SOGAR: Global Actionable Recourse Summaries

• SOGAR is a method for deriving compact, globally optimal recourse summaries by partitioning negatively classified groups using an exact bi-objective decision tree formulation.
• It employs dynamic programming and cost-loss caching to efficiently compute the entire Pareto front of non-dominated summary trees without repeated retraining.
• SOGAR supports practical auditing by revealing trade-offs between recourse cost and loss, facilitating subgroup comparisons and systematic bias detection.

Searching arXiv for the SOGAR paper and closely related recourse-summary work. SOGAR, short for Summaries of Optimal and Global Actionable Recourse, is a method for learning recourse summaries: compact, interpretable rules that assign a shared actionable recourse plan to groups of negatively classified individuals. It was introduced to move beyond individualized recourse, where each person receives a separate action, toward a global representation of recourse that is useful for auditing, subgroup comparison, and bias detection. Formally, SOGAR casts recourse-summary learning as an exact bi-objective optimal decision tree problem with objectives given by recourse cost and recourse loss, and computes the entire Pareto front of non-dominated summaries, enabling post-hoc trade-off selection without retraining (Chatzis et al., 8 May 2026).

1. Problem formulation and motivation

SOGAR is defined in the setting of a classifier $f:\mathcal{X}\to\{0,1\}$, where label $1$ is the desired outcome and $0$ is the undesired one. For an instance $x$, an action $a\in\mathcal{A}$ transforms it into $a(x)$. Individual recourse seeks the minimum-cost action that flips the label:

$$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$

For a population $D=\{\langle x^{(i)}, f(x^{(i)})\rangle\}_{i=0}^N$, the affected set is

$D_0 = \{ x^{(i)} : f(x^{(i)})=0 \}.$

A recourse summary partitions $D_0$ into subgroups $1$0 and assigns one shared action $1$1 to each subgroup. The summary is therefore a mapping $1$2 such that $1$3 if $1$4. Group-wise and global cost/loss are defined as

$1$5

$1$6

Within this formulation, $1$7 measures total recourse effort and $1$8 measures the number of affected individuals for whom the assigned action fails to flip the decision. The method is motivated by the observation that recourse summaries are useful for global auditing and bias detection, whereas aggregating local actions is costly and often inconsistent. Existing recourse-summary methods are described as either computationally expensive, inconsistent, reliant on arbitrary scalarization weights, or not globally optimal. SOGAR was introduced to produce globally optimal summaries, expose the full cost–effectiveness trade-off, keep summaries interpretable and small, avoid repeated reruns for different hyperparameters, and support practical auditing of classifier behavior across subgroups (Chatzis et al., 8 May 2026).

2. Recourse summaries as shallow decision trees

SOGAR represents a summary as a recourse summary tree, i.e., a decision tree $1$9. Each internal node tests a single feature with an axis-parallel split of the form

$0$0

These binary splits partition the affected population into leaf regions $0$1, and each leaf $0$2 is assigned a single action $0$3, so $0$4 for all $0$5.

This representation serves two explicit purposes. First, interpretability follows from shallow, axis-parallel trees that yield simple subgroup descriptions. Second, action consistency is enforced because every individual in a leaf receives exactly one shared action. The tree is constrained by a maximum depth $0$6 and maximum number of leaves $0$7, which controls the number of subgroups and the description length. Leaf actions are constrained to be sparse, meaning each action can modify at most $0$8 features; in the default setup, $0$9 (Chatzis et al., 8 May 2026).

The action space is also specified structurally. For binary features, actions flip value; for categorical features, actions change category; for numerical features, actions move up or down by $x$0 bins. These edits can be combined into $x$1-edit actions, and users may mark features as immutable. This design ties interpretability to operational feasibility: the subgroup description is given by the path in the tree, and the recommended recourse is given by a sparse leaf action.

3. Bi-objective optimization and Pareto optimality

The central optimization problem in SOGAR is bi-objective. For a summary tree $x$2, the two objectives are $x$3, total recourse cost over the affected population, and $x$4, total recourse loss over the affected population. A summary tree $x$5 strictly dominates another tree $x$6 if

$x$7

The Pareto front of a candidate set $x$8 is

$x$9

The significance of this construction is explicit: each point on $a\in\mathcal{A}$0 is a summary for which one cannot improve cost without worsening loss, or vice versa. SOGAR therefore computes not one solution but the entire Pareto front of non-dominated summaries, so the preferred compromise can be chosen after optimization rather than by fixing a scalarization coefficient in advance (Chatzis et al., 8 May 2026).

The paper’s key theoretical claim is Proposition 1: for any fixed maximum depth $a\in\mathcal{A}$1, the complete Pareto front of recourse summary trees of depth $a\in\mathcal{A}$2 can be computed exactly by dynamic programming. This is the basis for describing SOGAR as an optimal recourse-summary learner rather than a heuristic. The bi-objective formulation is especially relevant for auditing because a single operating point can hide disparities that become visible only when the full frontier is examined.

4. Exact solution via STreeD, caching, and separability

To fit the STreeD framework, SOGAR defines the task as a separable optimization problem with states, transitions, solution values, and Pareto comparison. It uses a binarized representation

$a\in\mathcal{A}$3

where $a\in\mathcal{A}$4 maps the original data to binary features $a\in\mathcal{A}$5. The binarized features are used for branching, while the original features are retained for accurate cost evaluation.

A state is

$a\in\mathcal{A}$6

where $a\in\mathcal{A}$7 is the subset of instances in the current subproblem and $a\in\mathcal{A}$8 is the set of features already used along the path. A solution value is a pair

$a\in\mathcal{A}$9

representing aggregated cost and loss, and the initial state is

$a(x)$0

For a feature split $a(x)$1, the transitions are

$a(x)$2

$a(x)$3

Branching itself has zero cost,

$a(x)$4

while assigning a leaf action $a(x)$5 yields

$a(x)$6

Pareto comparison is

$a(x)$7

and the combine operator is component-wise addition,

$a(x)$8

The recursive computation is defined abstractly as

$a(x)$9

with

$$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$0

$$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$1

$$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$2

$$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$3

A key implementation detail is the cost-loss cache: SOGAR precomputes $$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$4 and $$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$5 for every candidate action-instance pair before tree search, so lookup becomes $$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$6 during optimization. The cost function is the Maximum Percentile Shift (MPS),

$$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$7

where $$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$8 is the CDF of feature $$\arg\min_{a \in \mathcal{A}} c(a,x)\quad \text{s.t.}\quad f(a(x)) = 1.$$9. This makes the cost scale-invariant because it measures movement in percentile space rather than raw units (Chatzis et al., 8 May 2026).

5. Computational behavior and empirical performance

SOGAR produces the entire Pareto front in one run, after which a user can choose a cost-leaning, loss-leaning, or balanced summary post hoc, without retraining or rerunning optimization with new scalarization coefficients. If computational budget is limited, the method is also anytime: a timeout can be imposed, and the algorithm returns the best non-dominated solutions found so far, although optimality guarantees are then lost (Chatzis et al., 8 May 2026).

The practical stability reported for SOGAR is attributed to five design choices: shallow axis-parallel trees, sparse leaf actions, dynamic programming with caching and pruning, precomputed cost-loss caches, and CPU/GPU parallelization of leaf evaluation. The exact Pareto-front computation is also presented as reducing sensitivity to arbitrary hyperparameter tuning relative to single-solution methods.

The experiments evaluate SOGAR on four tabular datasets—Employee Attrition, German Credit, Bank Marketing, and Adult Income—using LightGBM, XGBoost, and a DNN as the audited classifiers. Baselines include CET, AReS, GLOBE-CE, GLANCE, and T-CREx. The reported metrics are recourse cost (MPS), recourse loss, invalidity $D=\{\langle x^{(i)}, f(x^{(i)})\rangle\}_{i=0}^N$0 cost + loss, and runtime.

Dataset	Audited classifiers	Reported observations
Employee Attrition	LightGBM, XGBoost, DNN	Full Pareto front visualized; held-out test points lie very close to the training Pareto frontier, with an average Euclidean distance of about $D=\{\langle x^{(i)}, f(x^{(i)})\rangle\}_{i=0}^N$1
German Credit	LightGBM, XGBoost, DNN	With DNN, SOGAR achieves extremely low cost and loss; GPU acceleration drops runtime dramatically
Bank Marketing	LightGBM, XGBoost, DNN	SOGAR beats the baselines on invalidity while remaining feasible at the full-frontier scale
Adult Income	LightGBM, XGBoost, DNN	SOGAR returns a strong balanced solution and can produce thousands of Pareto-optimal summaries in a single run

Across these experiments, the headline quantitative finding is that SOGAR achieves the best invalidity on all datasets, and often the best or near-best cost. GLOBE-CE and GLANCE often achieve very low loss but at much higher cost; CET and AReS are frequently slower and sometimes hit the one-hour timeout, especially on larger datasets; T-CREx is competitive in some cases but often worse on invalidity and with weaker consistency. Ablations show the expected trade-offs: lower depth gives much faster but slightly worse summaries, lower sparsity $D=\{\langle x^{(i)}, f(x^{(i)})\rangle\}_{i=0}^N$2 reduces runtime and memory but increases loss, and fewer discretization bins reduce runtime with only small quality loss.

6. Auditing capability, disparity analysis, and nomenclature

A major use case for SOGAR is global auditing and bias detection. Because it returns the full Pareto front, subgroup disparities can be inspected over all attainable cost–loss trade-offs rather than at a single chosen operating point. On the Adult Income dataset, the classifier exhibited gender bias: females received adverse predictions much more often than males, and the disparate impact ratio was below the common $D=\{\langle x^{(i)}, f(x^{(i)})\rangle\}_{i=0}^N$3 threshold. SOGAR was then used to compare recourse summaries for females versus males across the entire Pareto front (Chatzis et al., 8 May 2026).

The reported result is that across the 4,142 Pareto-optimal summaries, females had consistently worse recourse than males in 96.4\% of the solutions. The average gap in invalidity was

$D=\{\langle x^{(i)}, f(x^{(i)})\rangle\}_{i=0}^N$4

with the loss gap $D=\{\langle x^{(i)}, f(x^{(i)})\rangle\}_{i=0}^N$5 larger than the cost gap $D=\{\langle x^{(i)}, f(x^{(i)})\rangle\}_{i=0}^N$6. This indicates that the disparity was not only that actions were more expensive for women, but also that they were more often ineffective. This suggests that full-frontier analysis can reveal systematic recourse disparities that methods returning only a single solution may obscure.

The acronym should also be distinguished from SoGAR, short for Self-supervised Spatiotemporal Attention-based Social Group Activity Recognition, a separate 2023 method in video understanding that addresses weakly supervised group activity recognition rather than actionable recourse (Chappa et al., 2023).