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% Our packages
\usepackage{algorithm}
\usepackage[noend]{algpseudocode}

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\newtheorem{theorem}{Theorem}
\newtheorem{example}{Example}
\newtheorem{lemma}{Lemma}
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\newcommand{\deltaexp}{\ensuremath{\delta}-robust AXp}
\newcommand{\gaxp}{\ensuremath{\delta}-gAXp}
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\title{Abductive Explanations for Groups of Similar Samples}

\begin{document}

\maketitle

\begin{abstract}
    Explaining the decisions of machine learning models is crucial as their use becomes widespread.
    While many approaches to explanation are based on heuristics or surrogate models without formal guarantees, formal explanations provide reasoning for a particular decision that is guaranteed to be valid.
    We focus on abductive explanations (AXp) that identify sufficient subsets of input features for a given classification.
    We extend AXp to not only cover a particular sample, but to cover all of the samples whose features are within a given interval, providing explanations that remain valid even when the features in the explanation vary by up to $\delta$.
    In addition to applying this notion of \emph{\deltaexp} to a single sample, we also consider \emph{group explanations} (\gaxp), which give a common explanation for a group of samples that share the same classification.
    We evaluate our approach by producing explanations for a neural network model with the help of Marabou, a neural network verifier.
    The evaluation shows that, compared to a recent approach for finding a maximally ``inflated'' explanation, a \deltaexp\ covers a significant volume of the inflated explanation with a dramatically lower runtime.
    Our evaluation also provides evidence that group explanations capture important features for all the samples within the group much faster than computing explanations for each sample separately.
\end{abstract}

%\clearpage

\section{Introduction}

Many real world applications are increasingly relying on decision-making systems based on machine learning, including in sensitive contexts such as health care, law enforcement and finance~\citep{DBLP:journals/csur/KarimiBSV23}.
Those affected by the decisions of automated decision-making systems demand a right to explanation.
OECD AI Principles recommend that AI systems should ``provide information that enable those adversely affected by an AI system to challenge its output''\footnote{\url{https://oecd.ai/en/dashboards/ai-principles/P7}} while the data protection regulation of the European Union GDPR requires that users subject to automatic decision-making have the right to ``meaningful information about the logic involved''\footnote{\url{https://gdpr-text.com/read/article-13/}}.
Indeed, research into methods for explaining machine learning models has blossomed in recent years~\citep{DBLP:journals/widm/ConfalonieriCWB21,DBLP:journals/air/MinhWLN22,DBLP:journals/datamine/BodriaGGNPR23,DBLP:conf/isola/MarquesSilva24}.

Many proposed explanation methods, however, produce misleading explanations sometimes~\citep{DBLP:conf/icml/KumarVSF20,DBLP:journals/corr/abs-2302-08160,DBLP:journals/cacm/MarquesSilvaH24}.
Approaches such as the popular SHAP~\citep{DBLP:conf/nips/LundbergL17} and LIME~\citep{DBLP:conf/kdd/Ribeiro0G16} are heuristic and thus do not provide formal guarantees.
\emph{Formal} explanations overcome these shortcomings~\citep{DBLP:conf/ijcai/ShihCD18,DBLP:conf/aaai/IgnatievNM19,DBLP:conf/aaai/0001I22}. In this paper, we focus especifically on formal \emph{abductive} explanations (AXps) \citep{DBLP:conf/aaai/0001I22}, which explain why the model classifies a sample as it does.
More concretely, an abductive explanation is a subset of an input sample’s feature values that is sufficient to guarantee that the classifier assigns the same label to any sample sharing these values. AXps, in other words, offer a local feature selection explanation.

AXps suffer from a notable limitation.
They are fragile to small perturbations in feature values.
An AXp is only valid for the exact feature values of a given sample $\sample$, so after a minimal perturbation to a single feature $x_i$ it may no longer be valid, even though the perturbed sample is almost identical to the original.
This locality can be a limiting factor in real-world applications where measurements are noisy or uncertain.
For instance, in medical diagnosis, an explanation for a classification based on  a patient's blood pressure being 140.0 mmHg might be invalid for a reading of 140.1 mmHg, although the difference is a slight measurement error.
Such a brittle explanation is intuitively inferior to an explanation that holds for all similar values.
Furthermore, one might be interested in explanations for a \emph{group} of similar samples.
For example, given five patients with similar health records and a similar classification given by the model, it might be more instructive to see explanations for the group than for the individuals.

In this work we consider the extension of AXps to a neighbourhood of samples rather than to a single sample.
We introduce \deltaexp s, where $\delta$ is a robustness bound that ensures that an explanation remains valid when the feature values vary within $\delta$.
A separate $\delta_i$ can also be set for each feature $i$.
With these robustness bounds, we lift AXps from a local to semi-local explanations where explanations are valid for a group of samples.
The use cases for \deltaexp s are, among others, real-world applications with measurement uncertainty, or \gaxp s, which explain groups of similar samples by providing a \deltaexp s for a representative of the group.
When applied to a single sample, our approach is similar to a recent approach to compute \emph{inflated} AXps~\citep{izza_2024,DBLP:journals/corr/abs-2505-10991}, which give a maximal interval for each feature within which an explanation is valid.
In contrast, our approach first sets a desired interval and then finds a valid explanation for this potentially smaller set of intervals, hugely reducing the computational cost.
Given the difficulty of producing formal explanations, it is crucial to find the most efficient methods for obtaining the desired results.

We make the following contributions.
We introduce \deltaexp s, an extension of formal explanations that provide formal guarantees while being robust to small perturbations, and \gaxp, a formalization of explanations for groups of samples.
We implement our approach using Marabou~\citep{DBLP:conf/cav/WuIZTDKRAJBHLWZKKB24}, a framework for verifying properties of neural networks.
We show in a comprehensive empirical evaluation that our approach scales effectively to high-dimensional datasets.
We show that, compared to (maximally) inflated explanations, \deltaexp s cover a significant portion of the feature space at a fraction of the computational cost.
Finally, we evaluate \gaxp s based on their induced formal feature attribution scores (FFA)~\citep{yu_2023} and show that they effectively capture the feature importance within a group.
Using the FFA scores of each individual sample of the group as the ground truth, \gaxp s perform better than SHAP and LIME, and comparable to a brute-force method of computing the average of the FFAs for all samples of the group, with an orders of magnitude lower runtime.
Unlike approaches to similar problems, we show that computing \deltaexp s is not more computationally expensive than computing a regular AXp.

\section{Related Work}

Explainable AI has gained significant attention in recent years~\citep{DBLP:journals/widm/ConfalonieriCWB21,DBLP:journals/air/MinhWLN22,DBLP:journals/datamine/BodriaGGNPR23}, leading to the development of diverse methodologies including heuristic explanations, surrogate models, feature attribution, feature selection, and interpretable models.
In this work, we focus on formal feature selection explanations.

Researchers have approached formal explanations from various perspectives.
Among these, feature selection methods, which identify subsets of features that constitute explanations, can be roughly divided into \emph{abductive} and \emph{counterfactual} explanations~\citep{WachterMR2018,DBLP:conf/ijcai/ShihCD18,DBLP:conf/aaai/IgnatievNM19,DBLP:conf/ijcai/Izza021,DBLP:conf/sat/IgnatievS21,DBLP:conf/icml/ParmentierV21,DBLP:conf/nips/Wu0B23,DBLP:conf/cav/WuIZTDKRAJBHLWZKKB24,DBLP:conf/aistats/KarimiBBV20,DBLP:conf/aistats/AudemardLMS23,DBLP:conf/aaai/0001I22,DBLP:conf/isola/MarquesSilva24,DBLP:journals/csur/KarimiBSV23,DBLP:journals/datamine/Guidotti24}.
Intuitively, abductive explanations answer the question \emph{why this classification} and counterfactual explanations answer \emph{how to obtain a different classification}.
Formally, an abductive explanation is a subset of a sample's features such that any sample fixing these features to their respective values receives the same classification as the original sample.
A counterfactual explanation, conversely, is a subset of features such that any sample with different values for these features receives a different classification than the original sample.

Robust explanations have been studied for counterfactual explanations~\citep{DBLP:conf/aaai/JiangL0T23,DBLP:conf/ijcai/JiangL0T24}, but they have received less attention for AXps.
For AXps, a type of robustness called distance-restricted AXps~\citep{DBLP:conf/kr/IzzaH0PI024,DBLP:journals/corr/abs-2306-03048} and $\epsilon$-robust explanations~\citep{wu_2023} has been considered, where the \emph{relevant} features, features inside the explanation, remain fixed, and the \emph{irrelevant} features, features outside the explanation, can only vary within a given bound $\epsilon$.

In contrast to this concept of robustness, our \deltaexp s extend AXps by remaining valid when the relevant features vary by up to $\delta$ (regardless of how much the irrelevant features vary).
In the parlance of~\citet{DBLP:conf/ijcai/JiangL0T24}, our \deltaexp s are robust against noisy execution.
As mentioned above, the closest to this sense of robustness for AXps are inflated explanations \citet{izza_2024}, which give bounds to the relevant features so that explanations remain valid when the input variations occur inside the bounds.
The algorithm to generate inflated explanations maximizes one at a time the bounds of each feature in a given AXp, with multiple calls to a solver that checks whether the current explanation is valid.
This approach has two drawbacks compared to our \deltaexp s.
First, the feature(s) whose bounds are maximized last can be left with tiny bounds or no bounds at all.
Second, they are computationally costly since they require multiple calls to a validity checker.

Group explanations present a separate but related challenge, since they attempt to explain simultaneously multiple samples with the same classification.
They have recently gained attention for counterfactual explanations~\citep{DBLP:journals/corr/abs-2303-09297,DBLP:journals/eor/CarrizosaRM24,DBLP:journals/eswa/CarrizosaRM24,DBLP:journals/corr/abs-2405-17642}, but, to the best of our knowledge, group explanations have not been considered for AXps. Our work addresses these gaps in the literature by considering \emph{robust abductive} explanations, and applying them to produce abductive explanations for \emph{groups of samples}.

\section{Formal Explanations}

Let $f: \mathbb{R}^d \rightarrow \{1, \ldots, k\}$ be a classifier that maps samples $\sample=\{\samp{1},\dots,\samp{d}\} \in \mathbb{R}^d$ to classes $c \in \{1, \ldots, k\}$.
An abductive explanation (AXp) answers the question \emph{why does $f$ classify $\sample$ as $c$} by identifying a subset of features $E$, such that the classifier assigns the label $c$ to any sample that shares all values of $E$ with $\sample$.
Formally, a subset of features $E\subseteq \feats$ is an AXp of $\sample$ if
\[
    \forall \difsample \in \R^d : \bigwedge_{i \in E} (\difsamp{i} = \samp{i}) \implies f(\difsample) = f(\sample)
\]
and there is no proper subset of $E$ satisfying this condition.

\subsection{$\delta$-robust Explanations}

A shortcoming of AXps is that a tiny change to one of the relevant features can make an explanation invalid.
In real world scenarios, measures have error bounds, so the fragility of formal explanations makes them impractical.
To address this limitation, we propose a variant of formal explanations that is valid when relevant features vary within specified bounds.
This new formulation not only makes explanations robust to input variation, but it allows us to define formal explanations for groups of samples, clarifying further how classifiers behave.

A \deltaexp\ for a sample $\sample$ with classification $f(\sample) = c$ is a subset of features $E \subseteq \feats$ and a set of robustness bounds $\{[\samp{i} - \delta_i, \samp{i} + \delta_i] \mid i \in E\}$, such that $f(\difsample) = c$ for any input $\difsample \in \mathbb{R}^d$ where $\difsamp{i} \in [\samp{i} - \delta_i, \samp{i} + \delta_i]$ for all $i \in E$. The robustness parameter $\delta$ controls the maximum allowed variation in each relevant feature. Formally, a pair $(E, \{\delta_i\}_{i \in E})$ is a \deltaexp\ if
\[
\forall \difsample \in \mathbb{R}^d: \bigwedge_{i \in E} (\difsamp{i} \in [\samp{i}  - \delta_i, \samp{i} + \delta_i]) \implies f(\difsample) = f(\sample)
\]
and there is no proper subset of $E$ satisfying this condition.

This definition ensures that explanations remain valid when relevant features vary within their specified bounds, addressing the fragility to input variation of formal explanations.
For convenience, when $\delta_i=v$ for each $i\in E$, we may write that $\delta=v$ and refer to a  $v$-robust AXp.
Note that \deltaexp s are a generalization of AXps.
If $(E, \{\delta_i\}_{i \in E})$ is a \deltaexp\ for a sample $\sample$, then $E$ is a formal explanation for $\sample$.
Similarly, an AXp is a 0-robust AXp.

\begin{example}
    Figure \ref{fig:robust_explanation} illustrates an AXp and a \deltaexp\ with $\delta=0.09$.
    In both, the irrelevant features are the same and can take whatever value without changing the classification.
    In the relevant features, however, the AXp shows a precise point where the explanation is valid, represented by the narrow green line correspoding to the feature values of the input sample, while the \deltaexp\ shows intervals within which the relevant features can take any value without the classification changing, represented by the large green rectangles corresponding to the intervals of validity.
\end{example}

To compute a \deltaexp, we assume that features are normalized to $[0,1]$ and that we have a formal model of the classifier $f$ as a set of logical constraints and an oracle for checking whether a given set of constraints is satisfiable.
In our implementation we use Marabou, a formal verifier of properties in neural networks based on constraint solving~\citep{DBLP:conf/cav/WuIZTDKRAJBHLWZKKB24}.
We impose further constraints as the formula $\phi$ during the course of the algorithm.
The algorithm takes as input a sample $\sample$ and a set of robustness bounds $\deltavec=\{\delta_i\}_{i=1}^d$, and finds a set $E$ such that $(E,\{\delta_i\}_{i \in E})$ is a \deltaexp\ for $\sample$.

In Algorithm~\ref{alg:robust_explanation}, we modify the standard deletion-based algorithm for computing minimal unsatisfiable sets of constraints~\citep{DBLP:journals/informs/ChinneckD91} to compute a \deltaexp.
First, if the classification is not guaranteed even with all features fixed to be within their respective bounds, there are no \gaxp s and the algorithm terminates (lines \ref{li:precheck}--\ref{li:no-gaxp}).
Otherwise, the loop iterates over each feature $i$ to determine whether it should include it in the explanation (lines \ref{li:start-loop}--\ref{li:end-loop}).
The order of iteration can be arbitrary, but some orders might result in smaller or otherwise more desirable explanations~\citep{wu_2023}.
Inside the loop the algorithm sets the constraints for each feature $j$ based on whether it is in an over-approximation of an explanation $E$ or already deemed irrelevant.
The features in $E$ are the potentially relevant features, so they must be within their robustness bounds (line \ref{li:fixed}).
An exception is the feature $i$, which we allow to take any value in order to check whether it is irrelevant (lines \ref{li:fixed}--\ref{li:free}).
Features not in $E$ are irrelevant features, so they can take any value in $[0,1]$ (line \ref{li:free}).
In line \ref{li:check} the function \textsc{Guaranteed}$(\phi, f(\hat{x}) = f(\sample))$ uses a constraint-solving oracle and returns \texttt{True} iff $f(\hat{x}) = f(\sample)$ holds under all values of $\difsample$ that satisfy the constraints $\phi$.
If this is the case, then the feature $i$ can demonstrably take any value without changing the classification, so $i$ is irrelevant and it is removed from $E$ (\ref{li:remove}).

\begin{algorithm*}[b]
    \begin{algorithmic}[1]
        \Function{ComputeRobustAXp}{$\sample$, $\deltavec$}

        \If{$\neg\textsc{Guaranteed}(\{(|\difsamp{j}-\samp{j}|\leq \delta_j) \mid j\in \feats\},f(\difsample) = f(\sample))$} \phantomsection\label{li:precheck}
        \State \Return \texttt{False}\label{li:no-gaxp}
        \EndIf

        \State $E \gets \feats$

        \For{each feature $i$ in $\feats$} \phantomsection\label{li:start-loop}
            \State $\phi \gets \{(|\difsamp{j}-\samp{j}|\leq \delta_j) \mid j\in E\setminus\{i\}\}$ \phantomsection\label{li:fixed}
            \State $\phi \gets \phi\cup \{(\difsamp{j}\in [0,1]) \mid j\in \feats\setminus (E \setminus \{i\})\}$ \phantomsection\label{li:free}
            \If{$\textsc{Guaranteed}(\phi,f(\difsample) = f(\sample))$} \phantomsection\label{li:check}
                \State $E \gets E \setminus \{i\}$ \phantomsection\label{li:remove}
            \EndIf
        \EndFor \phantomsection\label{li:end-loop}

        \State \Return $E$
        \EndFunction
    \end{algorithmic}
    \caption{Computing a \deltaexp\ for sample $\sample$}
    \label{alg:robust_explanation}
\end{algorithm*}

\begin{figure}[!t]
    \centering
    \begin{minipage}{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/delta_robust/examples/robust_explanation_glass_label_5_sample_3_delta_0.00.pdf}
        %\label{fig:formal_explanation}
    \end{minipage}
    %\hfill
    \begin{minipage}{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/delta_robust/examples/robust_explanation_glass_label_5_sample_3_delta_0.09.pdf}
    \end{minipage}
        \caption{Comparison of two explanations for the first sample with class 5 of the Glass dataset. An AXp (left) and a \deltaexp\ for $\delta=0.09$ (right).}
        \label{fig:robust_explanation}
\end{figure}

Inflated explanations \citep{izza_2024} are a related approach to our \deltaexp s.
They maximize the bounds of relevant features one feature at a time and separately for the lower and upper bound.
They achieve this by first computing an AXp and then considering each feature $i$ in the AXp in turn.
They increase the bound $b_i$ for feature $i$ by a small, predetermined amount $\eta$ (precision) and check whether the explanation is still valid with a call to an oracle (typically a solver for a constraint language such as propositional satisfiability or mixed-integer linear programming).
When the explanation is no longer valid, they take the previous bound (i.e. $b_i-\eta$) and repeat the process for the lower bound of feature $i$.
The result is an explanation together with a set of maximal bounds on each relevant feature that guarantee that varying the values of the relevant feature within these bounds does not change the classification.
In other words, inflated explanations are a variant of \deltaexp s with a separate $\delta$ for each feature and each direction (i.e. increasing the value and decreasing the value) where all the $\delta$s are maximal.
\begin{example}
    In Figure \ref{fig:inflated_explanation_comparison} we illustrate an inflated and a \deltaexp\ sharing the same underlying AXp.
    The bounds of the inflated explanation are larger in general, but since inflated explanations are maximized one feature at a time, some features have smaller bounds.
    Notably, the feature `K' could not be inflated at all and must remain fixed in the inflated explanation.
    Conversely, the \deltaexp\ guarantees given bounds for each feature, so every feature has flexibility.
\end{example}

\begin{figure}[!t]
    \centering
    \begin{minipage}{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/delta_robust/examples/inflated_explanation_glass_label_5_sample_3_delta_0.09.pdf}
    \end{minipage}
    \hfill
    \begin{minipage}{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/delta_robust/examples/robust_explanation_glass_label_5_sample_3_delta_0.09.pdf}
    \end{minipage}
    \caption{Comparison of two explanations for the first sample with class 5 of the Glass dataset. A an inflated AXp (left) generated by inflating a \deltaexp\ for $\delta=0.09$ (right). \red{The \deltaexp\ covers 43\% of the area of the inflated AXp.}}
    \label{fig:inflated_explanation_comparison}
\end{figure}

\subsection{Group Explanations}

While AXps explain the classification of a single sample, in many cases it is insightful to explain the classification of a group of samples.
In medical diagnosis, for example, researchers want to understand what features are important to diagnose a condition in general, not just for a particular patient.
Group explanations address this need by identifying the features that explain why the samples in a group receive the same classification.
Group explanations can be thought of as semi-local, since they do not apply to a single sample (local) or to the whole model (global), but to a group of samples.
They are most effective with samples that are naturally grouped together, such as patients with similar symptoms, customers with similar profiles, or products with similar characteristics.

\begin{algorithm*}[b]
    \begin{algorithmic}[1]
        \Function{ComputeGroupAXp}{$X = \{\sample_1, \ldots, \sample_n\}$, $\deltavec$}
        %\State $d \gets \text{number of features}$
        %\State $n \gets |X|$

        \State $\bar{\sample} \gets \mathrm{Mean}(X)$ \phantomsection\label{li:representative}
        %\State $\bar{\sample} \gets \mathrm{Mean}(\{\sample_1, \ldots, \sample_n\})$

        \State $\gammavec \gets \deltavec \cdot \mathrm{StandardDeviation}(X)$ \phantomsection\label{li:std}
        %\State $\gammavec \gets \deltavec \cdot \mathrm{StandardDeviation}(\{\sample_1, \ldots, \sample_n\})$

        %\For{each feature $i$ in $\feats$}
        %\State $\sigma_i \gets \mathrm{StandardDeviation}(\{x_{1,i}, x_{2,i}, \ldots, x_{n,i}\})$
        %\State $\gamma_i \gets \delta \cdot \sigma_i$
        %\EndFor

        \State $E \gets$ \textsc{ComputeRobustAXp}$(\sample, \gammavec)$ \phantomsection\label{li:get-gaxp}
        %\State $(E, \{\gamma_i\}_{i \in E}) \gets$ \textsc{ComputeRobustAXp}$(\sample, \{\gamma_i\}_{i=1}^d)$
        \If{$E=\text{\texttt{False}}$}
        \State \Return \texttt{False}
        \EndIf

        \State \Return $(E,\gammavec)$
        \EndFunction
    \end{algorithmic}
    \caption{Computing a \gaxp\ for a Group of Samples}
    \label{alg:group_explanation}
\end{algorithm*}

In Algorithm \ref{alg:group_explanation}, we illustrate how to compute a type of group explanation, \gaxp, by computing \deltaexp s on a representative of the group with robustness bounds that depend on the variance of feature values occurring in the group.
The algorithm takes a group of samples $X=\{\sample_1,\dots,\sample_n\}$ that share the same classification and a vector $\deltavec$ as input.
First, it computes a representative of the group by averaging the samples in the group (line \ref{li:representative}).
Then it computes the robustness bounds $\delta_i$ that are proportional to the standard deviation of each feature $i$ within the group for the representative (line \ref{li:std}).
Finally, it computes a \deltaexp\ with Algorithm~\ref{alg:robust_explanation} using these bounds (line \ref{li:get-gaxp}).
If the representative receives a different classification than the samples in $X$, the group is too dissimilar and has no \gaxp.
If, on the other hand, the corresponding \deltaexp\ exists, we call the output a $\text{\gaxp} = (E,\gammavec)$.
For any sample $\difsample=\{\difsamp{1},\dots,\difsamp{d}\}$, if $|\difsamp{i}-\bar{x}_{i}|\leq \gamma_i$ for every feature $i\in E$, then $f(\difsample)=f(\bar{\sample})$.

The parameters $\delta_i$ control the ratio of the standard deviation of each feature in the group that is taken as the bound for that feature.
This formulation sets wider bounds for features that have diverse values in the group and narrower bounds for features that have similar values in the group.
If $\delta = 0$, the \gaxp\ is a AXp for the group representative, while larger values of $\delta$ increase the robustness bounds, making the explanation valid for a wider range of samples.

\begin{figure}[!t]
    \centering
    \begin{minipage}{0.20\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/cluster/examples/individual_explanation_mnist_label_5_sample_0.pdf}
    \end{minipage}
    \hfill
    \begin{minipage}{0.20\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/cluster/examples/individual_explanation_mnist_label_5_sample_1.pdf}
    \end{minipage}
    \hfill
    \begin{minipage}{0.20\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/cluster/examples/individual_explanation_mnist_label_5_sample_2.pdf}
    \end{minipage}
    \hfill
    \hspace{25pt}
    \begin{minipage}{0.20\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/cluster/examples/cluster_explanation_mnist_label_5_cluster_0_size_3_delta_0.3.pdf}
    \end{minipage}
    \caption{From left to right, the first three images represent individual AXps for three images in MNIST, while the last image represents a \gaxp\ for $\delta = 0.3$ for the former images. The green pixels represent the relevant features in the explanations.}
    \label{fig:cluster_explanation}
\end{figure}

\begin{example}
In Figure \ref{fig:cluster_explanation}, we compute an AXp for three MNIST samples and their common \gaxp\ (for $\delta = 0.3$). The individual explanations show different patterns for each sample, reflecting their specific characteristics, while the \gaxp\ captures the common features that are important across the three samples, reflecting the general characteristics that make them belong to the same class.
\end{example}

\section{Empirical Evaluation}

We conducted an empirical evaluation using 2x12 core Xeon E5 2680 v3 2.50GHz CPUs with 128GB DDR4-2133 of RAM.
We leave details and code in the supplementary material \red{TODO!}.

\paragraph{Datasets} We evaluate our methods on both \emph{classic machine learning datasets} (Glass~\citep{openml_glass}, Leaf~\citep{openml_leaf}, Parkinsons~\citep{openml_parkinsons}, Diabetes~\citep{sklearn_diabetes} and Wine~\citep{sklearn_wine}) and \emph{synthetic datasets} generated using scikit-learn's \texttt{make\_classification} function.
The classic datasets represent real-world scenarios with varying dimensionality and complexity, while the synthetic datasets allow us to control dimensionality and feature characteristics for scalability analysis.
In the synthetic datasets, we have $10,000$ samples, $2$ classes, and $10$ to $25$ features.
The \texttt{make\_classification} function generates datasets with informative and noisy features.
Informative features are those that contribute to the target variable, while noisy features are random and do not provide useful information for classification.
We set the number of informative features to $\lfloor n/2 \rfloor$, where $n$ is the total number of features.

\paragraph{Comparisons} We compare \deltaexp s against inflated explanations \citep{izza_2024} as the primary baseline, since they represent the most relevant alternative approach.
We evaluate the volume of feature values covered compared to inflated explanations.
For \gaxp s, we compare the formal feature attribution (FFA) induced by our approach against SHAP, LIME, and a brute-force approach of computing explanations separately for each sample in the group and aggregating them afterwards.
We consider the FFA scores of the individual samples as the ground-truth.
We also evaluate the runtime since all the explanation problems we consider are computationally hard.

\paragraph{Machine Learning Model} We train a fully-connected neural network (NN) with $2$ hidden layers of $10$ neurons each and ReLU as their activation function for each dataset.
We train each NN for $500$ epochs with a learning rate of $0.001$, a batch size of $16$, and a patience of $10$.
We use $0$ as the seed.
We compute explanations with Marabou~\citep{DBLP:conf/cav/WuIZTDKRAJBHLWZKKB24}, which encodes the NN into a logical formula and allows for formally verifying properties of the NN.

\subsection{Evaluation of $\delta$-robust Explanations}

\begin{figure}[!t]
    \centering
    \begin{minipage}{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/delta_robust/classic_datasets/robustness.pdf}
    \end{minipage}
    \centering
    \begin{minipage}{0.48\textwidth}
        \centering
        \resizebox{\textwidth}{!}{
        \begin{tabular}{cccccc}
        & \textbf{Diabetes} & \textbf{Glass} & \textbf{Leaf} & \textbf{Parkinsons} & \textbf{Wine} \\
        \toprule
        $\delta=0.00$ & 1.1574 & 1.1088 & \textbf{1.6919} & \textbf{1.2455} & \textbf{1.0777} \\
        $\delta=0.01$ & 1.1400 & \textbf{1.0847} & 1.7071 & 1.3002 & 1.1018 \\
        $\delta=0.02$ & 1.1345 & 1.1936 & 1.7253 & 1.4251 & 1.1106 \\
        $\delta=0.03$ & \textbf{1.1272} & 1.1345 & - & 1.4121 & 1.1217 \\
        $\delta=0.04$ & 1.1592 & - & - & 1.3869 & 1.1181 \\
        $\delta=0.05$ & 1.1600 & - & - & 1.4919 & 1.1250 \\
        $\delta=0.06$ & 1.1511 & - & - & 1.4730 & 1.1665 \\
        $\delta=0.07$ & 1.1578 & - & - & 1.4327 & 1.1723 \\
        $\delta=0.08$ & 1.2342 & - & - & - & 1.1515 \\
        $\delta=0.09$ & - & - & - & - & 1.1874 \\
        \midrule
        Inflated & 12.1076 & 4.5886 & 14.5489 & 34.9010 & 14.9624 \\
        \bottomrule
        \end{tabular}
        }
    \end{minipage}
    \caption{Across 100 samples, average area of inflated AXps covered by \deltaexp s (left) and average runtime (right).}
    \label{fig:robustness}
\end{figure}

We evaluate our approach across 100 samples by comparing the volume of a \deltaexp\ to the volume of an inflated explanation based on the same underlying AXp (i.e. sharing the same relevant features), and by comparing their runtimes.
We use a precision of $\eta=0.01$ for the inflated explanations.
To present cases where applications could realistically use \deltaexp s, we only show data for values of $\delta$ where at least 50\% of the samples have a \deltaexp.

In Figure \ref{fig:robustness}, we see on the left that \deltaexp s achieve up to 20--50\% of the coverage of inflated explanations, and we see on the right that they are 4 to 30 times faster to compute than inflated explanations.
Thus, even though \deltaexp s explanations do not maximize the validity bounds of each feature, they provide valid explanations covering a substantial percentage of the area of inflated explanations at a fraction of the computational cost.
Therefore, \deltaexp s offer an attractive tradeoff between bound-size and runtime that might be crucial in real-time decision support systems or in  applications with limited computational resources, where they aim to find explanations for a large, but not necessarily maximal, feature intervals.
Note that the coverage of \deltaexp s increases monotonically with $\delta$ across all datasets without any computational overhead.
One can thus disregard the computational cost in choosing $\delta$ and instead choose it according to the needs of the application in question, or to maximize the are covered while still finding enough explanations.

\subsection{Evaluation of Group Explanations}

To the best of our knowledge, there is no standard dataset or approach to evaluate group abductive explanations, so we evaluate the quality of \gaxp s with the so-called weighted FFA~\citep{yu_2023}.
The weighted FFA score of a feature for a sample is, intuitively, the proportion of explanations in which this feature occurs in, weighted so that shorter explanations have additional importance.
To compute the FFA, we generate all possible explanations with the MARCO algorithm~\citep{liffiton_2015,DBLP:conf/sat/IgnatievS21} and compute the weighted FFA~\citep{yu_2023}.
We compare the FFA scores generated using \gaxp s to the FFA scores from each individual sample in the group, and evaluate how close they are.

To put the result in context, we provide the same evaluation for three further approaches: ``Aggregated group FFA'', SHAP, and LIME.
The Aggregated group FFA is the brute-force approach of computing all explanations for each sample separately and taking the mean of the FFA score for each feature.
As metrics, we first use the \emph{feature importance error}, which is the average normalized mean absolute error between the group-FFA and the FFA of each sample in the group.
We normalize the mean absolute error to provide a scale-invariant measure of accuracy, and we do it by dividing the mean absolute error by the mean of the target feature importance values.
Secondly, we evaluate the \emph{feature importance correlation}, which is the average spearman correlation between the ranking induced by the group-FFA and the ranking induced by the FFA of each sample in the group.
We compute the ranking by ordering features according to their FFA score.
We also compare our approach to the heuristic explanation methods SHAP and LIME by ordering the features in the order of importance given by SHAP and LIME and computing the correlation between their order and the ranking induced by the FFA of each sample.
For SHAP and LIME, we do not compare the feature importance scores given by their FFA since the scores themselves are not commensurable.

\begin{table}[!t]
    \centering
    \begin{tabular}{lcccccccccccc}
        & \multicolumn{2}{c}{\textbf{Diabetes}} & \multicolumn{2}{c}{\textbf{Glass}} & \multicolumn{2}{c}{\textbf{Leaf}} & \multicolumn{2}{c}{\textbf{Parkinsons}} & \multicolumn{2}{c}{\textbf{Wine}} & \multicolumn{2}{c}{\textbf{Synthetic}} \\
        & \textbf{Err.} & \textbf{Cor.} & \textbf{Err.} & \textbf{Cor.} & \textbf{Err.} & \textbf{Cor.} & \textbf{Err.} & \textbf{Cor.} & \textbf{Err.} & \textbf{Cor.} & \textbf{Err.} & \textbf{Cor.} \\
        \toprule
        Aggr. & \textbf{.232} & \textbf{.727} & \textbf{.202} & .563 & \textbf{.095} & .817 & \textbf{.109} & .907 & \textbf{.114} & \textbf{.852} & \textbf{.176} & \textbf{.746} \\
        LIME & -- & .493 & -- & .164 & -- & .200 & -- & .482 & -- & .586 & -- & .529 \\
        SHAP & -- & .588 & -- & .355 & -- & .374 & -- & .592 & -- & .525 & -- & .634 \\
        $\delta=.00$ & .277 & .684 & .279 & .712 & \textbf{.098} & \textbf{.855} & .117 & .905 & \textbf{.130} & \textbf{.824} & \textbf{.239} & \textbf{.670} \\
        $\delta=.02$ & \textbf{.271} & .699 & .288 & .712 & .105 & .840 & \textbf{.116} & .907 & .133 & .818 & .240 & .661 \\
        $\delta=.04$ & .272 & .716 & .277 & .706 & .110 & .828 & .117 & .906 & .135 & .813 & .246 & .657 \\
        $\delta=.06$ & .273 & \textbf{.723} & .280 & .725 & .106 & .832 & .118 & .906 & .135 & .811 & .250 & .659 \\
        $\delta=.08$ & .274 & .712 & .271 & \textbf{.746} & .105 & .837 & .118 & .906 & .135 & .810 & .254 & .664 \\
        $\delta=.10$ & .276 & .708 & .258 & .738 & .106 & .827 & .119 & .907 & .138 & .817 & .256 & .664 \\
        $\delta=.12$ & .292 & .695 & .258 & .738 & .107 & .826 & .119 & .907 & .138 & .819 & .257 & .655 \\
        $\delta=.14$ & .284 & .720 & \textbf{.250} & .739 & .111 & .828 & .121 & .908 & .134 & .820 & .259 & .655 \\
        $\delta=.16$ & .286 & .702 & .253 & .737 & .112 & .829 & .117 & \textbf{.910} & .134 & .817 & .260 & .652 \\
        $\delta=.18$ & .278 & .701 & .265 & .729 & .111 & .818 & .121 & .902 & .137 & .821 & .261 & .640 \\
        \bottomrule
    \end{tabular}
    \caption{Average feature importance error (Err.) and ranking correlation (Cor.) across 10 different groups for aggregated explanations (Aggr.), SHAP, LIME, and \gaxp s for $\delta \in \{.00, .02, ..., .18\}$. We denote in bold the best scores overall and among \gaxp s.}
    \label{tab:cluster_feature_importance_error_and_correlation}
\end{table}

In Table \ref{tab:cluster_feature_importance_error_and_correlation} we see that \gaxp s are more accurate than SHAP and LIME. The FFA from \gaxp s have at least 10 percentage points higher correlation to the ground-truth than SHAP on every dataset except the synthetic, and on some datasets more than 30 percentage points higher, while they have 10 to 40 percentage points higher correlation than LIME.
The brute-force aggregation approach generally produces a lower error and higher correlation than \gaxp, but in most cases not by much.
In some cases \gaxp s produce \emph{higher} correlation than the aggregation approach, which is notable since the aggregation considers (and thus has to compute) all explanations for all samples separately, while computing \gaxp s is much more tractable.

Figure~\ref{fig:cluster_scalability} illustrates the computational advantage of \gaxp s.
We report three runtime measurements.
`Individual' refers to the mean runtime for computing all explanations for each individual sample, `Group' refers to the mean runtime for computing all \gaxp s across all values of $\delta$ (as listed in Table~\ref{tab:cluster_feature_importance_error_and_correlation}), and `Aggregated' refers to the mean runtime for the naive aggregation method that computes all explanations for all samples in a group.
In the plot on the left, we see that \gaxp s are an order of magnitude faster to compute than aggregated explanations, and that, as the number of features increases, the runtime of the naive aggregation method increases faster than the runtime of \gaxp s.
We also see that the runtime of \gaxp s is similar to the runtime of computing all explanations for a single sample (compare Group to Individual).
In the plot on the right, we see that the runtime of \gaxp s remains constant as the group size increases, while the runtime of aggregated explanations increases linearly.
    Regarding the runtime of computing individual explanations (right side of Figure~\ref{fig:cluster_scalability}), the mean runtime is 183 seconds and the standard deviation is 1384 seconds (note that these are independent of the group size).
This points to another benefit of \gaxp s: they seem to avoid some outliers with a huge number of explanations that make computing individual explanations slower, as can be seen by the fact that computing all \gaxp s takes well below 100 seconds for each group size.

\begin{figure}[!t]
    \centering
    \begin{minipage}{0.48\textwidth}
    \includegraphics[width=\textwidth]{../figures/cluster/scalability/number_of_features/computation_time.pdf}
\end{minipage}
    %\label{fig:cluster_scalability_computation_time_by_num_features}
    \centering
    \begin{minipage}{0.48\textwidth}
    \includegraphics[width=\textwidth]{../figures/cluster/scalability/cluster_size/computation_time.pdf}
    \end{minipage}
    \caption{Average runtime in seconds (log scale) to generate \gaxp s. On the left, we evaluate groups of 10 samples on synthetic datasets with different numbers of features (10-25). On the right, we evaluate groups of different sizes (3-20 samples) on synthetic datasets with 10 features. To avoid variance, each point represents the average of 25 instances (5 groups × 5 datasets).}
    \label{fig:cluster_scalability}
\end{figure}

\paragraph{Summary of the evaluation}

\deltaexp s achieve up to 20--50\% of the coverage of inflated explanations and are 4 to 30 times faster to compute.
The FFA scores induced by \gaxp s are comparable to (and sometimes better than) the FFA scores induced by the aggregated method, while \gaxp s are an order of magnitude faster to compute than aggregated explanations.
Furthermore, the runtime of \gaxp s is independent of the size of the groups of samples, and it scales similarly to the runtime of individual explanations with respect to dataset dimensionality.

\subsection{Limitations}

A limitation of \deltaexp s is that, with larger values for $\delta$, there might be only a very limited number of explanations, or no explanations at all.
In Figure~\ref{fig:robustness} (left), for example, less than 50\% of samples have an explanation for some datasets with $\delta=0.03$.
There is an inherent trade-off between the size $\delta$ and number of explanations.
Additionally, the possible bounds for different features might be lopsided, and in that case finding close to maximal values for $\delta$ might be difficult and one would need to revert to computing inflated explanations.

A limitation of \gaxp s is that, in our experiments, \gaxp s are not formally valid for each individual sample in the group.
This is because, in our datasets, the features within a group vary by more than the values of $\delta$ under which some formal explanations can still be found.
This limitation might limit the usefulness of \gaxp s in some applications, although our evaluation based on FFA values shows that the given explanations correctly identify features that are important for the obtained classification of the samples in a group.
To overcome this limitation, one could consider dividing groups into smaller subgroups of more similar samples, or even combining \deltaexp s with a distance restriction on the irrelevant features~\citep{DBLP:conf/kr/IzzaH0PI024}, since the lower the distance restriction, the easier it is to find an explanation.

A limitation of our empirical evaluation is that, in the evaluation of \deltaexp s against inflated explanations, we compute a single inflated explanation among multiple possible ones (evaluating the features in a different order could produce a different inflated explanation), so an inflated explanation with more volume might this exist.
Very recently, it was shown that computing a maximum-size inflated explanation is much more computationally expensive than computing an arbitrary inflated explanation~\citep{DBLP:journals/corr/abs-2505-10991}, so \deltaexp s would cover less area compared to a maximum inflated explanation, but they would be even faster.

\section{Conclusion}

We introduced \deltaexp s, extending robustness analysis from counterfactual to abductive explanations in a way that complements existing robustness notions for abductive explanations such as inflated and distance-restricted explanations.
The notion of \deltaexp\ defines formal abductive explanations that are robust to input variations, making them more practical for real-world applications with measurement noise, such as healthcare, finance, and autonomous systems.
Furthermore, we introduced \gaxp s, semi-local explanations that explain groups of samples instead of individual samples.
Our empirical evaluation demonstrated that \deltaexp s achieve 20--50\% of the area coverage of inflated explanations for relevant features while requiring 4 to 30 times less runtime.
We also demonstrated that \gaxp s capture commonly important features within groups of samples, achieving an accuracy comparable to a brute-force method that aggregates all explanations for all samples in the group separately, but at a fraction of the computational cost.

\section{Reproducibility Statement}
\todo{see \url{https://iclr.cc/Conferences/2026/AuthorGuide}}

\bibliography{references}

\appendix
\section{Appendix}

\citep{wu_2023} proposed a sensitivity-based ordering to compute small, formal explanations. The idea of their approach is to sort features by how much they influence the classifier's output. The feature that influences the classifier's output the least is the first feature in the ordering. This approach leads to small explanations since it is easier to remove features to which the classifier is not sensitive. The problem, however, is that the explanations look close to random. To achieve better looking explanations, we smoothed the sensitivity of a feature by averaging it with the sensitivity of its neighbors. We call our approach smoothed sensitivity ordering.

In Figure \ref{fig:ordering_explanation} we can see the explanations generated using different orderings. The VeriX ordering produces explanations that are small but are unintuitive, while the smoothed sensitivity ordering produces explanations that are concentrated towards the center and capture the shape of the digit better.

\begin{figure}[!t]
    \centering
    \begin{minipage}{0.2\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/cluster/examples/individual_explanation_mnist_label_5_sample_1_ordering_linear.pdf}
    \end{minipage}
    \hfill
    \begin{minipage}{0.2\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/cluster/examples/individual_explanation_mnist_label_5_sample_1_ordering_random.pdf}
    \end{minipage}
    \hfill
    \begin{minipage}{0.2\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/cluster/examples/individual_explanation_mnist_label_5_sample_1_ordering_verix.pdf}
    \end{minipage}
    \hfill
    \begin{minipage}{0.2\textwidth}
        \centering
        \includegraphics[width=\textwidth]{../figures/cluster/examples/individual_explanation_mnist_label_5_sample_1_ordering_sensitivity.pdf}
    \end{minipage}
    \caption{The first explanation generated using different orderings. From left to right, the orderings are: linear, random, VeriX, and smoothed sensitivity. The green pixels represent the relevant features in the explanations.}
    \label{fig:ordering_explanation}
\end{figure}

\end{document}
