SlowGuess/NeurIPS_2023_OCR huggingface dataset:
SHAP-1Q: Unified Approximation of any-order Shapley Interactions
Fabian Fumagalli*
Bielefeld University, CITEC
D-33619, Bielefeld, Germany
ffumagalli@techfak.uni-bielefeld.de
Maximilian Muschalik*
LMU Munich, MCML Munich
D-80539, Munich, Germany
maximilian.muschalik@ifi.lmu.de
Patrick Kolpaczki
Paderborn University
D-33098, Paderborn, Germany
patrick.kolpaczki@upb.de
Eyke Hullermeier
bhammer@techfak.uni-bielefeld.de
Predominately in explainable artificial intelligence (XAI) research, the Shapley value (SV) is applied to determine feature attributions for any black box model. Shapley interaction indices extend the SV to define any-order feature interactions. Defining a unique Shapley interaction index is an open research question and, so far, three definitions have been proposed, which differ by their choice of axioms. Moreover, each definition requires a specific approximation technique. Here, we propose SHAPley Interaction Quantification (SHAP-IQ), an efficient sampling-based approximator to compute Shapley interactions for arbitrary cardinal interaction indices (CII), i.e. interaction indices that satisfy the linearity, symmetry and dummy axiom. SHAP-IQ is based on a novel representation and, in contrast to existing methods, we provide theoretical guarantees for its approximation quality, as well as estimates for the variance of the point estimates. For the special case of SV, our approach reveals a novel representation of the SV and corresponds to Unbiased KernelSHAP with a greatly simplified calculation. We illustrate the computational efficiency and effectiveness by explaining language, image classification and high-dimensional synthetic models.
Feature attributions are a prevalent approach to interpret black box machine learning (ML) models [2, 7, 26]. However, in many real-world applications, such as understanding drug-drug interactions, mutational events or complex language models, quantifying interactions between features is essential, too [48, 23, 43]. Feature interactions provide a more comprehensive explanation, which can be seen as an enrichment of feature attributions [3, 39, 40]. While feature attributions quantify the contribution of single features to the model's prediction or performance, feature interactions quantify the contribution of a group of features to the model's prediction or performance.
Figure 1: Interaction scores for a movie review excerpt presented to a sentiment analysis model.
In this work, we are interested in feature interactions that make use of the Shapley value (SV) and its extension to Shapley interactions. The SV is a concept from cooperative game theory that has been used, apart from feature attributions [7], as a basis for many Shapley-based explanations [21, 15, 49]. It distinguishes itself through uniqueness given a set of intuitive axioms. A number of approaches extend Shapley-based explanations to feature interactions [17, 3, 39, 40]. Yet, in contrast to the SV, a "natural" extension of the intuitive set of axioms for a unique Shapley interaction index is less clear. Moreover, its efficient computation is challenging and, so far, approximation approaches are specifically tailored to the particular definition.
In this paper, we consider a more general class of interaction indices, known as cardinal interaction indices (CII) [17], which covers all currently proposed definitions and all other that satisfy the (generalized) linearity, symmetry and dummy axiom. We present SHAPley Interaction Quantification (SHAP-IQ), a sampling-based unified approximation method. It is substantiated by mathematical guarantees and can be applied to any CII to approximate any-order interaction scores efficiently.
Contribution. Our main contributions include:
- We consider a general form of interaction indices, known as CII (Definition 3.3) and establish a novel representation (Theorem 4.1), which we utilize to construct SHAP-IQ (Definition 4.2), an efficient sampling-based estimator.
- We show that SHAP-IQ is unbiased, consistent and provide a general approximation bound (Theorem 4.3). We further prove that SHAP-IQ maintains the efficiency condition for n-Shapley Values [3] and the Shapley Taylor Interaction Index [39] (Theorem 4.7).
- For the SV, we find a novel representation (Theorem 4.4). We further prove that SHAP-IQ is linked to Unbiased KernelSHAP [8] (Theorem 4.5) and greatly simplifies its representation.
- We use SHAP-IQ to compute any-order n-Shapley Values on different ML models and demonstrate that it outperforms existing baseline methods. We further contrast different existing CIs and compare SHAP-IQ to the corresponding baseline approximation method.
The Shapley Interaction Index (SII) [17], its efficiency preserving aggregation as n-Shapley Values (n-SII) [3], the Shapley Taylor Interaction (STI) [39] and the Faithful Shapley Interaction Index (FSI) [40] offer different ways of extending the SV to interactions, which extend on the linearity, symmetry and dummy axiom to provide a uniquely defined interaction index. SII and STI extend on axiomatic properties of the weighted sum for SV [35], whereas FSI extends on the axiomatic properties of the Shapley interaction as the solution to a weighted least square solution [32, 33].
2The shapiq package extends on the well-known shap library and can be found at https://pypi.org/project/shapiq/.
In the field of cooperative game theory, interactions have also been studied from a theoretical perspective as the solution of a weighted least square problem and the sum of marginal contributions with constant weights, which both yield a generalized Banzhaf value [18, 16].
In the ML community, interactions of features have been studied from a practical perspective for text [29] and image [41] data, for specific models, such as neural networks [42, 10, 36, 20] or tree based models [25]. Other concepts of interactions have been discussed in [43] using marginal contributions, from a statistical perspective with functional decomposition [28] and improved white box models with interaction terms [24].
To approximate SII and STI, a permutation-based method [39, 40], as an extension of ApproShapley [5], was suggested, whereas FSI relies on a kernel-based approximation, similar to KernelSHAP [26], which utilizes a representation of the SV as the solution of a weighted least square problem [6]. Unlike these specific approaches, we consider general CIs, which subsume all of the above mentioned measures, and we propose a generic approximation technique, which can be accompanied by mathematical guarantees. In case of the SV, our approximation of the CII is related to Unbiased KernelSHAP [8], which is a variant of KernelSHAP [26]. It is further related to stratified sampling approximations for the SV and our sampling approach can be seen as flexible framework to find the optimum allocation for each stratum [4].
3 The Cardinal Interaction Index (CII) and Shapley-based Explanations
In this section, we review Shapley-based explanations and introduce the CII, which we aim to approximate in Section 4. We further introduce existing baseline methods for specific CIIs and Unbiased KernelSHAP for the SV, which is linked to our proposed method.
Notations. We refer to the model behavior on a set of features  $D = \{1, \dots, d\}$  as a function  $\nu: \mathcal{P}(D) \to \mathbb{R}$ , where  $\mathcal{P}(D)$  refers to the power set of  $D$ . We denote  $\nu_0(T) := \nu(T) - \nu(\emptyset)$ , which is the default setting in game theory and also known as set function [17, 14]. The subsets  $S \subseteq D$  refer to the set of features (or players in game theory) of which the interaction is computed, where we use lower case letters for the cardinality, i.e.  $s := |S|$ . The maximum order interaction of interest is denoted with  $s_0$  and we use the set  $\mathcal{T}_k := \{T \subseteq D : k \leq t \leq d - k\}$  and the set of interactions  $S_{s_0} := \{S \subseteq D \mid s \leq s_0, S \neq \emptyset\}$ . For a subset  $T \subseteq D$ , we refer to the binary representation as  $Z_T = (z_1, \ldots, z_d) \in \{0, 1\}^d$  with  $z_i = \mathbf{1}(i \in T)$  for  $i = 1, \ldots, d$ , where  $\mathbf{1}$  refers to the indicator function. We further denote the Shapley kernel [6, 26] as  $\mu(t) := \frac{1}{d-1} \binom{d-2}{t-1}^{-1}$ .
Removal-based explanations [9] consider a model that is trained on  $d$  features, where the goal is to examine a model behavior that is defined on a subset of features. The model behavior  $\nu$  for a subset of features could, for instance, be a particular model prediction for one input (local explanation) or an overall measure of model performance (global explanation), if only this subset of features is known [9]. To quantify the contribution for individual features, the change in model behavior is evaluated, if the feature is removed from the model. To restrict a ML model on a subset of features, different feature removal techniques have been proposed, such as marginalization of features or retraining the model [9]. To quantify the impact of a single feature  $i \in D$  on the model behavior  $\nu$  it is then intuitive to compute the difference  $\delta_{\{i\}}^{\nu}(T) = \nu(T \cup \{i\}) - \nu(T)$  for subsets  $T \subseteq D \setminus \{i\}$ . For a distinct pair of features  $(i,j)$  with  $i,j \in D$ , a natural extension is  $\delta_{\{i,j\}}^{\nu}(T) = \nu(T \cup \{i,j\}) - \nu(T) - \delta_{\{i\}}^{\nu}(T) - \delta_{\{j\}}^{\nu}(T)$  for  $T \in D \setminus \{i,j\}$ , i.e. subtracting the contribution of single features from the joint impact of both features. The following definition generalizes this recursion and is known as discrete derivative or  $S$ -derivative [14].
Definition 3.1 (Discrete Derivative [14]). For  $S \subseteq D$  the S-derivative of  $\nu$  at  $T \subseteq D \setminus S$  is
$$
\delta_ {S} ^ {\nu} (T) := \sum_ {L \subseteq S} (- 1) ^ {s - l} \nu (T \cup L).
$$
To obtain an attribution score, the marginal contributions on different subsets  $T \subseteq D \setminus S$  are aggregated using a specific summary technique. In this work, we are interested in the approximation and extension of one particular summary technique for single features  $i \in D$ , called the Shapley value [35], independent of model behavior and feature removal.
Definition 3.2 (Shapley Value (SV) [35]). The SV is  $I^{SV}(i) = \sum_{T \subseteq D \setminus \{i\}} \frac{(d - t - 1)!t!}{d!} \delta_{\{i\}}^{\nu}(T)$ ,  $i \in D$ .
The SV is the unique attribution method that fulfills the axioms: symmetry (attributions are independent of feature ordering), linearity (in terms of the model behavior  $\nu$ ), dummy (if a feature does not change  $\nu$  then its attribution is zero) and efficiency (the sum of attributions are equal to  $\nu_0(D)$ ) [35]. However, the SV does not give any information about the interactions between two or more features. A suitable extension of the SV for interactions of features  $S \subseteq D$  remains an open question, as different axiomatic extensions have been proposed [17, 39, 40]. In this work, we thus consider a broad class of interaction indices, known as CIIs [17, 40], which subsumes popular choices.
Definition 3.3 (Cardinal Interaction Index (CII) [17]). A CII is an interaction index of the form
$$
I ^ {m} (S) := \sum_ {T \subseteq D \backslash S} m _ {s} (t) \delta_ {S} ^ {\nu} (T) \text {w i t h w e i g h t s} m _ {s} (t) \text {f o r} s = 1, \dots , s _ {0} \text {a n d} t = 0, \dots , d - s.
$$
Remark 3.4. It was shown that every interaction index satisfying the generalized linearity, symmetry and dummy axioms can be represented as a CII [17]. If  $\sum_{t=0}^{d-s}\binom{d-s}{t}m_s(t)=1$ , then the CII is also referred to as a cardinal-probabilistic interaction index (CPII) [14].
In this paper, we present a unified approximation technique for arbitrary CIIs.
3.1 Shapley Interaction Index (SII) and other CIIIs
In the following, we introduce prominent examples of CIIs. For further details on the axioms and exact definitions, we refer to the appendix. The SII [17] is a direct extension of the SV, that relies on an additional recursive axiom to obtain a unique CII.
Definition 3.5 (Shapley Interaction Index (SII)[17]). The SII is a CII defined as
$$
I ^ {S I I} (S) := \sum_ {T \subseteq D \backslash S} m _ {s} ^ {S I I} (t) \delta_ {S} ^ {\nu} (T) a n d m _ {s} ^ {S I I} (t) := \frac {(d - t - s) ! t !}{(d - s + 1) !}.
$$
It has been shown that the SII is a CPII [14]. In contrast to the SV, the SII does not fulfill the efficiency axiom, which is a desirable property in the context of ML. Therefore an extension of SII, as well as other interaction indices have been proposed.
n-Shapley Values (n-SII) and other interaction indices. The efficiency axiom for interaction indices of maximum interaction order  $1 \leq s_0 \leq d$  requires that the sum of  $I^m(S)$  up to order  $s_0$  equals  $\nu_0(D)$ .
Definition 3.6 (Efficiency [39, 40]). A CII is efficient of order  $s_0$ , if  $\sum_{S \in S_{s_0}} I^m(S) = \nu_0(D)$ , where  $S_{s_0}$  is the set of interactions up to order  $s_0$ .
In [3], an aggregation of SII was proposed to obtain n-SII  $I_{s_0}^{\mathrm{n - SII}}(S)$  of order  $s_0$  that satisfies efficiency. Other axiomatic approaches directly require efficiency together with the linearity, symmetry and dummy axioms, and omit the recursive axiom of SII. However, in contrast to the SV, this axiom alone does not yield a unique interaction index [39, 40]. The STI [39] requires the efficiency axiom and an additional interaction distribution axiom. On the other hand, the FSI [40] requires the efficiency axiom and the faithfulness property, that relates the interaction index to a solution of a constrained weighted least square problem. The choice of axioms of SII (n-SII), STI and FSI yield a unique interaction index that reduces to the SVs for  $s_0 = 1$ . For FSI, it was shown that the top-order interactions define a CPII [40, Proposition 21], which is also easily verified for STI. All orders of interactions of FSI and STI can in general be represented as a CII, as they fulfill the linearity, symmetry and dummy axioms [17, Proposition 5]. However, it was noted that for FSI a simple closed-form solution for lower-order interactions in terms of discrete derivatives remains unclear [40, Lemma 70].
3.2 Baseline Approximations of SII, STI and FSI.
By definition, the number of evaluations of  $\nu$  in  $I$ , which constitutes the limiting factor in ML, grows exponentially with  $d$  and thus, in practice, approximation methods are required. Currently, there does not exist an approximation for the general CII definition, as each index (SII, STI, FSI) requires a specifically tailored technique. Approximations of CII can be distinguished into permutation-based approximation (SII and STI) and kernel-based approximation (FSI). Both extend on existing methods
for the SV, namely permutation sampling [5] for SII and STI, and KernelSHAP [26] for FSI. For a comprehensive overview of the original SV methods, we refer to the appendix. We now briefly discuss existing approaches, which will be used as baselines in our experiments.
Permutation-based (PB) Approximation for STI and SII [39, 40]. The permutation-based (PB) approximation computes estimates of SII and STI based on a representation of uniformly sampled random permutations  $\pi \sim \mathrm{unif}(\mathfrak{S}_D)$ , where  $\mathfrak{S}_D$  is the set of all permutations, i.e. the set of all ordered sequences of the elements in  $D$ . Then,
$$
I ^ {\mathrm {S I}} (S) = \mathbb {E} _ {\pi \sim \operatorname {u n i f} (\mathfrak {S} _ {D})} \left[ \mathbf {1} (S \in \pi) \delta_ {S} ^ {\nu} \left(u _ {S} ^ {-} (\pi)\right) \right] \text {a n d} I ^ {\mathrm {S T I}} (S) = \mathbb {E} _ {\pi \sim \operatorname {u n i f} (\mathfrak {S} _ {D})} \left[ \delta_ {S} ^ {\nu} \left(u _ {S} ^ {-} (\pi)\right) \right].
$$
Here,  $u_{S}^{-}(\pi)$  refers to the set of indices in  $\pi$  preceding the first occurrence of any element of  $S$  in  $\pi$  and  $S \in \pi$  is fulfilled, if all elements of  $S$  appear as a consecutive sequence in  $\pi$ . The estimators for SII and STI then compute an approximation by Monte Carlo integration by sampling  $\pi \sim \mathrm{unif}(\mathfrak{S}_D)$ .
Kernel-based (KB) Approximation for FSI [40, 8]. Kernel-based (KB) approximation estimates FSI based on the representation of  $I$  as a solution to a constrained weighted least square problem
$$
I ^ {\mathrm {F S I}} = \underset {\beta \in \mathbb {R} ^ {d s _ {0}}} {\arg \min } \mathbb {E} _ {T \sim p (T)} [ (\nu (T) - \sum_ {S \in \mathcal {S} _ {s _ {0}}} \beta (S)) ^ {2} ] \text {s . t .} \sum_ {S \in \mathcal {S} _ {s _ {0}}} \beta (S) = \nu (D) \text {a n d} \beta (\emptyset) = \nu (\emptyset), \tag {1}
$$
where  $p(T) \propto \mu(t)$ , is a probability distribution over  $\mathcal{T}_1$  and  $d_{s_0} \coloneqq |\mathcal{S}_{s_0}|$ . KB approximation for FSI estimates the expectation using Monte Carlo integration by sampling from  $p(T)$  and solves the approximated least-squares problem explicitly, similar to KernelSHAP [26, 8, 40]. For more details and pseudo code, we refer to the appendix.
3.3 Unbiased KernelSHAP (U-KSH) for the SV
U-KSH constitutes a variant of KernelSHAP (KSH) [26], which relies on KB approximation for the SV. In contrast to KSH, U-KSH is theoretically well understood and it was shown that the estimator is unbiased and consistent [8]. U-KSH finds an exact solution to (1) with  $s_0 = 1$  as
$$
I ^ {\mathrm {S V}} = A ^ {- 1} \left(b - \mathbf {1} \frac {\mathbf {1} ^ {T} A ^ {- 1} b - \nu_ {0} (\mathbf {1})}{\mathbf {1} ^ {T} A ^ {- 1} \mathbf {1}}\right) \text {w h e r e} A := \mathbb {E} [ Z Z ^ {T} ], b = \mathbb {E} [ Z \nu_ {0} (Z) ] \text {a n d} p (Z) \propto \mu (t).
$$
U-KSH then approximates this solution using Monte Carlo integration.
Definition 3.7 (Unbiased KernelSHAP (U-KSH) [8]). Given  $T_{1},\ldots ,T_{K}\sim p(T)\propto \mu (t)$  with binary representation  $Z_{1},\dots,Z_{K}\in \{0,1\}^{d}$ ,  $U$ -KSH is defined as
$$
\hat {I} _ {U} ^ {S V} := A ^ {- 1} \left(\hat {b} - \mathbf {1} \frac {\mathbf {1} ^ {T} A ^ {- 1} \hat {b} - \nu_ {0} (\mathbf {1})}{\mathbf {1} ^ {T} A ^ {- 1} \mathbf {1}}\right), w h e r e \hat {b} := \frac {1}{K} \sum_ {k = 1} ^ {K} Z _ {k} \nu_ {0} (Z _ {k}).
$$
The main idea of U-KSH is that  $A$  can be computed explicitly independent of  $\nu$  and only  $b$  has to be estimated [8]. By linking U-KSH to our method (Theorem 4.5), we will show that  $\hat{I}_U^{\mathrm{SV}}$  can be greatly simplified to a weighted sum.
4 SHAP-IQ: Unified Approximation of any-order CII
So far, there exists no unified approximation technique for the general CII. In particular, it is unknown if existing approximation techniques, such PB and KB, generalize to other indices [39, 40, 13]. Furthermore, PB approximation for SII and STI is very inefficient as each update of all estimates requires a significant number of model evaluations. KB approximation for FSI efficiently computes estimates, where one model evaluation can be used to update all interaction scores. It is, however, impossible to compute only a selection of interaction estimates and theoretical results for the estimator are difficult to establish. In the following, we introduce SHAP-IQ (Section 4.1), a unified sampling-based approximation method that can be applied to any CII. SHAP-IQ is based on a Monte Carlo estimate of a novel representation of the CII and well-known statistical results are applicable. In the special case of SV, we find a novel representation of the SV and show that SHAP-IQ is linked to
U-KSH (Section 4.2). SHAP-IQ therefore greatly reduces the computational complexity of U-KSH. We further show (Section 4.3), that the sum of interaction estimates of SHAP-IQ remains constant and therefore maintains the efficiency property for STI and SII. Interestingly, for FSI this property does not hold, which should be investigated in future research. All proofs can be found in the appendix.
4.1 SHAPley Interaction Quantification (SHAP-IQ)
A key challenge in approximating the CII efficiently is that the sum changes for every interaction subset  $S$ . We thus first establish a novel representation of the CII. Based on this representation, we construct SHAP-IQ, an efficient estimator of the CII. We show that SHAP-IQ is unbiased, consistent and provide a general approximation bound.
Our novel representation of the CII is defined as a sum over all subsets  $T \subseteq D$ . In previous works, it was shown that such a representation does exist for games with  $\nu(\emptyset) = 0$ , if the linearity axiom is fulfilled [17, Proposition 1]. We now explicitly specify this representation and show that the weights, for a CII, only depend on the sizes of  $T$  and the intersection  $T \cap S$ .<sup>3</sup>
Theorem 4.1. It holds  $I^{m}(S) = \sum_{T\subseteq D}\nu_{0}(T)\gamma_{s}^{m}(t,|T\cap S|)$  with  $\gamma_s^m (t,k)\coloneqq (-1)^{s - k}m_s(t - k)$ .
Theorem 4.1 yields a novel representation of the CII, where the model evaluations  $\nu_0(T)$  are independent of  $S$ . This allows to utilize every model evaluation to compute all CII scores simultaneously by properly weighting with  $\gamma_s^m$ . Notably, our representation relies on  $\nu_0$  instead of  $\nu$ , which constitutes an important choice for approximation, on which we elaborate in the appendix.
To approximate  $I$ , we introduce a sampling order  $k_{0} \geq s_{0}$ , for which we split the sum in Theorem 4.1 to subsets with  $T \in \mathcal{T}_{k_0}$  and  $T \notin \mathcal{T}_{k_0}$  and rewrite
$$
I ^ {m} (S) = c _ {k _ {0}} (S) + \mathbb {E} _ {T \sim p _ {k _ {0}} (T)} \left[ \nu_ {0} (T) \frac {\gamma_ {s} ^ {m} (t , | T \cap S |)}{p _ {k _ {0}} (T)} \right] \text {w i t h} c _ {k _ {0}} (S) := \sum_ {T \notin \mathcal {T} _ {k _ {0}}} \nu_ {0} (T) \gamma_ {s} ^ {m} (t, | T \cap S |),
$$
where  $p_{k_0}$  is over  $\mathcal{T}_{k_0}$ . SHAP-IQ then estimates the CII by Monte Carlo integration.
Definition 4.2 (SHAP-IQ). The Shapley Interaction Quantification (SHAP-IQ) of order  $k_{0}$  with  $K$  samples is
$$
\hat {I} _ {k _ {0}} ^ {m} (S) := c _ {k _ {0}} (S) + \frac {1}{K} \cdot \sum_ {k = 1} ^ {K} \nu_ {0} (T _ {k}) \frac {\gamma_ {s} ^ {m} (t _ {k} , | T _ {k} \cap S |)}{p _ {k _ {0}} (T _ {k})} w i t h T _ {1}, \ldots , T _ {K} \sim p _ {k _ {0}} (T).
$$
SHAP-IQ is outlined in the appendix and we establish the following important theoretical guarantees.
Theorem 4.3. SHAP-IQ is unbiased,  $\mathbb{E}\left[\hat{I}_{k_0}^m (S)\right] = I^m (S)$ , and consistent,  $\hat{I}_{k_0}^m (S)\stackrel {K\to \infty}{\rightarrow}I^m (S)$ . With  $\sigma^2 (S)\coloneqq \mathbb{V}\left[\nu_0(T)\frac{\gamma_s^m(|T|,|T\cap S|)}{p_{k_0}(T)}\right]$  and  $\epsilon >0$ , it holds  $\mathbb{P}(|\hat{I}_{k_0}^m (S) - I^m (S)| > \epsilon)\leq \frac{1}{K}\frac{\sigma^2(S)}{\epsilon^2}$ .
SHAP-IQ provides efficient estimates of all CII scores with important theoretical guarantees. The sample variance  $\hat{\sigma}^2$  can further be used for statistical analysis of the estimates.
Finding the sampling order  $k_{0}$  and distribution  $p_{k_{0}}$ . In line with KSH and U-KSH [26, 8], we find  $k_{0}$  in an iterative procedure, outlined in the appendix. We consider sampling weights  $q(t) \geq 0$  for  $0 \leq t \leq d$  that grow symmetrically towards the center and consider a distribution  $p_{k_{0}}(T) \propto q(t)$ . Given a budget  $M$  and initial  $k_{0} = 0$ , we consider  $I^{m}(S) = \mathbb{E}_{T \sim p_{k_{0}}(T)}\left[\nu_{0}(T)\frac{\gamma_{s}^{m}(t,|T \cap S|)}{p_{k_{0}}(T)}\right]$  and iteratively increase  $k_{0}$ , if for a subset  $T$  of size  $k_{0}$  and  $d - k_{0}$ , the condition  $M \cdot p_{k_{0}}(T) \geq 1$  is fulfilled. The budget is then decreased by the number of subsets of that size, i.e.  $2\binom{d}{k_0}$ . This essentially verifies iteratively, if the expected number of subsets exceeds the total number of subsets. For more details and possible choices of sampling weights  $q$ , we refer to the appendix.
Computational Complexity. In contrast to PB approximations, SHAP-IQ allows to iteratively update all interaction estimates with one single model evaluation for any-order interactions. The