in a much larger percentage—over 62% of the cases. It remains dominated in the same four cases, maintaining the percentage of dominated cases at a low 4%.

### F.4 Comperative Evaluation with different Counterfactual Generation Methods

**Table 13: Evaluation of the effectiveness and cost of C-GLANCE and T-GLANCE, utilizing different candidate action generation methods for $s$-GCE when $s=4$. The best metrics are shown in bold.**
| Models | Algorithms | Datasets |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  | Adult | COMPAS | Default Credit | German Credit | HELOC |  |  |  |  |  |
|  |  | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ |
| dnn | C-GLANCE - Dice | 100.0 ± 0.0 | 4.6 ± 0.73 | 100.0 ± 0.0 | 2.34 ± 0.43 | 100.0 ± 0.0 | 1.2 ± 0.4 | 99.46 ± 1.08 | 1.22 ± 0.41 | 99.94 ± 0.05 | 11.24 ± 1.37 |
|  | C-GLANCE - NearestNeighbors | 100.0 ± 0.0 | 6.04 ± 0.35 | 100.0 ± 0.0 | 2.27 ± 0.31 | 100.0 ± 0.0 | 4.07 ± 0.08 | 100.0 ± 0.0 | 4.9 ± 0.69 | 99.82 ± 0.15 | 20.18 ± 3.28 |
|  | C-GLANCE - NearestNeighborsScaled | 99.99 ± 0.03 | 5.04 ± 0.51 | 100.0 ± 0.0 | 1.74 ± 0.38 | 100.0 ± 0.0 | 3.02 ± 0.58 | 100.0 ± 0.0 | 2.1 ± 0.4 | 99.79 ± 0.3 | 18.88 ± 2.11 |
|  | C-GLANCE - RandomSampling | 98.01 ± 0.66 | 2.03 ± 0.01 | 100.0 ± 0.0 | 2.08 ± 0.21 | 100.0 ± 0.0 | 1.0 ± 0.0 | 99.43 ± 1.14 | 1.0 ± 0.2 | 94.02 ± 4.28 | 2.53 ± 1.12 |
|  | T-GLANCE - Dice | 100.0 ± 0.0 | 4.43 ± 0.43 | 99.53 ± 0.22 | 2.62 ± 0.24 | 100.0 ± 0.0 | 1.48 ± 0.43 | 96.97 ± 3.48 | 1.59 ± 0.8 | 99.8 ± 0.24 | 10.9 ± 1.22 |
|  | T-GLANCE - NearestNeighbors | 99.96 ± 0.08 | 9.4 ± 1.65 | 99.52 ± 0.29 | 2.71 ± 0.19 | 99.95 ± 0.06 | 4.13 ± 0.38 | 100.0 ± 0.0 | 8.54 ± 1.9 | 93.49 ± 2.03 | 17.71 ± 2.06 |
|  | T-GLANCE - NearestNeighborsScaled | 99.97 ± 0.04 | 5.92 ± 1.06 | 99.88 ± 0.16 | 2.47 ± 0.35 | 99.73 ± 0.37 | 3.22 ± 0.32 | 100.0 ± 0.0 | 2.35 ± 0.44 | 95.32 ± 1.19 | 18.97 ± 2.08 |
|  | T-GLANCE - RandomSampling | 97.57 ± 0.36 | 2.91 ± 0.05 | 99.65 ± 0.08 | 2.28 ± 0.4 | 99.97 ± 0.05 | 1.0 ± 0.0 | 99.53 ± 0.93 | 1.06 ± 0.46 | 95.9 ± 1.85 | 6.45 ± 1.31 |
| lr | C-GLANCE - Dice | 100.0 ± 0.0 | 1.04 ± 0.07 | 100.0 ± 0.0 | 2.33 ± 0.38 | 100.0 ± 0.0 | 1.05 ± 0.11 | 100.0 ± 0.0 | 1.21 ± 0.06 | 100.0 ± 0.0 | 1.55 ± 0.54 |
|  | C-GLANCE - NearestNeighbors | 100.0 ± 0.0 | 3.76 ± 0.96 | 100.0 ± 0.0 | 2.53 ± 0.32 | 100.0 ± 0.0 | 1.78 ± 0.39 | 100.0 ± 0.0 | 4.21 ± 0.56 | 99.94 ± 0.05 | 21.52 ± 1.33 |
|  | C-GLANCE - NearestNeighborsScaled | 100.0 ± 0.0 | 3.19 ± 0.51 | 100.0 ± 0.0 | 2.14 ± 0.15 | 100.0 ± 0.0 | 1.11 ± 0.14 | 100.0 ± 0.0 | 2.02 ± 0.27 | 99.94 ± 0.12 | 20.95 ± 3.47 |
|  | C-GLANCE - RandomSampling | 100.0 ± 0.0 | 0.72 ± 0.12 | 100.0 ± 0.0 | 2.21 ± 0.2 | 99.94 ± 0.12 | 1.0 ± 0.0 | 100.0 ± 0.0 | 1.31 ± 0.15 | 100.0 ± 0.0 | 1.4 ± 0.27 |
|  | T-GLANCE - Dice | 100.0 ± 0.0 | 0.69 ± 0.01 | 99.79 ± 0.22 | 2.19 ± 0.1 | 100.0 ± 0.0 | 1.43 ± 0.36 | 99.58 ± 0.83 | 1.54 ± 0.32 | 100.0 ± 0.0 | 1.58 ± 0.37 |
|  | T-GLANCE - NearestNeighbors | 99.92 ± 0.04 | 3.52 ± 1.37 | 99.79 ± 0.22 | 3.51 ± 0.19 | 100.0 ± 0.0 | 2.98 ± 0.28 | 100.0 ± 0.0 | 9.66 ± 1.55 | 99.31 ± 0.92 | 19.48 ± 2.53 |
|  | T-GLANCE - NearestNeighborsScaled | 98.64 ± 1.3 | 2.98 ± 0.88 | 99.83 ± 0.15 | 3.45 ± 0.25 | 100.0 ± 0.0 | 2.1 ± 0.18 | 100.0 ± 0.0 | 2.14 ± 0.26 | 99.54 ± 0.68 | 18.15 ± 2.02 |
|  | T-GLANCE - RandomSampling | 100.0 ± 0.0 | 0.71 ± 0.12 | 99.71 ± 0.16 | 2.3 ± 0.28 | 99.97 ± 0.06 | 1.04 ± 0.07 | 98.71 ± 1.05 | 1.46 ± 0.25 | 100.0 ± 0.0 | 1.57 ± 0.66 |
| xgb | C-GLANCE - Dice | 99.85 ± 0.12 | 5.98 ± 4.22 | 99.51 ± 0.46 | 2.96 ± 0.82 | 98.13 ± 1.05 | 3.68 ± 1.64 | 100.0 ± 0.0 | 1.05 ± 0.02 | 98.94 ± 0.66 | 19.99 ± 1.91 |
|  | C-GLANCE - NearestNeighbors | 100.0 ± 0.01 | 4.53 ± 0.56 | 99.6 ± 0.34 | 2.95 ± 0.58 | 100.0 ± 0.0 | 3.8 ± 0.44 | 100.0 ± 0.0 | 3.67 ± 0.58 | 99.51 ± 0.42 | 19.3 ± 2.66 |
|  | C-GLANCE - NearestNeighborsScaled | 99.98 ± 0.03 | 4.12 ± 0.31 | 99.44 ± 0.54 | 3.0 ± 0.33 | 100.0 ± 0.0 | 3.09 ± 0.42 | 100.0 ± 0.0 | 1.75 ± 0.2 | 99.68 ± 0.13 | 20.92 ± 3.96 |
|  | C-GLANCE - RandomSampling | 99.85 ± 0.19 | 2.1 ± 0.19 | 99.88 ± 0.16 | 2.83 ± 0.76 | 91.52 ± 4.44 | 2.25 ± 1.06 | 100.0 ± 0.0 | 1.14 ± 0.07 | 82.75 ± 5.2 | 9.6 ± 1.84 |
|  | T-GLANCE - Dice | 99.86 ± 0.14 | 1.8 ± 0.51 | 99.02 ± 0.93 | 2.6 ± 0.46 | 94.6 ± 3.13 | 2.75 ± 1.42 | 99.46 ± 1.08 | 1.33 ± 0.42 | 96.42 ± 2.45 | 24.85 ± 8.4 |
|  | T-GLANCE - NearestNeighbors | 99.87 ± 0.22 | 9.44 ± 1.71 | 98.99 ± 0.54 | 3.25 ± 0.33 | 100.0 ± 0.0 | 4.35 ± 0.47 | 100.0 ± 0.0 | 7.8 ± 1.26 | 94.94 ± 1.91 | 18.4 ± 1.41 |
|  | T-GLANCE - NearestNeighborsScaled | 99.68 ± 0.25 | 9.04 ± 1.75 | 99.24 ± 0.67 | 3.25 ± 0.41 | 100.0 ± 0.0 | 3.93 ± 0.53 | 100.0 ± 0.0 | 2.35 ± 0.36 | 96.07 ± 1.85 | 20.39 ± 1.69 |
|  | T-GLANCE - RandomSampling | 99.61 ± 0.66 | 2.03 ± 0.3 | 98.67 ± 1.0 | 2.53 ± 0.34 | 87.04 ± 7.36 | 2.18 ± 1.65 | 99.33 ± 1.33 | 1.23 ± 0.21 | 79.63 ± 7.83 | 8.02 ± 1.43 |

**Table 14: Evaluation of the effectiveness and cost of C-GLANCE and T-GLANCE, utilizing different candidate action generation methods for $s$-GCE when $s=8$. The best metrics are shown in bold.**
| Models | Algorithms | Datasets |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  | Adult | COMPAS | Default Credit | German Credit | HELOC |  |  |  |  |  |
|  |  | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ |
| dnn | C-GLANCE - Dice | 100.0 ± 0.0 | 3.96 ± 0.56 | 100.0 ± 0.0 | 1.77 ± 0.16 | 100.0 ± 0.0 | 1.2 ± 0.4 | 99.46 ± 1.08 | 0.78 ± 0.27 | 99.98 ± 0.04 | 9.42 ± 2.22 |
|  | C-GLANCE - NearestNeighbors | 100.0 ± 0.01 | 4.99 ± 0.62 | 100.0 ± 0.0 | 1.71 ± 0.18 | 100.0 ± 0.0 | 3.84 ± 0.26 | 100.0 ± 0.0 | 3.94 ± 0.39 | 99.75 ± 0.26 | 16.07 ± 1.32 |
|  | C-GLANCE - NearestNeighborsScaled | 100.0 ± 0.0 | 4.44 ± 0.15 | 100.0 ± 0.0 | 1.08 ± 0.08 | 100.0 ± 0.0 | 2.5 ± 0.24 | 100.0 ± 0.0 | 1.65 ± 0.14 | 99.71 ± 0.15 | 16.17 ± 1.72 |
|  | C-GLANCE - RandomSampling | 95.36 ± 3.8 | 2.84 ± 0.8 | 100.0 ± 0.0 | 1.74 ± 0.22 | 99.82 ± 0.29 | 1.07 ± 0.14 | 100.0 ± 0.0 | 0.7 ± 0.18 | 98.55 ± 0.45 | 10.01 ± 0.53 |
|  | T-GLANCE - Dice | 100.0 ± 0.0 | 3.77 ± 0.27 | 99.65 ± 0.15 | 2.19 ± 0.15 | 100.0 ± 0.0 | 1.25 ± 0.38 | 96.65 ± 4.08 | 0.94 ± 0.3 | 99.78 ± 0.21 | 9.42 ± 1.83 |
|  | T-GLANCE - NearestNeighbors | 99.95 ± 0.08 | 7.71 ± 1.11 | 99.79 ± 0.14 | 2.53 ± 0.17 | 100.0 ± 0.0 | 4.18 ± 0.29 | 100.0 ± 0.0 | 7.54 ± 0.97 | 96.48 ± 0.9 | 19.11 ± 1.76 |
|  | T-GLANCE - NearestNeighborsScaled | 99.81 ± 0.16 | 7.56 ± 1.74 | 99.88 ± 0.1 | 2.24 ± 0.39 | 99.97 ± 0.05 | 3.47 ± 0.72 | 100.0 ± 0.0 | 2.11 ± 0.57 | 97.48 ± 0.89 | 20.98 ± 1.93 |
|  | T-GLANCE - RandomSampling | 96.76 ± 0.39 | 2.87 ± 0.06 | 99.58 ± 0.22 | 2.18 ± 0.45 | 100.0 ± 0.0 | 1.16 ± 0.31 | 100.0 ± 0.0 | 0.86 ± 0.38 | 96.76 ± 4.58 | 7.7 ± 2.89 |
| lr | C-GLANCE - Dice | 100.0 ± 0.0 | 1.03 ± 0.07 | 100.0 ± 0.0 | 1.69 ± 0.06 | 100.0 ± 0.0 | 1.05 ± 0.11 | 100.0 ± 0.0 | 1.15 ± 0.03 | 100.0 ± 0.0 | 1.2 ± 0.12 |
|  | C-GLANCE - NearestNeighbors | 100.0 ± 0.0 | 3.22 ± 0.49 | 100.0 ± 0.0 | 1.9 ± 0.12 | 100.0 ± 0.0 | 1.39 ± 0.11 | 100.0 ± 0.0 | 3.98 ± 0.55 | 99.88 ± 0.14 | 17.3 ± 1.9 |
|  | C-GLANCE - NearestNeighborsScaled | 100.0 ± 0.0 | 2.96 ± 0.62 | 100.0 ± 0.0 | 1.81 ± 0.18 | 100.0 ± 0.0 | 1.01 ± 0.0 | 100.0 ± 0.0 | 1.7 ± 0.2 | 99.88 ± 0.15 | 17.39 ± 0.69 |
|  | C-GLANCE - RandomSampling | 100.0 ± 0.0 | 0.67 ± 0.07 | 100.0 ± 0.0 | 1.78 ± 0.25 | 99.94 ± 0.12 | 1.0 ± 0.0 | 100.0 ± 0.0 | 1.19 ± 0.05 | 100.0 ± 0.0 | 1.19 ± 0.15 |
|  | T-GLANCE - Dice | 100.0 ± 0.0 | 0.69 ± 0.01 | 99.71 ± 0.16 | 1.94 ± 0.11 | 100.0 ± 0.0 | 1.13 ± 0.18 | 99.55 ± 0.91 | 1.34 ± 0.22 | 100.0 ± 0.0 | 1.52 ± 0.32 |
|  | T-GLANCE - NearestNeighbors | 99.92 ± 0.04 | 3.52 ± 1.37 | 99.79 ± 0.22 | 3.51 ± 0.19 | 100.0 ± 0.0 | 2.98 ± 0.28 | 100.0 ± 0.0 | 7.47 ± 2.24 | 99.31 ± 0.92 | 19.48 ± 2.53 |
|  | T-GLANCE - NearestNeighborsScaled | 98.64 ± 1.3 | 2.98 ± 0.88 | 99.83 ± 0.15 | 3.45 ± 0.25 | 100.0 ± 0.0 | 2.1 ± 0.18 | 100.0 ± 0.0 | 1.82 ± 0.29 | 99.54 ± 0.68 | 18.15 ± 2.02 |
|  | T-GLANCE - RandomSampling | 100.0 ± 0.0 | 0.63 ± 0.09 | 99.71 ± 0.16 | 2.46 ± 0.32 | 99.97 ± 0.06 | 1.02 ± 0.02 | 100.0 ± 0.0 | 1.27 ± 0.17 | 100.0 ± 0.0 | 1.62 ± 0.6 |
| xgb | C-GLANCE - Dice | 99.87 ± 0.09 | 3.85 ± 2.33 | 99.83 ± 0.24 | 2.06 ± 0.46 | 98.87 ± 0.59 | 2.14 ± 0.18 | 100.0 ± 0.0 | 0.97 ± 0.08 | 99.0 ± 0.55 | 16.65 ± 2.44 |
|  | C-GLANCE - NearestNeighbors | 100.0 ± 0.0 | 4.43 ± 0.55 | 99.6 ± 0.34 | 1.8 ± 0.16 | 100.0 ± 0.0 | 3.2 ± 0.45 | 100.0 ± 0.0 | 3.63 ± 0.58 | 99.49 ± 0.47 | 15.76 ± 1.71 |
|  | C-GLANCE - NearestNeighborsScaled | 99.98 ± 0.03 | 3.9 ± 0.41 | 99.48 ± 0.57 | 1.88 ± 0.09 | 100.0 ± 0.0 | 2.4 ± 0.31 | 100.0 ± 0.0 | 1.62 ± 0.13 | 99.72 ± 0.17 | 18.58 ± 1.88 |
|  | C-GLANCE - RandomSampling | 99.76 ± 0.37 | 2.13 ± 0.17 | 99.84 ± 0.24 | 2.06 ± 0.21 | 93.86 ± 2.69 | 1.51 ± 0.52 | 100.0 ± 0.0 | 1.06 ± 0.1 | 84.76 ± 4.77 | 7.11 ± 1.13 |
|  | T-GLANCE - Dice | 99.94 ± 0.04 | 1.65 ± 0.39 | 99.39 ± 0.26 | 2.41 ± 0.4 | 94.1 ± 3.57 | 1.85 ± 0.16 | 99.46 ± 1.08 | 1.07 ± 0.16 | 98.25 ± 1.85 | 18.75 ± 4.92 |
|  | T-GLANCE - NearestNeighbors | 99.87 ± 0.22 | 9.44 ± 1.71 | 98.99 ± 0.54 | 3.25 ± 0.33 | 100.0 ± 0.0 | 4.35 ± 0.47 | 100.0 ± 0.0 | 5.39 ± 1.33 | 94.94 ± 1.91 | 18.4 ± 1.41 |
|  | T-GLANCE - NearestNeighborsScaled | 99.68 ± 0.25 | 9.04 ± 1.75 | 99.24 ± 0.67 | 3.25 ± 0.41 | 100.0 ± 0.0 | 3.93 ± 0.53 | 100.0 ± 0.0 | 2.09 ± 0.28 | 96.07 ± 1.85 | 20.39 ± 1.69 |
|  | T-GLANCE - RandomSampling | 99.67 ± 0.52 | 1.7 ± 0.57 | 98.3 ± 0.83 | 2.61 ± 0.48 | 87.39 ± 5.65 | 2.55 ± 1.29 | 99.33 ± 1.33 | 1.33 ± 0.57 | 79.59 ± 7.41 | 9.37 ± 1.06 |

Tables 14 and 13 present a comprehensive comparison of C-GLANCE and T-GLANCE utilizing the counterfactual generation methods described in Appendix E, i.e., DiCE, NearestNeighbors, NearestNeighborsScaled, and RandomSampling, for the $s$-GCE problem with $s=4$ and $s=8$, respectively. Across all datasets and models, both C-GLANCE and T-GLANCE consistently exhibit superb performance in terms of effectiveness, frequently achieving near 100%.
Since the performance in terms of effectiveness is comparable, we will focus on the cost our algorithms achieved under the different generation methods.
The best performance is observed with the RandomSampling method, achieving the best results in 41 out of 60 cases. DiCE, which is regarded as state-of-the-art, follows, achieves the lowest cost in 15 out of 60 cases. NearestNeighbors and NearestNeighborsScaled achieve the best cost in only
1 out of 60 and 3 out of 60 cases respectively. In general, C-GLANCE and T-GLANCE, can demonstrate great performance under all action generation methods, maintaining high effectiveness
and low costs in almost all cases.

### F.5 Detailed results

**Table 15: Detailed results for the solution of $s$-GCE ($s=4$ and $s=8$) for COMPAS dataset. The table reports effectiveness, cost, size, and runtime, including their standard deviations for each method.**
| method / model | DNN | LR | XGB |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | $\operatorname{\mathsf{eff}}$ | Cost | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | Cost | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | Cost | $\operatorname{\mathsf{size}}$ | RunTime |
| Fast AReS-4 | 55.0 ± 0.86 | 1.21 ± 0.09 | 4 | 59.26 ± 26.71 | 62.5 ± 1.82 | 1.24 ± 0.14 | 4 | 11.52 ± 5.39 | 59.83 ± 3.12 | 1.1 ± 0.05 | 4 | 12.88 ± 5.5 |
| CET - 4 | 63.62 ± 10.35 | 0.96 ± 0.24 | 7.0 | 1339.42 ± 247.04 | 73.18 ± 4.34 | 1.24 ± 0.15 | 4.0 | 663.22 ± 45.34 | 58.4 ± 9.3 | 1.06 ± 0.24 | 3.0 | 3280.17 ± 834.28 |
| GroupCF - 4 | 100.0 ± 0.0 | 4.48 ± 2.53 | 4 | 173.24 ± 2.53 | 100.0 ± 0.0 | 3.97 ± 2.38 | 4 | 45.04 ± 2.38 | 100.0 ± 0.0 | 4.06 ± 2.1 | 4 | 36.07 ± 2.1 |
| GLOBE-CE - 4 | 100.0 ± 0.0 | 4.54 ± 3.31 | 3.0 | 1.09 ± 0.06 | 95.74 ± 8.52 | 5.14 ± 3.77 | 2.6 | 0.42 ± 0.01 | 87.13 ± 11.14 | 9.75 ± 7.2 | 1.6 | 1.06 ± 0.01 |
| dGLOBE-CE - 4 | 100.0 ± 0.0 | 7.96 ± 3.91 | 3.4 | 1.55 ± 0.07 | 100.0 ± 0.0 | 6.71 ± 0.23 | 4.0 | 0.61 ± 0.01 | 99.84 ± 0.31 | 12.46 ± 3.42 | 3.4 | 1.53 ± 0.03 |
| C-GLANCE-4 | 100.0 ± 0.0 | 2.34 ± 0.43 | 4.0 | 177.41 ± 0.0 | 100.0 ± 0.0 | 2.33 ± 0.38 | 4.0 | 82.09 ± 0.0 | 99.51 ± 0.46 | 2.96 ± 0.82 | 4.0 | 286.26 ± 0.0 |
| T-GLANCE-4 | 99.53 ± 0.22 | 2.62 ± 0.24 | 3.0 | 72.01 ± 0.0 | 99.79 ± 0.22 | 2.19 ± 0.1 | 3.0 | 50.14 ± 0.0 | 99.02 ± 0.93 | 2.6 ± 0.46 | 3.0 | 156.0 ± 0.0 |
| Fast AReS-8 | 63.02 ± 1.58 | 1.11 ± 0.04 | 8 | 65.84 ± 29.93 | 68.19 ± 0.6 | 1.03 ± 0.01 | 8 | 15.2 ± 6.9 | 66.07 ± 2.5 | 1.14 ± 0.1 | 8 | 17.69 ± 8.05 |
| CET - 8 | 74.66 ± 5.33 | 1.02 ± 0.07 | 8.0 | 656.64 ± 119.51 | 82.51 ± 2.97 | 1.51 ± 0.23 | 6.0 | 442.76 ± 64.47 | 68.62 ± 15.67 | 1.3 ± 0.52 | 6.0 | 1787.18 ± 360.63 |
| GroupCF - 8 | 100.0 ± 0.0 | 5.65 ± 3.14 | 8 | 210.34 ± 3.68 | 100.0 ± 0.0 | 4.67 ± 2.29 | 8 | 48.53 ± 2.29 | 100.0 ± 0.0 | 4.08 ± 2.15 | 8 | 37.53 ± 2.15 |
| GLOBE-CE - 8 | 100.0 ± 0.0 | 4.31 ± 2.84 | 6.8 | 1.06 ± 0.04 | 97.74 ± 8.52 | 5.22 ± 3.3 | 5.8 | 0.45 ± 0.02 | 89.17 ± 11.09 | 9.78 ± 6.35 | 3.8 | 1.08 ± 0.03 |
| dGLOBE-CE - 8 | 100.0 ± 0.0 | 7.28 ± 1.25 | 6.8 | 2.3 ± 0.09 | 100.0 ± 0.0 | 6.71 ± 0.23 | 8.0 | 0.88 ± 0.01 | 99.96 ± 0.08 | 9.17 ± 2.01 | 5.6 | 2.23 ± 0.03 |
| C-GLANCE-8 | 100.0 ± 0.0 | 1.77 ± 0.16 | 8.0 | 177.35 ± 0.0 | 100.0 ± 0.0 | 1.69 ± 0.06 | 8.0 | 84.11 ± 0.0 | 99.83 ± 0.24 | 2.06 ± 0.46 | 8.0 | 173.1 ± 0.0 |
| T-GLANCE-8 | 99.65 ± 0.15 | 2.18 ± 0.15 | 4.0 | 21.48 ± 0.64 | 99.71 ± 0.16 | 1.94 ± 0.11 | 4.0 | 13.06 ± 0.46 | 99.39 ± 0.26 | 2.41 ± 0.4 | 4.0 | 40.05 ± 4.5 |

Table 15 presents a comparative analysis of all algorithms for the COMPAS dataset, revealing distinct performance patterns. CET and Fast AReS, both with 4 and 8 actions, show very low costs across all models; however, this is largely attributable to their relatively low effectiveness scores.

Methods like GLOBE-CE, dGLOBE-CE display significantly higher effectiveness, mostly above 90% across all models, GroupCF yields perfect effectiveness of 100%. However, these methods come with higher costs, indicating that increased effectiveness is often tied to a rise in cost.

On the other hand, C-GLANCE and T-GLANCE methods consistently achieve near-perfect or perfect effectiveness ($\sim$ 100%) across all models while maintaining much lower costs.
This balance of high effectiveness and low cost makes C-GLANCE and T-GLANCE the optimal methods for the COMPAS dataset.

**Table 16: Detailed results for the solution of $s$-GCE ($s=4$ and $s=8$) for German Credit dataset. The table reports effectiveness, cost, size, and runtime, including their standard deviations for each method.**
| method / model | DNN | LR | XGB |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime |
| Fast AReS-4 | 52.39 ± 1.63 | 1.0 ± 0.0 | 4 | 119.34 ± 54.6 | 75.27 ± 2.96 | 1.0 ± 0.0 | 4 | 47.88 ± 23.06 | 51.27 ± 1.57 | 1.0 ± 0.0 | 4 | 55.01 ± 27.27 |
| CET - 4 | 97.3 ± 2.46 | 1.58 ± 0.54 | 3.0 | 230.36 ± 47.15 | 96.5 ± 2.85 | 2.42 ± 0.24 | 3.0 | 237.17 ± 11.2 | 100.0 ± 0.0 | 2.73 ± 0.49 | 3.0 | 347.41 ± 30.21 |
| GroupCF - 4 | 97.8 ± 4.4 | 1.85 ± 0.13 | 4 | 25.54 ± 0.13 | 97.6 ± 2.94 | 9.34 ± 3.85 | 4 | 7.91 ± 3.85 | 100.0 ± 0.0 | 5.78 ± 4.11 | 4 | 5.87 ± 4.11 |
| GLOBE-CE - 4 | 93.31 ± 3.48 | 2.0 ± 1.55 | 2.0 | 1.02 ± 0.01 | 57.09 ± 20.03 | 0.75 ± 1.04 | 1.0 | 0.44 ± 0.0 | 77.05 ± 11.26 | 1.14 ± 1.24 | 1.0 | 1.34 ± 0.03 |
| dGLOBE-CE - 4 | 97.36 ± 0.82 | 2.49 ± 0.27 | 4.0 | 1.57 ± 0.01 | 69.89 ± 15.35 | 2.47 ± 0.23 | 3.2 | 0.7 ± 0.01 | 86.96 ± 9.79 | 2.66 ± 0.77 | 3.6 | 2.09 ± 0.04 |
| C-GLANCE-4 | 95.31 ± 3.15 | 1.25 ± 0.33 | 4.0 | 78.45 ± 0.0 | 100.0 ± 0.0 | 1.21 ± 0.06 | 4.0 | 61.58 ± 0.0 | 100.0 ± 0.0 | 1.06 ± 0.03 | 4.0 | 67.36 ± 0.0 |
| T-GLANCE-4 | 96.97 ± 3.48 | 1.59 ± 0.8 | 3.0 | 85.12 ± 0.0 | 99.58 ± 0.83 | 1.54 ± 0.32 | 3.0 | 75.64 ± 0.0 | 99.46 ± 1.08 | 1.33 ± 0.42 | 3.0 | 185.11 ± 0.0 |
| Fast AReS-8 | 65.47 ± 2.2 | 1.0 ± 0.0 | 8 | 415.74 ± 175.82 | 71.9 ± 2.65 | 1.0 ± 0.0 | 8 | 59.8 ± 32.65 | 68.88 ± 1.27 | 1.0 ± 0.0 | 8 | 64.77 ± 29.67 |
| CET - 8 | 100.0 ± 0.0 | 1.27 ± 0.37 | 1.0 | 142.29 ± 27.72 | 98.75 ± 1.67 | 2.02 ± 0.33 | 6.0 | 156.14 ± 15.7 | 100.0 ± 0.0 | 2.18 ± 0.37 | 5.0 | 239.2 ± 25.84 |
| GroupCF - 8 | 99.6 ± 0.8 | 2.49 ± 0.82 | 8 | 28.93 ± 0.82 | 98.9 ± 1.65 | 10.56± 2.43 | 8 | 15.6 ± 2.33 | 100.0 ± 0.0 | 4.05 ± 2.55 | 8 | 7.84 ± 2.55 |
| GLOBE-CE - 8 | 95.37 ± 2.9 | 1.89 ± 0.38 | 2.0 | 1.03 ± 0.01 | 58.42 ± 18.45 | 2.25 ± 0.33 | 1.0 | 0.45 ± 0.0 | 80.25 ± 9.62 | 2.51 ± 0.33 | 1.0 | 1.36 ± 0.05 |
| dGLOBE-CE - 8 | 98.09 ± 1.72 | 2.34 ± 0.17 | 7.8 | 2.34 ± 0.03 | 69.89 ± 15.35 | 2.47 ± 0.23 | 5.6 | 1.02 ± 0.04 | 88.17 ± 8.2 | 2.64 ± 0.78 | 6.2 | 3.06 ± 0.04 |
| C-GLANCE-8 | 96.43 ± 3.6 | 0.86 ± 0.18 | 8.0 | 78.38 ± 0.0 | 100.0 ± 0.0 | 1.18 ± 0.05 | 8.0 | 63.1 ± 0.0 | 100.0 ± 0.0 | 1.02 ± 0.04 | 8.0 | 68.96 ± 0.0 |
| T-GLANCE-8 | 96.65 ± 4.08 | 0.94 ± 0.3 | 4.0 | 32.19 ± 1.34 | 99.55 ± 0.91 | 1.34 ± 0.22 | 3.8 | 29.11 ± 1.18 | 99.46 ± 1.08 | 1.07 ± 0.16 | 3.8 | 68.67 ± 1.51 |

Table 16 presents a comparative analysis of all algorithms for the German Credit dataset. Fast AReS remains the weakest method overall, continuing to exhibit poor effectiveness. Notably, while CET incurs a slightly higher cost than on the COMPAS dataset, it achieves significantly greater effectiveness, and the increase in cost is quite reasonable.
The GLOBE-CE and dGLOBE-CE methods demonstrate mixed results. They perform relatively well under the DNN model, achieving effectiveness scores above 90%, but their effectiveness declines when applied to the LR and XGB models.
On the other hand, T-GLANCE and C-GLANCE continue to excel due to their robustness and consistent high performance. These methods consistently reach near-perfect effectiveness ($\sim$100%, and only for DNN 95%-97%) across all models while maintaining very low costs. Notably, C-GLANCE-8 achieves the lowest cost of 0.86 under the DNN model, while still maintaining high effectiveness.

**Table 17: Detailed results for the solution of $s$-GCE ($s=4$ and $s=8$) for Default Credit dataset. The table reports effectiveness, cost, size, and runtime, including their standard deviations for each method.**
| method / model | DNN | LR | XGB |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime |
| Fast AReS-4 | 18.88 ± 2.16 | 1.0 ± 0.0 | 4 | 789.48 ± 306.69 | 10.85 ± 5.45 | 1.07 ± 0.13 | 4 | 244.78 ± 128.93 | 31.86 ± 5.12 | 1.05 ± 0.04 | 4 | 226.08 ± 107.9 |
| CET - 4 | 98.87 ± 0.62 | 6.32 ± 2.28 | 3.0 | 4404.58 ± 453.7 | 100.0 ± 0.0 | 3.79 ± 1.31 | 1.0 | 3711.21 ± 164.94 | 86.29 ± 9.94 | 4.5 ± 2.64 | 3.0 | 7982.05 ± 668.76 |
| GroupCF - 4 | 79.6 ± 20.79 | 1.53 ± 0.62 | 4 | 2415.7 ± 0.62 | 95.4 ± 9.2 | 1.94 ± 1.2 | 4 | 306.76 ± 1.2 | 95.2 ± 1.6 | 1.41 ± 0.64 | 4 | 991.53 ± 0.64 |
| GLOBE-CE- 4 | 76.94 ± 37.55 | 5.14 ± 0.35 | 1.4 | 3.33 ± 0.2 | 99.94 ± 0.07 | 3.42 ± 1.99 | 1.6 | 1.76 ± 0.04 | 82.7 ± 7.26 | 20.82 ± 1.73 | 2.6 | 2.44 ± 0.05 |
| dGLOBE-CE - 4 | 87.38 ± 18.69 | 5.96 ± 4.14 | 3.4 | 4.98 ± 0.21 | 99.94 ± 0.07 | 10.38 ± 7.76 | 1.2 | 2.54 ± 0.07 | 97.47 ± 0.82 | 42.58 ± 3.57 | 4.0 | 3.48 ± 0.09 |
| C-GLANCE-4 | 100.0 ± 0.0 | 1.2 ± 0.4 | 4.0 | 290.89 ± 0.0 | 100.0 ± 0.0 | 1.05 ± 0.11 | 4.0 | 243.45 ± 0.0 | 98.13 ± 1.05 | 3.68 ± 1.64 | 4.0 | 319.45 ± 0.0 |
| T-GLANCE-4 | 100.0 ± 0.0 | 1.48 ± 0.43 | 3.0 | 775.09 ± 0.0 | 100.0 ± 0.0 | 1.43 ± 0.36 | 3.0 | 876.08 ± 0.0 | 94.6 ± 3.13 | 2.75 ± 1.42 | 3.0 | 688.19 ± 0.0 |
| Fast AReS-8 | 23.75 ± 1.81 | 1.02 ± 0.03 | 8 | 428.04 ± 199.29 | 11.67 ± 6.18 | 1.04 ± 0.08 | 8 | 272.64 ± 120.89 | 36.88 ± 5.46 | 1.07 ± 0.06 | 8 | 280.88 ± 135.37 |
| CET - 8 | 98.53 ± 2.11 | 3.68 ± 1.46 | 4.0 | 2500.02 ± 328.94 | 100.0 ± 0.0 | 2.0 ± 1.36 | 2.0 | 1795.08 ± 182.55 | 90.01 ± 10.44 | 2.61 ± 1.06 | 5.0 | 5105.11 ± 923.27 |
| GroupCF - 8 | 84.8 ± 13.23 | 2.15 ± 0.98 | 8 | 2476.21 ± 0.82 | 100.0 ± 0.0 | 3.98 ± 2.86 | 8 | 279.86 ± 2.86 | 97.8 ± 0.75 | 1.92 ± 1.33 | 8 | 631.31 ± 1.33 |
| GLOBE-CE - 8 | 93.25 ± 11.26 | 2.73 ± 0.85 | 3.6 | 3.49 ± 0.27 | 99.94 ± 0.07 | 2.24 ± 0.98 | 1.8 | 1.75 ± 0.05 | 84.94 ± 6.59 | 11.21 ± 1.29 | 5.2 | 2.47 ± 0.06 |
| dGLOBE-CE - 8 | 99.95 ± 0.04 | 6.77 ± 3.67 | 4.8 | 6.79 ± 0.16 | 99.94 ± 0.07 | 10.38 ± 7.76 | 1.2 | 3.51 ± 0.07 | 99.08 ± 0.66 | 28.73 ± 9.8 | 8.0 | 5.05 ± 0.12 |
| C-GLANCE-8 | 100.0 ± 0.0 | 1.2 ± 0.4 | 8.0 | 289.46 ± 0.0 | 100.0 ± 0.0 | 1.05 ± 0.11 | 8.0 | 226.96 ± 0.0 | 98.87 ± 0.59 | 2.14 ± 0.18 | 8.0 | 257.39 ± 0.0 |
| T-GLANCE-8 | 100.0 ± 0.0 | 1.25 ± 0.38 | 3.8 | 177.22 ± 0.46 | 100.0 ± 0.0 | 1.13 ± 0.18 | 3.8 | 167.72 ± 7.88 | 94.1 ± 3.57 | 1.85 ± 0.16 | 4.0 | 191.86 ± 4.82 |

Table 17 provides a comparative analysis of all algorithms for the Default Credit dataset. C-GLANCE and T-GLANCE methods exhibit perfect effectiveness scores across both the DNN and LR models while maintaining the lowest costs, second only to Fast AReS, highlighting their superior cost-effectiveness.

In the XGB model, GroupCF-4 dominates T-GLANCE-4 and achieves smaller cost compared to C-GLANCE-4, which, however, has better performance in terms of effectiveness. Concerning the case $s=8$, GroupCF-8 has slightly better cost but worse effectiveness than C-GLANCE-8, and slightly better effectiveness but worse cost than T-GLANCE-8. Overall, our methods ensure competitive, robust and highly efficient performance.

**Table 18: Detailed results for the solution of $s$-GCE ($s=4$ and $s=8$) for HELOC dataset. The table reports effectiveness, cost, size, and runtime, including their standard deviations for each method.**
| method / model | DNN | LR | XGB |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime |
| Fast AReS-4 | 12.19 ± 0.58 | 1.03 ± 0.05 | 4 | 1562.21 ± 656.59 | 9.23 ± 1.24 | 1.12 ± 0.1 | 4 | 961.23 ± 541.41 | 8.49 ± 1.32 | 1.16 ± 0.13 | 4 | 963.0 ± 492.98 |
| CET - 4 | 86.78 ± 10.62 | 8.67 ± 3.25 | 1.0 | 11090.16 ± 4597.13 | 100.0 ± 0.0 | 3.57 ± 1.48 | 2.0 | 2662.77 ± 739.88 | 86.78 ± 6.7 | 12.51 ± 2.75 | 2.0 | 17339.7 ± 1363.44 |
| GroupCF - 4 | 80.4 ± 10.17 | 3.09 ± 0.91 | 4 | 468.52 ± 0.91 | 90.6 ± 3.93 | 2.4 ± 1.38 | 4 | 209.26 ± 1.38 | 78.4 ± 5.82 | 5.63 ± 1.93 | 4 | 201.24 ± 1.93 |
| GLOBE-CE - 4 | 42.72 ± 46.97 | 11.77 ± 15.87 | 1.4 | 1.43 ± 0.13 | 99.9 ± 0.0 | 0.6 ± 0.54 | 3.0 | 0.65 ± 0.06 | 27.66 ± 5.06 | 12.52 ± 32.48 | 2.4 | 1.18 ± 0.02 |
| dGLOBE-CE - 4 | 99.96 ± 0.05 | 11.07 ± 8.6 | 2.0 | 2.2 ± 0.14 | 99.9 ± 0.0 | 1.63 ± 0.35 | 4.0 | 0.94 ± 0.05 | 77.64 ± 11.51 | 128.0 ± 0.0 | 4.0 | 1.77 ± 0.03 |
| C-GLANCE-4 | 99.94 ± 0.05 | 11.24 ± 1.37 | 4.0 | 307.06 ± 0.0 | 100.0 ± 0.0 | 1.55 ± 0.54 | 4.0 | 214.6 ± 0.0 | 98.94 ± 0.66 | 19.99 ± 1.91 | 4.0 | 183.45 ± 0.0 |
| T-GLANCE-4 | 99.8 ± 0.24 | 10.9 ± 1.22 | 3.0 | 135.37 ± 0.0 | 100.0 ± 0.0 | 1.58 ± 0.37 | 3.0 | 108.79 ± 0.0 | 96.42 ± 2.45 | 24.85 ± 8.4 | 3.0 | 72.95 ± 0.0 |
| Fast AReS-8 | 15.44 ± 1.91 | 1.05 ± 0.06 | 8 | 1070.19 ± 558.28 | 10.95 ± 1.73 | 1.11 ± 0.07 | 8 | 901.3 ± 493.42 | 11.22 ± 2.05 | 1.09 ± 0.09 | 8 | 829.62 ± 420.35 |
| CET - 8 | 99.36 ± 0.92 | 8.86 ± 2.99 | 4.0 | 6758.87 ± 3384.51 | 100.0 ± 0.0 | 2.99 ± 1.3 | 2.0 | 1628.08 ± 344.57 | 90.25 ± 2.51 | 12.99 ± 3.49 | 6.0 | 9108.94 ± 1624.21 |
| GroupCF - 8 | 84.8 ± 12.32 | 1.98 ± 0.5 | 8 | 501.45 ± 2.5 | 96.6 ± 1.96 | 1.98 ± 0.5 | 8 | 249.14 ± 0.5 | 81.4 ± 4.22 | 5.14 ± 1.46 | 8 | 197.68 ± 1.46 |
| GLOBE-CE - 8 | 45.49 ± 44.72 | 14.19 ± 14.14 | 3.4 | 1.44 ± 0.02 | 99.9 ± 0.0 | 0.58 ± 0.5 | 7.0 | 0.64 ± 0.02 | 28.88 ± 4.71 | 12.51 ± 30.88 | 3.0 | 1.18 ± 0.02 |
| dGLOBE-CE - 8 | 100.0 ± 0.0 | 6.65 ± 0.15 | 4.4 | 3.2 ± 0.22 | 99.9 ± 0.0 | 1.63 ± 0.35 | 8.0 | 1.35 ± 0.04 | 81.95 ± 10.38 | 128.0 ± 0.0 | 8.0 | 2.55 ± 0.04 |
| C-GLANCE-8 | 99.98 ± 0.04 | 9.42 ± 2.22 | 8.0 | 301.47 ± 0.0 | 100.0 ± 0.0 | 1.2 ± 0.12 | 8.0 | 217.9 ± 0.0 | 99.0 ± 0.55 | 16.65 ± 2.44 | 8.0 | 182.83 ± 0.0 |
| T-GLANCE-8 | 99.78 ± 0.21 | 9.92 ± 1.83 | 4.0 | 40.64 ± 0.43 | 100.0 ± 0.0 | 1.52 ± 0.32 | 3.0 | 39.0 ± 0.89 | 98.25 ± 1.85 | 18.75 ± 4.92 | 4.0 | 75.9 ± 0.8 |

Table 18 presents a comparative analysis of all algorithms for the Heloc dataset, which, due to its exclusively numeric features, poses challenges in achieving low costs. Despite this complexity, the methods T-GLANCE, C-GLANCE, and dGLOBE-CE stand out by delivering near-perfect or perfect effectiveness in the DNN models, while T-GLANCE, C-GLANCE, dGLOBE-CE, and GLOBE-CE excel in the LR models. dGLOBE-CE-8 provides the best combination of effectiveness and cost for DNN, and GLOBE-CE-8 achieves the same for LR. For the XGB models, T-GLANCE and C-GLANCE lead in effectiveness across all methods. Overall, our methods demonstrate strong and consistent performance, unlike other methods, where results fluctuate considerably.

**Table 19: Detailed results for the solution of $s$-GCE ($s=4$ and $s=8$) for Adult dataset. The table reports effectiveness, cost, size, and runtime, including their standard deviations for each method.**
| method / model | DNN | LR | XGB |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime | $\operatorname{\mathsf{eff}}$ | $\operatorname{\mathsf{cost}}$ | $\operatorname{\mathsf{size}}$ | RunTime |
| Fast ares - 4 | 12.39 ± 1.06 | 1.0 ± 0.0 | 4.0 | 266.51 ± 127.13 | 11.74 ± 2.4 | 1.0 ± 0.0 | 4.0 | 124.2 ± 57.64 | 6.13 ± 0.42 | 1.0 ± 0.0 | 4.0 | 108.66 ± 57.21 |
| CET - 4 | - | - | - | $>$20h or infeasible (on Gurobi) | - | - | - | $>$20h or infeasible (on Gurobi) | - | - | - | $>$20h or infeasible (on Gurobi) |
| GroupCF - 4 | 100.0 ± 0.0 | 10.08 ± 0.03 | 4.0 | 3224.75 ± 0.03 | 100.0 ± 0.0 | 1.71 ± 0.39 | 4.0 | 925.25 ± 0.39 | 96.8 ± 1.72 | 1.41 ± 0.54 | 4.0 | 596.57 ± 0.54 |
| GLOBE-CE - 4 | 99.92 ± 0.0 | 4.24 ± 0.42 | 3.0 | 6.6 ± 0.22 | 99.92 ± 0.0 | 2.68 ± 0.17 | 3.0 | 6.48 ± 0.17 | 82.87 ± 12.14 | 30.1 ± 10.39 | 2.4 | 5.44 ± 0.18 |
| dGLOBE-CE - 4 | 99.92 ± 0.0 | 10.89 ± 1.37 | 3.6 | 9.74 ± 0.21 | 99.92 ± 0.0 | 5.91 ± 0.93 | 4.0 | 9.36 ± 0.06 | 93.76 ± 1.98 | 64.76 ± 1.29 | 4.0 | 8.26 ± 0.14 |
| C-GLANCE - 4 | 100.0 ± 0.0 | 4.6 ± 0.73 | 4.0 | 246.82 ± 0.0 | 100.0 ± 0.0 | 1.04 ± 0.07 | 4.0 | 194.83 ± 0.0 | 99.85 ± 0.12 | 5.98 ± 4.22 | 4.0 | 134.27 ± 0.0 |
| T-GLANCE-4 | 100.0 ± 0.0 | 4.43 ± 0.43 | 3.0 | 410.47 ± 0.0 | 100.0 ± 0.0 | 0.69 ± 0.01 | 3.0 | 422.45 ± 0.0 | 99.86 ± 0.14 | 1.8 ± 0.51 | 3.0 | 334.14 ± 0.0 |
| Fast ares - 8 | 13.8 ± 0.92 | 1.0 ± 0.0 | 8.0 | 271.04 ± 126.28 | 12.51 ± 2.44 | 1.0 ± 0.0 | 8.0 | 157.94 ± 73.44 | 7.19 ± 0.3 | 1.0 ± 0.0 | 8.0 | 142.82 ± 67.84 |
| CET - 8 | - | - | - | $>$20h or infeasible (on Gurobi) | - | - | - | $>$20h or infeasible (on Gurobi) | - | - | - | $>$20h or infeasible (on Gurobi) |
| GroupCF - 8 | 100.0 ± 0.0 | 11.54 ± 2.98 | 8.0 | 3539.35 ± 2.98 | 100.0 ± 0.0 | 2.33 ± 1.91 | 8.0 | 872.1 ± 1.91 | 99.4 ± 1.2 | 1.88 ± 1.36 | 8.0 | 1705.8 ± 1.36 |
| GLOBE-CE - 8 | 100.0 ± 0.0 | 2.54 ± 0.14 | 6.6 | 6.56 ± 0.18 | 100.0 ± 0.0 | 1.67 ± 0.11 | 7.0 | 6.41 ± 0.08 | 85.76 ± 9.11 | 13.48 ± 4.71 | 3.8 | 5.58 ± 0.11 |
| dGLOBE-CE - 8 | 100.0 ± 0.0 | 11.85 ± 1.53 | 6.4 | 14.09 ± 0.4 | 100.0 ± 0.0 | 6.12 ± 0.64 | 8.0 | 13.57 ± 0.22 | 94.38 ± 2.01 | 62.37 ± 3.07 | 7.4 | 12.01 ± 0.33 |
| C-GLANCE - 8 | 100.0 ± 0.0 | 3.96 ± 0.56 | 8.0 | 240.16 ± 0.0 | 100.0 ± 0.0 | 1.03 ± 0.07 | 8.0 | 179.43 ± 0.0 | 99.87 ± 0.09 | 3.85 ± 2.33 | 8.0 | 126.65 ± 0.0 |
| T-GLANCE-8 | 100.0 ± 0.0 | 3.77 ± 0.27 | 4.0 | 119.66 ± 0.69 | 100.0 ± 0.0 | 0.79 ± 0.16 | 4.0 | 122.45 ± 0.63 | 99.94 ± 0.04 | 1.65 ± 0.39 | 4.0 | 128.22 ± 1.21 |

Table 19 presents a comparative analysis of all algorithms for the Adult dataset. The Fast AReS method consistently performs poorly, achieving notably low effectiveness scores across all models.

It is also noteworthy that in the Adult dataset which is the largest of all in terms of instances and dimensions, the CET method did not run properly due to exceedingly high runtimes or infeasibility determined by the Gurobi optimizer. This is indicated by the “$>$20h or Infeasible (on Gurobi)” note present in the runtime column for both CET-4 and CET-8 across all models.

On the other hand, GroupCF, C-GLANCE, T-GLANCE, and GLOBE-CE methods all perform notably well in terms of effectiveness. Under the DNN model, T-GLANCE-4 and T-GLANCE-8 achieve perfect effectiveness scores of 100%, with costs of 4.43 and 3.77, respectively. C-GLANCE-4 and C-GLANCE-8 mirror this perfect effectiveness, also achieving 100% with similarly good cost performance. Notably, GLOBE-CE-8 achieves the best cost performance under the DNN model with an effectiveness score of 100% and a cost of 2.54.

In the LR model, T-GLANCE-4 and T-GLANCE-8 perform exceptionally well, achieving 100% effectiveness with costs of 0.69 and 0.79, respectively. C-GLANCE also performs admirably, with 100% effectiveness and costs 1.04 and 1.03. Similarly, in the XGB model, T-GLANCE-4, T-GLANCE-8, C-GLANCE-4, and C-GLANCE-8 maintain high effectiveness scores of 99.86%, 99.94%, 99.85%, and 99.87%, with really good respective costs.

Upon evaluating the performance across the five datasets — COMPAS, German Credit, Default Credit, HELOC, and Adult Income — it is evident that our methods, T-GLANCE and C-GLANCE, consistently demonstrate concrete performance. They achieve near-perfect or perfect effectiveness scores while maintaining low costs across different models, such as DNN, LR, and XGB. This robustness and consistency underline their efficiency and practicality in handling diverse datasets and complexities.

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 5
presentation: 2
questions: Please, comment on weaknesses 2) and 3).

Furthermore, what is the definition of cost in the paper? It seems defined only for the toy example, is t the same employed in the whole paper?
soundness: 3
strengths: 1) The idea of clusters both based on the similarity in the feature space and the possible recourse actions is simple and very relevant.

2) Overall, the paper is clear and easy to follow.
summary: The paper introduces two heuristics for global counterfactual explanations, namely C-Glance and T-Glance. The idea of C-Glance is to cluster instances based on their similarity in the feature space or the possible recourse actions. T-Glance is a decision tree based method. For both the introduced techniques, the goal is to maximise effectiveness and minimise the cost, with a small number of actions.
weaknesses: 1) The novelty of the proposed work is very limited, particularly the methodological novelty.

2) It is unclear what are the main innovations and differences with the existing literature (e.g., how is T-Glance different from Kanamori et al., 2022.  Is the idea of clustering according to both features and actions novel?)

3) The discussion of the proposed methods lacks completeness. For example, C-Glance depends on the number of candidate actions, the initial number of clusters, and the number of actions, but there is no adequate discussion on their impact on the results and on how to select them.

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 2
questions: When calculating the distance between two clusters, why do you focus only on the distance between the average actions in each cluster? I’m curious if the average is the right summary statistic in this context.

Additionally, when merging clusters, do you retain all of their respective actions? Wouldn’t it be beneficial to prune the action set as part of the merging process?

I’m also unclear on why the evaluations of implicit and explicit GCE are not separated. Further elaboration here could help clarify the distinction between explicit and implicit GCE.

Finally, you might consider using additional metrics beyond Pareto dominance. For instance, Tables 2 and 3 suggest that Fast AReS and GLANCE perform similarly, but Table 1 gives the impression that GLANCE has notably better effectiveness.
soundness: 3
strengths: The proposed framework is intuitive, well-designed, and novel. Although it lacks theoretical guarantees, several design choices are particularly effective. For instance, in C-GLANCE, it is insightful to consider the similarity of actions rather than focusing solely on features.

I appreciate how the paper differentiates between explicit and implicit GCE.

The empirical results are strong.
summary: This paper introduces new algorithms for finding global counterfactual explanations (GCE) as a small set of actions that can offer recourse to most of the affected population. Any solution to the GCE problem must navigate a tradeoff among the number of actions, the cost of implementing these actions, and the proportion of the population that can effectively use them for recourse. This work specifically examines the case where the number of actions is fixed and explores the tradeoff between cost and effectiveness.

The paper differentiates between explicit and implicit GCE and proposes two algorithmic frameworks for each. Given that solving GCE with a fixed size is NP-hard in general, the proposed solutions focus on finding efficient GCEs. Empirical evaluations on standard CE datasets show that the proposed approach outperforms existing GCE methods in terms of Pareto optimality and is also computationally efficient.
weaknesses: I think the formulation of GCE as a set of actions rather than directions could benefit from further justification. My default understanding of GCE aligns with GLOBE-CE, which focuses on directions for recourse rather than the actions themselves.

The overall presentation of the paper could be significantly improved. The authors are encouraged to elaborate on the rationale behind certain design choices. I have asked clarification questions in the question section regarding this. In particular, I’m not sure I fully understand Section 4 (T-GLANCE).

I also found it challenging to understand average cost (avc) and the tradeoff with size. If an action is added to the action set C, the recourse cost (rc) for x in the previous C decreases. However, by definition, |X_C| increases, meaning new instances in X_C contribute positively to avc. As a result, it’s unclear how avc behaves as size(C) increases. I expected a definition that clearly illustrates this tradeoff. Given that the results hinge on this concept, further discussion on this choice would be beneficial.

Minor issues: page 3 (computationally efficiently -> efficient), page 5 (min in Problem 1 can be misread as a min of the set), page 8 (can you avoid using red and purple in Table 1? It's hard for me to distinguish them visually)

---

confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: - Q1. How were the candidates of split conditions of Algorithm 2 determined in the experiments?
- Q2. Could you show analyses of the time complexity of the proposed algorithms?
- Q3 (optional). As far as I remember, AReS and CET have several hyper-parameters. How were these values determined in the experiments?
- Q4 (optional). As the authors mention in Line 285, we can use any dissimilarity score as two distances $d_1$ and $d_2$ in Algorithm 1. What dissimilarity score could be considered appropriate, especially for $d_2$? Also, how do they affect the solution, i.e., partition and actions?
- Q5 (optional). Is there any way to compute the total effectiveness gain more efficiently? For example, in the case of the popular CART-like algorithms, we can compute the information gain for each candidate split condition in an amortized constant time if we have sorted training samples by the value of each feature in advance. In a similar way, can we efficiently calculate the total effectiveness gain?
soundness: 3
strengths: Overall, I think this is a good paper, and the authors make solid contributions to the research area of counterfactual explanation. 
- S1. This paper is well-written, well-organized, and easy to read. 
- S2. I think the proposed framework is simple but effective. The proposed algorithms have only intuitive hyper-parameters, which may be a  desirable advantage compared to AReS and CET whose hyper-parameters are hard to tune. 
- S3. The effectiveness of the proposed method was demonstrated through extensive experiments with major baselines for the global counterfactual explanation.
summary: This paper proposes a new framework for the global counterfactual explanation problem, which is a task of assigning actions to a given set of individuals by considering their effectiveness, cost, and size. The authors propose two algorithms: C-GLANCE and T-GRANCE. The former first partitions individuals into some clusters and then merges them by taking into account not only the dissimilarity of their centroids but also that of actions generated for the centroids. The latter algorithm recursively partitions individuals like CART, where the best split condition is determined by the improvement in the effectiveness score after splitting. By extensive experiments, the authors demonstrated the effectiveness of the proposed methods compared to the existing baselines, including AReS and CET.
weaknesses: While the efficacy of the proposed methods is well demonstrated empirically, I believe the following concerns should be addressed to improve the quality of this paper. 
- W1. One of my concerns is the computation cost of the proposed algorithms. For both algorithms, we need to generate diverse actions for each centroid many times. In particular, Algorithm 2 needs to compute the total effectiveness gain for all the candidates of split conditions in Line 9, which may incur a significantly high computational cost. To compute the total effectiveness gain for one split condition, we have to generate actions for two centroids after splitting and then compute each effectiveness value. In addition, we need to repeat such a heavy procedure for all the candidate split conditions. Although I recognize the average running times of the proposed algorithms in the experiments were less than roughly a few minutes in Appendix F, I am worried about their scalability when we have more candidates of split conditions. 
- W2. Another concern is that this paper includes no theoretical result on the proposed method. I am aware that Theorem 3 in Appendix A shows the hardness of the global counterfactual explanation problem. However, the proposed algorithms are heuristic without theoretical perspectives, and there is no guarantee for the quality of their obtained solution. In addition, there is no analysis of the time complexity of the proposed algorithms as well.

---

confidence: 4
contribution: 3
initial_rating: 3
presentation: 3
questions: Why did you prefer a clustering approach over a top-down division approach in order to create subspaces? Can you discuss the pros and cons of each approach for solving the recourse problem in explanations?

What would a user study design, to test the real world effectiveness of this algorithm as compared to competing algorithms look like?

If deployed in the real world today, what would bad edge cases for the proposed algorithm look like? Are there scenarios where the recourse algorithm can malfunction? Can it suggest incorrect recourse in any case?
soundness: 3
strengths: The paper addresses a problem relevant to many real-world users of explanation systems.

The work is easy to read and seems mathematically sound. The experiment section is sane and all the claims in the intially sections seem to be justified through the paper.

I like the premise of this paper. I'm willing to increase my score if the authors do the suggested work.
summary: This paper proposes GLANCE,  a technique used to provide actionable feedback to individuals impacted by machine learning decisions. The framework focuses on creating global counterfactual explanations (GCE) that can be applied across larger groups instead of individual cases, making it more useful for real-world applications like loan approvals or hiring. The GLANCE framework optimizes three factors for effective recourse: minimizing the number of actions to keep it interpretable, ensuring the actions are cost-effective, and maximizing coverage so that most individuals benefit from these actions.

GLANCE consists of two algorithms: C-GLANCE, which clusters individuals based on features and counterfactual actions to create generalizable solutions, and T-GLANCE, which uses a tree structure for more flexible and interpretable group-specific actions.
weaknesses: It seems that Equi-explanation maps[1] also divides the feature subspace into regions that have similar recourse options, but in a top down manner. They also seem to compute sub-space centroids, nearest decision boundary points etc and even merge the clusters under certain constraints. Given that these works are very similar, I think it is important that the authors place their work against Equi-explanation maps, at least in the related work section, and preferably treat it as a competing system. 

Additionally, I think it would be good to evaluate the effectiveness of such algorithms via a user study. While recourse cost is a good human-engineered metric, it would be interesting to see if the recourse computed by different algorithms were equally helpful to end users.

[1] Chowdhury, Tanya, Razieh Rahimi, and James Allan. "Equi-explanation maps: concise and informative global summary explanations." Proceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency. 2022.

Meta-review: The paper proposes two new counterfactual explanation methods, C-Glance and T-Glance where the former uses clustering over features and recommended recourses to enable counterfactual explanations (minimize cost, maximize individuals getting valid recourse) and T-glance uses decision trees to accomplish the same. Empirical demonstrates show improvement over existing baselines.

Overall the contribution is easy to follow and the paper is overall well written. However, reviewers have raised several concerns including some lack of clarity in the writing with very limited methodological novelty, especially when compared against existing prior works reviewers have highlighted. There are also concerns regarding computational cost, and lack of theoretical grounding.

During the rebuttal authors have responded to many of the reviewer concerns, including adding a user study demonstrating the benefit of their method compared to existing approaches, clarifying novelty, and distinguishing from prior work. While some reviewers have raised their score based on the rebuttal, during the discussion, reviewers feel that the contribution's novelty and overall improvements still are insufficient to make for a high-quality ICLR paper. As such in agreement with reviewers' comments, I recommend a rejection.

Decision: Reject