---
title: 'SOGAR: Global Actionable Recourse Summaries'
url: https://www.emergentmind.com/topics/sogar
type: topic
---

# SOGAR: Global Actionable Recourse Summaries

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 [2605.07598].

## 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 \(X_k\) and assigns one shared action \(a_k\) to each subgroup. The summary is therefore a mapping \(h\) such that \(h(x)=a_k\) if \(x\in X_k\). Group-wise and global cost/loss are defined as
$$
l(a,x)=\mathbb{I}[f(a(x))=0], \qquad l(a,X_k)=\sum_{x\in X_k} l(a,x),
$$
$$
C(h)=\sum_{x\in D_0} c(h(x),x), \qquad L(h)=\sum_{x\in D_0} l(h(x),x).
$$

Within this formulation, \(C(h)\) measures total recourse effort and \(L(h)\) 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 [2605.07598].

## 2. Recourse summaries as shallow decision trees

SOGAR represents a summary as a **recourse summary tree**, i.e., a decision tree \(\tau:\mathcal{X}\to\mathcal{A}\). Each internal node tests a single feature with an axis-parallel split of the form
$$
[x_{j(v)} \le b_v].
$$
These binary splits partition the affected population into leaf regions \(X_k\), and each leaf \(k\) is assigned a single action \(a_k\), so \(\tau(x)=a_k\) for all \(x\in X_k\).

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 \(d\) and maximum number of leaves \(m\), which controls the number of subgroups and the description length. Leaf actions are constrained to be **sparse**, meaning each action can modify at most \(k\) features; in the default setup, \(k=3\) [2605.07598].

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 \(n\) bins. These edits can be combined into \(k\)-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 \(\tau\), the two objectives are \(C(\tau)\), total recourse cost over the affected population, and \(L(\tau)\), total recourse loss over the affected population. A summary tree \(\tau\) **strictly dominates** another tree \(\tau'\) if
$$
C(\tau)\le C(\tau'), \quad L(\tau)\le L(\tau'), \quad \text{and at least one inequality is strict.}
$$
The Pareto front of a candidate set \(\mathcal{T}\) is
$$
P = \{ \tau \in \mathcal{T} \mid \nexists \tau' \in \mathcal{T}\; (\tau' \succ \tau)\}.
$$

The significance of this construction is explicit: each point on \(P\) 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 [2605.07598].

The paper’s key theoretical claim is Proposition 1: for any fixed maximum depth \(d\), the complete Pareto front of recourse summary trees of depth \(d\) 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
$$
D_B = \{ \langle \phi(x), x \rangle : x\in D_0 \},
$$
where \(\phi:\mathcal{X}\to\{0,1\}^{|\mathcal{F}|}\) maps the original data to binary features \(\mathcal{F}\). The binarized features are used for branching, while the original features are retained for accurate cost evaluation.

A state is
$$
s=\langle \mathcal{D}_s,\mathcal{F}_s\rangle,
$$
where \(\mathcal{D}_s\subseteq D_B\) is the subset of instances in the current subproblem and \(\mathcal{F}_s\subseteq \mathcal{F}\) is the set of features already used along the path. A solution value is a pair
$$
v=(v_C,v_L)\in \mathbb{R}_{\ge 0}\times \mathbb{Z},
$$
representing aggregated cost and loss, and the initial state is
$$
s_0=\langle D_B,\varnothing\rangle.
$$

For a feature split \(f\), the transitions are
$$
t(s,f)=\left\langle \{(\phi(x),x)\in\mathcal{D}_s:\phi(x)_f=0\},\; \mathcal{F}_s\cup\{f\}\right\rangle,
$$
$$
t(s,\bar f)=\left\langle \{(\phi(x),x)\in\mathcal{D}_s:\phi(x)_f=1\},\; \mathcal{F}_s\cup\{f\}\right\rangle.
$$
Branching itself has zero cost,
$$
g(s,f)=(0,0),
$$
while assigning a leaf action \(a\) yields
$$
g(s,a)=\left( \sum_{(\phi(x),x)\in\mathcal{D}_s} c(a,x), \sum_{(\phi(x),x)\in\mathcal{D}_s} l(a,x) \right).
$$
Pareto comparison is
$$
(v_C,v_L)\succ (v_C',v_L') \Leftrightarrow (v_C\le v_C')\wedge (v_L\le v_L')\wedge ((v_C,v_L)\neq (v_C',v_L')),
$$
and the combine operator is component-wise addition,
$$
(v_C,v_L)\oplus (v_C',v_L')=(v_C+v_C',\,v_L+v_L').
$$

The recursive computation is defined abstractly as
$$
T(s,d)=
\begin{cases}
\mathrm{opt}\!\left(\bigcup_{a\in\mathcal{A}}\{g(s,a)\},\,s\right), & d=0,\\[4pt]
\mathrm{opt}\!\left( \bigcup_{f\in\mathcal{F}} \mathrm{merge}\bigl(T(t(s,f),d-1),T(t(s,\bar f),d-1),s,f\bigr),\,s \right), & d>0,
\end{cases}
$$
with
$$
\mathrm{merge}(\Theta_1,\Theta_2,s) = \{v_1\oplus v_2\oplus g(s,f)\mid v_1\in \Theta_1,\; v_2\in \Theta_2\},
$$
$$
\mathrm{opt}(\Theta,s)=\mathrm{nondom}(\mathrm{feas}(\Theta,s)),
$$
$$
\mathrm{nondom}(\Theta)=\{v\in\Theta \mid \nexists v'\in\Theta,\; v'\succ v\},
$$
$$
\mathrm{feas}(\Theta,s)=\{v\in\Theta\mid c(v,s)=1\}.
$$

A key implementation detail is the **cost-loss cache**: SOGAR precomputes \(c(a,x)\) and \(l(a,x)\) for every candidate action-instance pair before tree search, so lookup becomes \(O(1)\) during optimization. The cost function is the **Maximum Percentile Shift (MPS)**,
$$
c_{\mathrm{MPS}}(a,x)=\max_j \left| Q_j(a(x)_j)-Q_j(x_j)\right|,
$$
where \(Q_j\) is the CDF of feature \(j\). This makes the cost scale-invariant because it measures movement in percentile space rather than raw units [2605.07598].

## 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 [2605.07598].

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 \(=\) 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 \(0.04\) |
| 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 \(k\) 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 \(4/5\) threshold. SOGAR was then used to compare recourse summaries for females versus males across the entire Pareto front [2605.07598].

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
$$
\Delta I = +0.093,
$$
with the loss gap \((+0.070)\) larger than the cost gap \((+0.023)\). 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 [2305.06310].

Source: https://www.emergentmind.com/topics/sogar