Prediction via Shapley Value Regression
Authors: Amr Alkhatib
Roman Bresson
Henrik Boström
Michalis Vazirgiannis
Venue: ICML 2025 poster (2025)
Keywords: Explainability
Shapley Value
Primary area: social_aspects->accountability_transparency_and_interpretability

Abstract: Shapley values have several desirable, theoretically well-supported, properties for explaining black-box model predictions. Traditionally, Shapley values are computed post-hoc, leading to additional computational cost at inference time. To overcome this, a novel method, called ViaSHAP, is proposed, that learns a function to compute Shapley values, from which the predictions can be derived directly by summation. Two approaches to implement the proposed method are explored; one based on the universal approximation theorem and the other on the Kolmogorov-Arnold representation theorem. Results from a large-scale empirical investigation are presented, showing that ViaSHAP using Kolmogorov-Arnold Networks performs on par with state-of-the-art algorithms for tabular data. It is also shown that the explanations of ViaSHAP are significantly more accurate than the popular approximator FastSHAP on both tabular data and images.

Prediction via Shapley Value Regression
Amr Alkhatib
Roman Bresson
Henrik Boström Michalis Vazirgiannis 2.3
Abstract
Machine learning explanation methods employ variety of strategies to produce explanations, e.g., the use of local inter- pretable surrogate models (Ribeiro et al., 2016), generation of counterfactual examples (Karimi et al., 2020; Dandl et al., 2020; Mothilal et al., 2020; Van Looveren & Klaise, 2021; Guo et al., 2021; Guyomard et al., 2022), selection of impor- tant features (Chen et al., 2018; Yoon et al., 2019; Jethani et al., 2021), and approximation of Shapley values (Lund- herg & Lee, 2017: Lundberg et al., 2020; Frye al., 2021; Covert & Lee, 2021; Jethani et al., 2022). Methods that gen- erate explanations based on Shapley values are prominent since they offer a unique solution that meets a set of theoret- ically established, desirable properties. The computation of Shapley values can, however, be computationally expensive. Recent work has therefore focused on reducing the running time (Lundberg & Lee, 2017; Lundberg et al., 2020; Jethani et al., 2022) and enhancing the accuracy of approximations (Frye et al., 2021; Aas et al., 2021; Covert & Lee, 2021; Mitchell et al., 2022; Kolpaczki et al., 2024), However, the Shapley values are computed post-hoc, and hence entail a computational overhead, even when approximated, e.g., as in the case of FastSHAP (Jethani et al., 2022). Generat- ing instance-based explanations learning pre-trained explainer always demands further data, time, and resources. Nevertheless, to the best of our knowledge, computing Shap- ley values as a means to form predictions has not yet been considered.
Shapley values have several desirable, theoret- ically well-supported, properties for explaining black-box model predictions. Traditionally, Shap- ley values are computed post-hoc, leading to ad- ditional computational cost at inference time. To overcome this, novel method, called ViaSHAP, is proposed, that learns a function to compute Shapley values, from which the predictions can be derived directly by summation. Two approaches to implement the proposed method are explored; one based the universal approximation theorem and the other on the Kolmogorov-Amon Armol repre- sentation theorem. Results from a large-scale em- pirical investigation are presented, showing that Networks
ViaSHAP using Kolmogorov- performs on par with sfte-off-hhe-u algorithms for tabular data. It is also shown that the explana- tions of ViaSHAP than the popular approximator FAstSHAP on both tabular data and images.
significantly more accurate
1. Introduction
The application of machine learning algorithms in some domains requires communicating the reasons behind predic- tions with the aim of building trust in the predictive models and, more importantly, addressing legal and ethical con- siderations (Lakkaraju et al., 2017; Goodman & Flaxman,
The main contributions of this study are:
a novel machine learning method, ViaSHAP, that trains a model to simultaneously provide accurate predictions and Shapley values
2017). Nevertheless, many state-of-the-an machine learn- ing algorithms result in black-box models, precluding the user's s ability to follow the reasoning behind the predictions. Consequently, explainable machine learning methods have gained notable attention as means acquire needed ex- plainability without sacrificing performance.
multiple implementations of the proposed method are explored, using both the universal approximation theo- rem and the Kolmogorov- representation theo- rem, which evaluated through large-scale empiri- cal investigation
'Örebro University, School Science and Technology, Swe- den *KTH Royal Institute of Technology, School of Electrical Engineering and Computer Science, Sweden "École Polytech- nique, IP Paris, France. Correspondence to: Amr Alkhatib <amr.akkhtib@omm @or.....iio
In the following section, cover fundamental concepts about the Shapley value and, along the way, introduce our notation. Section 3 describes the proposed method. In Section 4, results from a large-scale empirical investigation are presented and discussed. Section 5 provides a brief overview of the related work. Finally, in the concluding
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Prediction via Shapley Value Regression
remarks, we summarize the main conclusions and outline directions for future work.
Property 2. (Missingness): Players without impact on the prediction attributed value of zero. Let N:
4A _ N \ (7), v(S) = u(S (i)) - (1) =0
2. Preliminaries
Property 3. (Consistency)): The Shapley value grows or
2.1. The Shapley Value
remains the same if a player's contribution grows or stays the same. Let v and v' two games over N, let E N:
In game theory, a game in coalitional form is a formal model for a scenario in which players form coalitions, and the game's payoff shared between the coalition members. A coalitional game focuses on the behavior of the players and typically involves a finite set of players N = (1,2,.... ,n) (Manea, 2016). A coalitional game also involves charac- teristic set function v: 2N + IR that assigns a payoff, a real number, to a coalition 5 S N such that: v0) = 0 (Owen, 1995.). Different concepts can be employed to distribute the payoff among the players of a coalitional game to achieve fair and stable allocation. Such solution concepts include the Core, the Nucleolus, and the Shapley Value (Manea, 2016; Ferguson, 2018).
AA S N (1), VSU (i)) (() (S)
2 (()- - ( 2 2(0)
2.2. SHAP
In the context of explainable machine learning, the Shap- ley value is commonly computed post-hoc to explain the predictions of trained machine learning models. Let f be trained model whose inputs are defined on n features and whose output E Y S R. We also define a baseline or neutral instance, noted E X. For a given instance æ, the Shapley value is computed over each feature to explain the difference in output X and the baseline. The baseline may be determined depending on the context, but common examples include the average of all examples in the training set, or one that is commonly used as threshold (Izzo et al., 2021).
The Shapley Value is solution concept that allocates pay- offs to the players according to their marginal contribu- tions across possible coalitions. The Shapley value (v) of player i game v is given by (Shapley, 1953):
[ - | 1)!
6.(0) =
- v(S)).
n!
(CN)(
In this context, a coalitional game for can be derived from the model, where the players are the features, and the value function e represents how the prediction changes as different coalitions of features game, player getting picked for coalition $ means that its corresponding feature's 's value is vt, otherwise it remains at its baseline value 0,-
[S]](= nd
The term
is a combinatorial weighting fac- tor for the different coalitions that be formed for game The difference term (o(SU {i}) v(S) represents the additional value that player i contributes to the coalition S, i... the marginal contribution of player i.
masked out. In this
The Shapley values for this game can then be obtained as the solution of optimization problem. The objective is to determine set of values that accurately represent the marginal contributions of each feature while verifying properties 1 through 3. In the literature, they were obtained by minimizing the following weighted least squares loss function (Marichal & Mathonet, 2011; Lundberg & Lee, 2017; Patel al., 2021):
Given a game v, an additive explanation model is an interpretable approximation of e which can be written as (Lundberg & Lee, 2017; Covert & Lee, 2021):
u(S) = -o(vv= + 6,(w),
VES
with 60(u) constant and (v)) the payoff of player i. is a linear model whose weights are the payoffs of each player. Using the Shapley values as the payoffs is the only solution in the class of additive feature attribution methods that satisfies the following properties (Young, 1985):
[(urs = Σ wiS) (s(()) ((()))))))H) (I)
SCN
where w is a weighting kernel, the choice of the kernel can result in a solution equivalent to the Shapley value (Covert & Lee, 2021; Covert al., 2021). Therefore, (Lundberg Lee, 2017) proposed the Shapley kernel:
Property 1. (Local Accuracy): the solution matches the prediction of the underlying model:
((N) = Σ di(u)) = vNN)
(n 1)
EN
WShap(S)
(2)
(18) ||S| (n -S1)
2

---

Prediction via Shapley Value Regression
Note that, for a d-dimensional output with d 1, each different unidimensional model.
be solved separately for every prediction. Additionally, Ker- neISHAP requires many samples to converge accurate estimations for the Shapley values, and this problem is ex- acerbated with high dimensional data (Covert & Lee, 2021), Consequently, FaPtSHAP (Jethani et al., 2022) has been pro- posed to efficiently learn a parametric Shapley value func- tion and eliminate the need to solve a separate optimization problem for each prediction. The model @tast : X + I.", parameterized by 0 is then trained to produce the Shapley value for an input by minimizing the following loss function:
output is considered That is, each of the d dimensions will define a different game, and thus a different set of n Shapley values. The explanation of the output is thus an n x d matrix of Shapley values, providing the contribution of each input feature to each output game. This can trivially be obtained through the same optimization process by stacking d loss functions such in (1). Thus, we will consider in the following that y be unidimensional unless otherwise specified.
Un (0) - -1 ftast (z; ) ?
2.3. KernelSHAP
[(0) = E E ((2)
(vo(S) P(S)
Computing the exact Shapley values is demanding process as it requires evaluating all possible coalitions of feature val- ues. There are 2n 1 possible coalitions for a model with n features, each of which has to be evaluated to determine the features' marginal contributions, which renders the exact computation of Shapley values infeasible for models with a relatively large number of features. Consequently, (Lund- berg& & Lee, 2017) proposed KernelSHAP as more feasible method approximate the Shapley values. KernelSHAP samples a subset of coalitions instead of evaluating all possi- ble coalitions. The explanation model is learned by solving the following optimization problem (Covert & Lee, 2021; Jethani al., 2022):
(5)
where (zz) is the distribution of the input data, and p(S) is proportional to the Shapley kernel defined in (2). In the case of a multidimensional output, a uniform sampling is done over the possible output dimensions.
The accuracy of Utas in approximating the Shapley value depends on the expressiveness of the model class employed well as the data available for learning @iass as a post-hoc function.
3. ViaSHAP
We introduce ViaSHAP, a method that formulates predic- tions via Shapley values regression. In contrast to the pre- vious approaches, the Shapley values are not computed in post-hoc setup. Instead, the learning of Shapley values is integrated into the training of the predictive model and exploits every data example in the training data. Moreover, unlike (Chen et al., 2023b), ViaSHAP does not impose a specific neural network design or constrain the explanation to a subset of input features, as is in (Wang et al., 2021). At inference time, the Shapley values are used directly to generate the prediction. The following subsections outline how ViaSHAP is trained to simultaneously produce accurate predictions and their corresponding Shapley values.
d(vz) arg min
(vo(S)- (0) ( p(S)
E
- 01 )²
(3)
s.t. 1 02 v (x(())
(4)
where 1s is the mask corresponding to S, i.c., which takes value for features in S and () otherwise, and the distribution p(S) is proportional to the Shapley kernel (2) (Covert & Lee, 2021; Jethani et al., 2022). Equation (4) is referred to as the efficiency constraint. Different value functions (v) can be applied to marginalize features out, such as:
3.1. Predicting Shapley Values
1. Baseline Removal (Sundararajan & Najmi, 2020): (5) = f (=5 E [xM(5]: O)
Let X C R." and Y C Rd, respectively, the input and output spaces, and M = (1......d d} the set of output di- mensions. We define a model SHAP X + Y which, for a given instance x, computes both the Shapley values and the predicted output in single process.
2. Interventional/Marginn. Expectations (Chen et al., 2020): (S) = _ [f
3. Ohservational//Cnnitioon Expectations:
First, Via X + gaxd computes matrix of values pia (z: 9). Then, ViaSHAP predicts the output vector as SHAP (x) 1 &T (2; 0) i.e., summing column-wise. A link function a can be applied to accommodate a valid range of outputs (y = a(1 Dia (z; f)). e.g., the sigmoid function for binary classification or softmax for multi-class classification.
(S) = (x5 (x5 5:5
2.4. FaStSHAP
Although KerneISHAP provides a practical solution for the Shapley value estimation, the optimization problem 3 must
3

---

Prediction via Shapley Value Regression
ent largets y and y/. Then, given a feature E N:
X
Φ Via
Φ
1 TT
Φ
y
SA USCNY Mi), V(x50100) - 2 (xx))
0 (x:"((;;") o² 1/: 6*)
Figure 1. ViaSHAP generates predictions by first estimating the Shapley values, whose summation produces the final outcome.
Theorem 3.4. The global optimizer function Voa 0*) computes the exact Shapley values of the predictions of ViaSHAP (z).
ViaSHAP computes the Shapley values prior to each predic- tion formulation, as illustrated in Figure 1. Similar to Ker- nelSHAP and FaPtSHAP (in equation (3) and equation (5)), Vin is trained by minimizing the weighted least squares loss of the predicted Shapley values, as shown in equation (6). However, no pre-defined black-box model is available beforehand to train the of explainer. Instead, the ViA predictor is provided as a black box at each training step.
Theorem 3.4 directly follows from Lemma 3.1, Lemma 3.2, and Lemma 3.3, which demonstrate that df (ar; 0*) adheres to properties 1 through 3, as well as the fact that Shapley values provide the sole solution for assigning credit to players while satisfying the properties from I to 3 (Young, 1985; Lundberg & Lee, 2017).
3.2. Predictor Optimization
The parameters of ViaSHAP are optimized with the follow- ing dual objective: to learn an optimal function for produc- ing the Shapley values of the predictions and to minimize the prediction loss with respect to the true target. Therefore, the prediction loss is minimized using function suitable for the specific prediction task, e.g., binary cross-entropy for binary classification or mean squared error for regres- sion tasks. The following presents the loss function for multinomial classification:
(0) - -00-EE xEX JEM p(S)
Σ
E
SHAP (H:S) - Viia [ViagRAP
(x:; (x;)))();)))) 0.)
Given that the ground truth Shapley values are inaccessible during training, the learning process relies solely on sam- pling input features, based on the principle that unselected features should be assigned a Shapley value of zero, while the prediction formulated using the selected features should be equal the sum of their corresponding Shapley values. Since VVie and ViA are essentially the same model, coalition sampling for both functions is performed within the same model but at different locations. For SiAS (1*)) the sampling occurs on the input features before feeding them to the model. While 1 8 sampling is applied to the predicted Shapley values, given the original set of features as input to the model, as illustrated in Figure In the follow- ing, we show that the solution computed by the optimized (i 0*) function maintains the desirable properties of Shapley values for each output dimension. For ease of nota- tion, we drop the subscript j below and consider one output at a time. All proofs, unless otherwise specified, can be found in the Appendix.
ΣΣ (())
3
[ (paagip((s ViaahAP SHAP (0) p(S) p(S)L
E
DEX JEM
1 (z: (x: )) (0)))((1))
where 33 is a predefined scaling hyperparameter and új is the predicted probability of class V/ E Y by ViaSHAP. The optimization of ViaSHAP illustrated in Figure 2 and summarized in Algorithm 1.
The global optimizer of loss function 7 is restricted to pre- dict 0 if all features are marginalized out. However, this approach may not be suitable for all prediction tasks, e.g., regression problems. Therefore, we also propose a relaxed variant of the optimization problem 7, where a ViaSHAP model predicts y 1 T (z; 0) + å (further details on this approach are provided in Appendix G).
Lemma 3.1. Via (x: 0) satisfies the property of local accu- racy wrt SHAP.
3.3. ViaSHAP Approximator
According to the universal approximation theorem, a feed- forward network with at least one hidden layer and sufficient units in the hidden layer can approximate any continuous function over a compact input set to an arbitrary degree of accuracy, given a suitable activation function (Hornik et al., 1989; Cybenko, 1989; Hornik, 1991). Consequently, neural networks and multi-layer perceptrons (MLP) can be em- ployed to learn ViaSHAP for prediction tasks where there
Lemma 3.2. The global minimizer model, Via (z; 0*), of the loss function (6), assigns value zero to features that have no influence on the outcome predicted by ViaSHAP(ss (x) in the distribution p(S).
Lemma 3.3. Let two ViaSHAP models V and V whose respective o Via are parameterized by 6* and g', which globally optimize loss function (6) over two possibly differ-
4

---

Prediction via Shapley Value Regression
δ Via
Φ
(prec(0) (0. 1)
X
Via
1
Apply same of samples
Vía
ST
Φ
Figure 2. The optimization of ViaSHAP is conducted using dual-objective loss function that aims learn optimal function for generating the Shapley values while minimizing the prediction loss.
a continuous mapping function from the input dataset to the true targets, which also applies to the true Shapley values as a continuous function.
true Shapley values. This section begins with outlining the experimental setup. Then, the predictive performance of ViaSHAP is evaluated. Afterwards, we benchmark the simi- larity between the feature importance obtained by ViaSHAP and the ground truth Shapley values. We also evaluate the predictive performance and the accuracy of Shapley values on image data. Finally, we summarize the findings of the ablation study.
(Liu et al., 2024) recently proposed Kolmogorov-Arnold Networks (KAN), spired by the Kolmogorov-Amen Arool representation theorem. According to the Kolmogorov-Amon representation theo- rem, a multivariate continuous function on a bounded do- main can be represented by a finite sum compositions of continuous univariate functions (Kolmogorov, 1956; 1957; Liu et al., 2024), as follows:
alternative approach to MLPs in-
Algorithm VIASHAP
Data: training data X, labels Y, scalar / Result: model parameters
Initialize V Via (o (ovia (r; 0))
f(x) f(x1 f(
Inn
V
e (xp))
while not converged do
L 0
for each E X and y E Y do sample S ~ p(S)
where u.. : [0, R is univariate function and V, R - R is univariate continuous function. (Liu et al., 2024) defined KAN layer dimensional functions: V - (wap), with p 1,2,... Yin and q 2. Toout- Where 7in and neut represent the dimensions of the layer's input and output, respectively, and dgp are learnable functions parameterized as splines, A KAN network is composition of I. layers stacked together; subsequently, the output of KAN on instance x given by:
① + V(x)
matrix of one-
Epred + prediction loss(ū.y)
Ls (Vy(xS) - Vyy(0) ((x5))-Vy00
1 Vio (x: (x; (x:
C # Lpred + Ls end
Compute gradients VoL Update 0 &- v end
y KAN(x) 1
1-11 V 1
V Vo(x))
The degree of each spline and the number of splines for each function are both hyperparameters.
4.1. Experimental Setup
We employ 25 publicly available datasets in the experiments, each divided into training, validation, and test subsets The training set used train the model, the validation
4. Empirical Investigation
We evaluate both the predictive performance of ViaSHAP and the feature importance attribution with respect to the
'hhe details datasets are available Table 19
5

---

Prediction via Shapley Value Regression
used detect overfitting and determine early stopping, and the test set is used evaluate the model's performance. All the learning algorithms are trained using default settings without hyperparameter tuning. The training and validation sets are combined into a single training set for algorithms that do not utilize a validation set for performance tracking. During data preprocessing, categorical feature categories are tokenized with numbers starting from one, reserving zero for missing values. We use standard normalization the feature values are centered around O. ViaSHAP can be trained using the baseline removal approach or marginal expectations as a value function. However, the baseline removal approach is adopted as the default value function". We experimented with four different implementations of ViaSHAP, using Kolmogorov-Amnot Networks (KANs) and feedforward neural networks?:
tings without tuning, as it has been shown by (Shwartz-Ziv & Armon, 2022) that deep models typically require more extensive tuning on each tabular dataset to match the per- formance of tree ensemble models, e.g., XGBoost. If the model's performance varies with different random seeds, it will be trained using five different seeds, and the average result will be reported alongside the standard deviation. In binary classification tasks with imbalanced training data, the minority class in the training subset is randomly over- sampled to match the size of the majority class, common strategy to address highly imbalanced data (Koziarski et al., 2017; Huang et al., 2022). On the other hand, no oversam- pling is applied to multinomial classification datasets. The area under the ROC curve (AUC) used for measuring predictive performance since it invariant to classification thresholds. For multinomial classífication, we compute the AUC for each class versus the rest and then weighting it by the class support. If two algorithms achieve the same AUC score, the model with a smaller standard deviation across five repetitions with different random seeds is considered better. For the explainability evaluation, we generate ground truth Shapley values by running the unbiased KernelSHAP (Covert & Lee, 2021) until convergence. It has been demon- strated that the unbiased KernelSHAP will converge to the true Shapley values when given a sufficiently large number of data samples (Covert & Lee, 2021; Jethani et al., 2022). We measure the similarity of the approximated Shapley values by ViaSHAP to the ground truth using cosine simi- larity, Spearman rank correlation (Spearman, 1904), and the coefficient of determination (R²), where cosine similarity measures the alignment between two explanation vectors, while Spearman rank correlation measures the consistency in feature rankings. The results are presented as mean val- ues with standard deviations across all data instances in the test set.
1- KANVi Based on the method proposed by Liu et al. (2024) using computationally efficient implementation". Uses spline basis functions and consists of an input layer, two hidden layers, and an output layer. Layer dimensions: Input layer maps n features to 64 dimensions, the first hid- den layer 128 dimensions, the second hidden layer to 64 dimensions, and the output layer to nx (number of classes).
2- KANN ? Replaces the spline basis in the original KANs with Radial Basis Functions (RBFs)). The architecture matches that of KANV
3- VLP A multi-laye perceptron (MLP) with identical input and output dimensions per layer as the KAN-based implementations. Incorporates batch normalization after each layer and uses ReLU activation functions.
4- MLP)'' Similar to MLPVie but the number of units in the hidden layers is raised to match the total number of pa- rameters in the KAN models, as KiN always results in models with a greater number of parameters compared to the remaining implementations.
For image experiments, we use the CIFAR-10 dataset (Krizhevsky et al., 2014). We provide three ViaSHAP implementations for image classification: ResNer50 ResNer18 and U-Ner based on ResNet50, ResNet18 (He et al., 2016), and U-Net (Ronneberger et al., 2015), re- spectively. The accuracy of the Shapley values is estimated by measuring the effect of excluding and including the top important features on the prediction, similar to the approach followed by (Jethani et al., 2022).
The four implementations were trained with the 3 of (7) set to 10 and used 32 sampled coalitions per instance. The above hyperparameters were determined in a quasi-random manner.
For the evaluation of the predictive performance, the four ViaSHAP approximators (KAN KAN MLPVia, and MI are compared against XGBoost, Random Forests, and TabNet (Arik& & Plister, 2021). All the compared al- gorithms are trained using the default hyperparameters set-
4.2. Predictive Performance Evaluation
We evaluated the performance of the seven algorithms (KAN KANPPa MLP MLP)ii TabNet, Random Forests, and XGBoost) across the 25 datasets, with de- tailed results presented in Table 1. The results show that
Further details regarding the marginal expectations approach are provided in Appendix H.
'The source code is available here: https://github. com/ www.aaaaaatibb/iaaaataaaaaaaaaaa 'nttps://githut efficient-kan pattps://githin haaab..com/iyyaaa wwt.///yyoii//atttteeecoo
http:p////gittbb..om//caacoo shapley-
6

---

Prediction via Shapley Value Regression
KAN obtains the highest average rank with respect to AUC. KAN" came in second place, closely followed by XGBoost, with only slight difference between them. We employed the Friedman test (Friedman, 1939) to determine whether the observed performance differences are statisti- cally significant. We tested the null hypothesis that there no difference in predictive performance. The Friedman test allowed the rejection of the null hypothesis, indicating that there is indeed difference in predictive performance, as measured by AUC, at the 0.05 significance level. Subse- quently, the post-hoe Nemenyi (Nemenyi, 1963) test was applied to identify which pairwise differences are signif- icant, again at the 0.05 significance level. The results of the post-hoc test, summarized in Figure 3, indicate that the differences between ViaSHAP using KAN implementations and the tree ensemble models, i.e., XGBoost and Random Forests, are statistically insignificant, given the sample size of 25 datasets. However, the differences in predictive per- formance between KAN and MLP variants (MLP and MLP)'aa are statistically significant. It is also noticeable that the MLP variants of ViaSHAP underperform compared to all other competitors, even when the MLP models have an equivalent number of parameters to KANVie We also evalu- ated the impact of incorporating Shapley loss the predic- tive performance of a KAN model by comparing KAN to an identical KAN classifier trained without the Shapley loss. The results show that KAN significantly outper- forms identical KAN architecture that is not optimized to compute Shapley values. The detailed results are available in Appendix I.
Adult
Elevators
0.93
0.90
0.9
Moatt Value
0.8
0.00
0.7
0.75
0.6
Cocine Simlarity
0.70
Spearman Corretation
0
500
1000 1380 2000 2500
3000 3500
6003
500 1000 1509 200p 2008 3000 3300 sec0 Samples
*Semples
Figure 4. The similarity between KAN and KernelSHAP's approximations. KernelSHAP initially provides approximations that differ remarkably from the values of ViaSHAP. However, as KemelSSHAP refines its approximations with more samples, the similarity to ViaSHAP's values grows.
ilarity between the generated Shapley values and the ground truth as illustrated in Figure 4. The ranking of the com- pared implementations of ViaSHAP with respect to their cosine similarity to the ground truth Shapley values shows that MLP!'a is ranked first, followed by KAN KANP".. and MLPVia, respectively. However, the Friedman test does not indicate any significant difference between the different implementations of ViaSHAP. At the same time, the results of ranking the four implementations of ViaSHAP based their Spearman rank correlation with the ground truth Shapley values reveal that KAN ranks first, followed by tie for second place between KAN and MLP) and MLPVia placing last. In order to find out whether the differ- ences significant, the Friedman test is applied once again, which allows for the rejection of the null hypothesis, indicat- ing that there is indeed a difference between the compared models in their (> 0) correlations to the ground truth & at 0.05 significance level. The post-hoc Nemenyi test, at 0.05 level, indicates that differences between MLPVia and the remaining models are significant, as summarized in Figure 6. Overall, KAN is found to be relatively stable approximator across the 25 datasets when both similarity metrics (cosine similarity and Spearman rank correlation) are considered. Detailed results can be found in Tables 2 and 3 in Appendix E. We also compare the accuracy of the Shapley values generated by KAN to those produced by FastSHAP, with KAN utilized as black-box within FastSHAP. The results in Appendix K show that KiN significantly outperforms FastSHAP in terms of similarity to the ground truth.
CD
6
% K
MLP
MLPbu TabNet
XGBoost Random Forests
Figure 3. The average rank of the 7 predictors on the 25 datasets with respect to the AUC (the lower rank is better). The critical difference (CD) is the largest statistically insignificant difference.
4.3. Explainability Evaluation
The explainability of the various ViaSHAP implementations is evaluated by measuring the similarity of ViaSHAP's Shap- ley values Vax;; (a; 0) to the ground truth Shapley values (o). computed by the unbiased KerneISHAP, as discussed in Subsection 4.1, taking ViaSHAP the black-box model. We present results for models trained with the default values for the hyperparameters. The effect of these settings are further investigated in the ablation study.
4.4. Image Experiments
We evaluated the predictive performance of ResNer50 ResNer18Vi, and V-Ner on the CIFAR-10 dataset. All models were trained from scratch (without transfer learn- ing). The results, summarized in Table 4, demonstrate that ViaSHAP can perform accurately in image classification tasks. We also compared the accuracy of the explanations
The evaluation of the alignment between Φ (a; 6) and Φ using cosine similarity generally shows a high degree of sim-
7

---

Prediction via Shapley Value Regression
obtained by ViaSHAP implementations with those obtained by FastSHAP (where ViaSHAP models were treated as black boxes). The results in Table 5 and Figure 8 show that ViaSHAP models consistently provides more accurate Shap- ley value approximations than the explanations obtained using FastSHAP. Figure 5 provides two examples showing the explanations produced by ResNer18 and FaPt The experiment details can be found in Appendix F.
5. Related Work
In addition to KemeiSHAP and the real-time method Fast- SHAP, alternative approaches have been proposed to reduce the time required for Shapley value approximation. Meth- ods that exploit specific properties of the explained model can provide faster computations, e.g., TreeSHAP (Lundberg et al., 2020) and DASP (Ancona et al., 2019), while others limit the scope to specific problems, e.g., image classifica- tions or text classification (Chen et al., 2019; Teneggi et al., 2022). Additionally, directions to improve Shapley value approximation by enhancing data sampling have also been explored (Frye al., 2021; Aas et al., 2021; Covert et al., 2021; Mitchell et al., 2022; Chen al., 2023a; Kolpaczki et al., 2024). Nevertheless, traditional methods for comput- ing Shapley values have typically been considered post-hoc solutions for explaining predictions, requiring additional time, data, and resources to generate explanations. In con- trast, ViaSHAP computes Shapley values during inference, eliminating the need for separate post-hoe explainer.
ReN
FASISHAR )
Dog
Horse
Figure 5. The explanations for the predicted class generated by ViaSHAP and FastSHAP using two randomly selected images from the CIFAR- 10 dataset.
Research on generating explanations using pre-trained mod- els has explored several approaches. (Chen et al., 2018), (Yoon et al., 2019), and (Jethani et al., 2021) trained models for important features selection. (Schwab & Karlen, 2019) trained a model to estimate the influence of different in- puts on the predicted outcome. (Situ et al., 2021) proposed to distill any explanation algorithm for text classification. Pretrained explainers, similar to other post-hoc methods, re- quire further resources for training, and the fidelity of their explanations to the underlying black-box model can vary.
4.5. Ablation Study
The ablation study was conducted after the empirical evalua- tion to ensure that no prior knowledge of the data or models influenced the experimental setup. The detailed results of the ablation study are provided in Appendex J. We began by examining the effect of 3 on both predictive performance and the accuracy of Shapley values. The results demonstrate that the predictive performance remains robust changes in 8, unless /3 is raised an exceptionally large value, e.g.- 200-foldd. A more remarkable observation is that the accu- racy of the computed Shapley values improves as / grows without sacrificing predictive performance. However, the model fails to Iearn properly with substantially large 3. Af- terwards, we evaluated the effect of the number of sampled coalitions. The results indicate that the number of samples has little impact on predictive performance and the accuracy of Shapley values, especially if compared to the impact of 3. We also study the effect of a link function on both pre- dictive performance and the accuracy of Shapley values of ViaSHAP. The results show that removing the link function significantly improves the accuracy of the Shapley values while maintaining the high predictive performance. Then, we assessed the impact of the efficiency constraint. The re- sults indicate that the efficiency constraint has significant impact on the predictive performance or the accuracy of the explanations of ViaSHAP. Finally, we examined the impact of 3 on the progression of training and validation loss during training. The results indicate that ViaSHAP tends to require a longer time to converge as 3 values increase.
Many approaches for creating explainable neural networks have been proposed. Such approaches not only generate predictions but also include an integrated component that provides explanations, which is trained alongside the predic- tor (Lei et al., 2016 6; Alvarez Melis & Jaakkola, 2018; Guo et al., 2021; Al-Shedivat et al., 2022: Sawada & Nakamura, 2022; Guyomard et al., 2022). Explainable graph neural net- works (GNNs) have also been studied for graph-structured data, which typically exploit the internal properties of their models to generate explanations, e.g., the similarity between nodes (Dai & Wang, 2021), finding patterns and common graph structures(Feng et al., 2022; Zhang et al., 2022; Cui et al., 2022), or analyzing the behavior of different compo- nents of the GNN (Xuanyuan et al., 2023). GNNs have also been employed to learn explainable models for data types beyond graphs, e.g., tabular data (Alkhatib et al., 2024; Alkhatib & Boström, 2025) and images (Chaidos et al., 2025). However, explanations generated by explainable neural networks do not always correspond to Shapley val- ues, in contrast to ViaSHAP. Moreover, the explanations lack fidelity guarantees and do not elaborate on how exactly the predictions are computed, whereas ViaSHAP generates predictions directly from their Shapley values.
8

---

Prediction via Shapley Value Regression
6. Concluding Remarks
References
We have proposed ViaSHAP, an algorithm that computes Shapley values during inference. We evaluated the perfor- mance of ViaSHAP using implementations based on the uni- versal approximation theorem and the Kolmogorov-Amold representation theorem. We have presented results from large-scale empirical investigation, in which ViaSHAP was evaluated with respect to predictive performance and the accuracy of the computed Shapley values. ViaSHAP using Kolmogorov-Aran Networks showed superior predictive performance compared to multi-layer perceptron variants while competing favorably with state-of-the art algorithms for tabular data XGBoost and Random Forests. ViaSHAP estimations showed high similarity to the ground truth Shapley values, which can be controlled through the hy- perparameters. One natural direction for future research is to implement ViaSHAP using state-of-the-e -fet---rt- algorithms. Another direction use ViaSHAP to study possible ad- versarial attacks predictive model.
Aas, K., Jullum, M., and Løland, A. Explaining individual predictions when features are dependent: More accurate approximations to shapley values. Artificial Intelligence, 103502, 2021.
Al-Shedivat, M., Dubey, A., and Xing, E. Contextual expla- nation networks, J. Mach. Learn. Res., 21(1), 2022.
Alkhatib, A. and Boström, H. Interpretable graph neu- ral networks for heterogeneous tabular data. In Discov- ery Science, pp. 310-324, Cham, 2025. Springer Nature Switzerland.
Alkhatib, A., Ennadir, S., Boström, H., and Vazirgiannis, M. Interpretable graph neural networks for tabular data. ECAI 2024 27th European Conference on Artificial Intelligence, Including 13th Conference on Prestigious Applications of Intelligent Systems, PAIS 2024, Proceed- ings :, pp. 1848-1855, IOS Press, 2024.
Acknowledgements
Alvarez Melis, D. and Jaakkola, Towards robust in- terpretability with self-explaining neural networks. In Advances in Neural Information Processing Systems, vol- ume 31. Curran Associates, Inc., 2018.
This work was partially supported by the Wallenberg AI, Au- tonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. The com- putations (on GPUs) were enabled by resources provided by the National Academic Infrastructure for Supercomput- ing in Sweden (NAISS), partially funded by the Swedish Research Council through grant agreement no. 2022-06725.
Ancona, M., Oztireli, C., and Gross, M. Explaining deep neural networks with a polynomial time algorithm for shapley value approximation. In Proceedings of the 36th International Conference Machine Learning, vol- ume 97, pp. 272-281, Long Beach, California, USA, 09-15 Jun 2019, PMLR.
Impact Statement
This paper presents work whose goal advance the field of Explainable Machine Learning. The proposed method could be deployed across variety of domains where inter- pretability is crucial. Unlike traditional explanation methods that add latency and can be harder to scale, our approach provides explainable models that are not restricted by a specific design. Furthermore, improvements in explana- tion accuracy could enhance users' trust and understand- ing of the employed model. The availability of accurate real-time explanations through one model alongside pre- dictions could also enable its use in assource-constraiig settings. However, there are also potential risks, for exam- ple, accessible and fast explanations could lead to superficial trust in model predictions, specifically if the explanations are misinterpreted. Moreover, even when accurate in the Shapley value sense, explanations might still fail capture causal domain-relevant reasoning and potentially mislead users. More importantly, malicious actors might misuse the availability of explanations to reverse- reverse-enginee engineer models or identify vulnerabilities for certain adversarial attacks.
Arik, S. O. and Pfister, T. Tabnet: Attentive interpretable tabular learning. Proceedings of the AAAI Conference on Artificial Intelligence, 35(8))6679-6687, 2021.
Chaidos, N., Dimitriou, A., Spanos, N., Voulodimos, A.,
and Stamou, G. Explaining vision gnns: A semantic and visual analysis of graph-based image classification, 2025. URL https: arxiv. www.ivorrrrge... abs wwtp/ooggg/a.aao.. 19682.
Chen, H., Janizek, J. D., Lundberg, S., and Lee, S.-I. True to the model or true to the data?, 2020. URL https: / //arxivv.co org/ abs /206. 1.766..6.. 16234.
Chen, H., Covert, I. C., Lundberg, S. M., and Lee, S.-I. Algorithms to estimate shapley value feature attributions. Nature Machine Intelligence, (16):590-601, Jun 2023a.
Chen, J., Song, L., Wainwright, M., and Jordan, M. Learn- ing to explain: An information -theoretcc perspective on model interpretation. In Proceedings of the 35th Interna- tional Conference on Machine Learning, volume 80, pp. 883-892, 10-15 Jul 2018

---

Prediction via Shapley Value Regression
Chen, J., Song, L.,, Wainwright, M. J., and Jordan, M. I. L- shapley and c-shapley: Efficient model interpretation for structured data. In International Conference on Learning Representations. 2019.
Goodman, B. and Flaxman, S. European union regulations on algorithmic decision-making and "right to explana- tion". Al Magazine, 38(3):50-57, 2017.
Guo, H., Nguyen, T., and Yadav, A. Counternet: End-to-end training of counterfactual aware predictions. In ICML 202/ Workshop on Algorithmic Recourse, 2021.
Chen, L., Lou, S., Zhang, K., Huang, J., and Zhang, Q- HarsanyiNet: Computing accurate shapley values in a single forward propagation. In Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pp- 1804-4825 PMLR, 23-29 Jul 2023b.
Guyomard, V., Fessant, F., Guyet, T., Bouadi, T., and Ter- mier, A. Venet: A self-explaining model for realistic counterfactual generation. In Machine Learning and Knowledge Discovery in Databases, ECML PKDD, 2022.
Covert, I. and Lee, S.-I. Improving kernelshap: Practi- cal shapley value estimation using linear regression. In Proceedings of The 24th International Conference on Ar- tificial Intelligncce and Statistics, volume 130, pp. 3457- 3465, April 2021.
He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learning for image recognition. 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770-778, 2016.
Covert, I., Lundberg, S., and Lee, S.-I. Explaining by remov- ing: A unified framework for model explanation. Journal of Machine Learning Research, 22((09):1 1-90, 2021.
Homik, K. Approximation capabilities of multilayer feed- forward networks. Neural Networks, 4(2):251-257, 1991.
Hornik, K.
Stincheombe, M., and White, H. Multilayer feedforward networks are universal approximators. Neu- ral Networks, 2(5): 1989.
Cui, H., Dai, W., Zhu, Y., Li, X., He, L., and Yang, C. In- terpretable graph neural networks for connectome-brtt based brain disorder analysis. In Medical Image Computing and Computer Assisted Intervention MICCAI 2022: 25th International Conference, Singapore, September 18-22, 2022, Proceedings, Part VIII, pp. 375-385, 2022.
Huang, 7.,, Sang. Y., Sun, Y., and Lv, J. A neural network learning algorithm for highly imbalanced data classifica- tion. Information Sciences, 612:496-511 513,,,1112000 2022.
Cybenko, G. Approximation by superpositions of a sig- moidal function. Mathematics of Control, Signals and Systems, 2(4):303-314, Dec 1989.
Izzo, C., Lipani, A., Okhrati, R., and Medda, A baseline for shapley values in mips: from missingness neutrality. In ESANN 2021 proceedings. European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning., pp. 605-610, 10 2021. doi: 10. 00.14422/esan//222111 FFS0021-122011188
Dai, H. and Wang, S. Towards self exilanalle graph neural network. In Proceedings of the 30th ACM International Conference on Information & Knowledge Management, pp. 302-312212222 2021.
Jethani, N., Sudarshan, M., Aphinyanaphongs, Y., and Ran- ganath, R. Have we learned to explain?: How inter- pretability methods can learn to encode predictions in their interpretations. In Proceedings of The 24th Interna- tional Conference on Artificial Intelligence and Statistics, volume 130, pp. 1459 1459-1467, 13-15 Apr 2021.
DandI, S., Molnar, C., Binder, M., and Bischl, B. Multi- objective counterfactual explanations. In Parallel Prob- lem Solving from Nature - PPSN XVI, pp. 448 069,469,222 2020.
Feng, A., You, C., Wang, S., and Tassiulas, I.. Kergnns: Interpretable graph neural networks with graph kernels. Proceedings of the AAAI Conference on Artificial Intelli- gence, (666):661466622 Jun. 2022.
Jethani, N., Sudarshan, M., Covert, I. C., Lee, S.-L., and Ran- ganath, R. FastSHAP: Real-time shapley value estimation. In International Conference on Learning Representations, 2022.
Ferguson, T. S. Game Theory. University of California at Los Angeles, 2018.
Karimi, A.,,., Barthe, G., Balle, B., and Valera, I. Model- agnostic counterfactual explanations for consequential decisions. In Proceedings of the Twenty Third Interna- tional Conference on Artificial Intelligence and Statistics, volume 108, pp. 895- 105-905,22 26-28 Aug 2020.
Friedman, M. A correction: The of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association, 34(205):109-109, 1939.
Frye, C., de Mijolla, D., Begley, T., Cowton, L., Stanley, M., and Feige, I. Shapley explainability on the data manifold. In International Conference on Learning Representations, 2021.
Kolmogorov, A. N. On the representation of continuous functions of several variables as superpositions of contin- uous functions of a smaller number of variables. Doklady Akademii Nauk, 108(2):179) 199-121799-182, 1956.
10

---

Prediction via Shapley Value Regression
Kolmogorov, A. N. On the representation of continuous functions of many variables by superposition of contin- uous functions of one variable and addition. Doklady Akademii Nauk, 1 11::953-955 956, 1957.
Mothilal, R. K., Sharma, A., and Tan, C. Explaining ma- chine learning classifiers through diverse counterfactual explanations. In Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency, pp. 607-617, 2020.
Kolpaczki, P., Bengs, V., Muschalik, M., and Hüllermeier, E. Approximating the shapley value without marginal contributions. Proceedings of the AAAI Conference on Artificial Intelligence, (8(12):13246-1333 3246-13255 Mar. 2024.
Nemenyi, P. B. Distribution multiple comparisons. PhD
thesis, Princeton University, 1963.
Owen, G. Game theory. Academic Press,, New York 3rd ed. edition, 1995.
Koziarski, M., Krawezyk, B., and Wožniak, M. Radial- based approach to imbalanced data oversampling. In Hybrid Artificial Intelligent Systems: 12th International Conference, HAIS 2017, La Rioja, Spain, June 21-23, 2017, Proceedings 12, pp. 318-327. Springer, 2017.
Patel, N., Strobel, M., and Zick, Y. High dimensional model explanations: An axiomatic approach. In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Trunsparency, pp. 401-411, 2021.
Krizhevsky, A., Nair, V., Hinton, G., et al. The eifar-10 dataset. online: http://ww.. cs. toronto. edu/kriz/cffar. huml, 55(5):2, 2014.
Ribeiro, M. T., Singh, S., and Guestrin, C. "why should i trust you?": Explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 16, pp. 1135-1144, 2016.
Lakkaraju, H., Kamar, E., Caruana, R., and Leskovec, J. Interpretable & explorable approximations of black box models. CoRR, abs/1707.01. 00 2017.
Ronneberger, O., Fischer, P., and Brox, T. U-net: Con- volutional networks for biomedical image segmentation. In Medical Image Computing and Computer-Assisted Intervention - MICCAI 2015. pp. 234-241. Springer In- ternational Publishing, 2015.
Lei, T., Barzilay, R., and Jaakkola, T. Rationalizing neural predictions. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pp. 107-117, Austin, Texas, November 2016. Association for Computational Linguistics.
Sawada, Y. and Nakamura, K. Concept bottleneck model with additional unsupervised concepts. IEEE Access, 10: 41758 41765, 2022.
Liu, Z., Wang, Y., Vaidya, S., Ruehle, F., Halverson, J., Soljačić, M., Hou, T. Y., and Tegmark, M. Kan: Kolmogorov-arnold networks, 2024. URL https:// arxiv. orgo abs 24004 19756.
Schwab, P. and Karlen, W. Cxplain: Causal explanations for model interpretation under uncertainty. In Advances in Neural Information Processing Systems, volume 32, 2019.
Lundberg, S. M. and Lee, S.-I. A unified approach to in- terpreting model predictions. In Proceedings of the 31st International Conference on Neural Information Process- ing Systems, pp. 4777,4777,20000 2017.
Shapley, L. S. A value for n-person games. In Kuhn, H. W. and Tucker, A. W. (eds.), Contributions to the Theory of Games 11, pp. 307-317. Princeton University Press, Princeton, 1953.
Lundberg, M., Erion, G., Chen, H., DeGrave, A., Prutkin, J. M., Nair, B., Katz, R., Himmelfarb, J., Bansal, N., and Lee, S.-I. From local explanations to global under- standing with explainable ai for trees. Nature Machine Intelligence, 676, Jan 2020.
Shwartz-Ziv, R. and Armon, A. Tabular data: Deep learning is not all you need. Information Fusion, 81:84-90 9049,0000 2022.
Manea, M. Cooperative Games. Massachusetts Institute of Technology, 2016.
Situ, X., Zukerman, I., Paris, C., Maruf, S., and Haffari, G. Learning to explain: Generating stable explanations fast. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 1 11th International Joint Conference Natural Language Pro- cessing (Volume 1: Long Papers), pp. 5340-5355, August 2021.
Marichal, J.-L. and Mathonet, Weighted banzhaf power and interaction indexes through weighted approximations of games. European Journal of Operational Research, 21 211(2):352-3588 2011
Mitchell, R., Cooper, J., Frank, E., and Holmes, G. Sam- pling permutations for shapley value estimation. Journal of Machine Learning Research, (3(1).jan 2022.
Spearman, C. The proof and measurement of association between two things. The American Journal of Psychology, 15(1):72-101, 1904.

---

Prediction via Shapley Value Regression
Sundararajan, M. and Najmi, A. The many shapley values for model explanation. In III H. D. and Singh, A. (eds.), Proceedings of the 37th International Conference on Ma- chine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 9269-9278. PMLR, 13-18 Jul 2020.
Teneggi, J., Luster, A., and Sulam, J. Fast hierarchical games for image explanations. IEEE Transactions on Pultern Analysis and Machine Intelligence, 2022.
Van Looveren, A. and Klaise, J. Interpretable counterfactual explanations guided by prototypes. In Joint European Conference on Machine Learning and Knowledge Dis- covery in Databases, pp. 650-665, Springer, 2021.
Wang, R., Wang, X., and Inouye, D. I. Shapley explana- tion networks. In International Conference on Learning Representations, 2021.
Wilcoxon, F. Individual comparisons by ranking methods. biometrics bulletin 1, 6 (1945), 80-83, URL http://www. jstor. org/stable/3001968, 1945.
Xuanyuan, H., Barbiero, P., Georgiev, D., Magister, L. C., and Liò, Global concept-basse based interpretability for graph neural networks via neuron analysis. Proceedings of the AAA/ Conference Artificial Intelligence, 37(9): 10675-10683, Jun. 2023.
Yoon, J., Jordon, J., and van der Schaar, M. INVASE: Instance-wise variable selection using neural networks. In International Conference on Learning Representations, 2019.
Young, H. Monotonic solutions of cooperative games. International Journal of Game Theory, 14(2), Jun 1985.
Zhang, Z., Liu, Q., Wang, H., Lu, C., and Lee, C.-K. Prot- gnn: Towards sell-explaining graph neural networks. AAAI, 2022.
12

---

Prediction via Shapley Value Regression
A. Proof of Lemma 3.1
By definition of ViaSHAPP;
Via (AAAHAP)) (x) (x;))
Σ Ps (z; 0)
This is the definition of local accuracy for the game : S + Via (x
B. Proof of Lemma 3.2
Assume that the global minimizer V (x;0*) 0*) of the loss function (6) does not satisfy the missingness property, i.e., there exists feature i that has no impact on the prediction:
Via Via (2 (ZS(i)))
Via V (i)
(8)
However, the Shapley value di assigned by of (2;; 0*) is not zero (Ø: + 0).
We recall the optimized loss function:
(0) Σ
E (ViASHAP(s5) - -Via SHAP SHAP(()) (0) 1 Vio 0)) )
M(S)
ZEX
This loss is non-negative, and is thus minimized for value of 0, implying all terms in the expectancy are equal to 0. In particular, for any set S C N (i), we have:
Via SHAP ViaSHAP((
Via ViASHAP SHAP (0)
su( Via (x;
0
ViasuuP (x3) Via (0) via (I; 8)
Via (A (1)t)
1 (1) Via (z: 0) Via (1:5) 15 Vio (x;
Via SHA (HAS 1 15 Vio (z: 0) = Via SHAP (I - 15 Uia (x;
SHAP
Σ o Vi (1; 0*) Σ of (z; 0")
jESUfI)
JES
(x; =0
Via
In practice, is unlikely for loss to exactly reach its global optimum. Instead, it approximates it. We assume here that the loss has reached a value 22 for an O. We propose an upper bound on (xx 0) conditioned on c.
Since the loss is composed only of non-negative terms, this means that:
SA 5 N, (Via SHAP (2:)) Via (0) 150 (2: (20 6))
Via SHAP (x:8) -iv SHAP
15 (x; 0) 5e
13

---

Prediction via Shapley Value Regression
Via (x)5(0)) - Via SHAP (0) T o
ViasHAP ViaSHAP (0) 1 Vio 0)
ViA (z50)))) Via SHAP SMAP(() (0) 1 @ vin (z; Via (ansHAP(s (x8)) Via SHAPP (0) + 150 via (x; 0)
s 2e
SHAP
(x 1 /
Pia (x; 0) Via Via
SHAP (505) (HHAP(ps))) + 0) s 2e by (8)
Σ
13 )x;0));)) 0)
Σ
0
Via (2; 0) 2e
jESU(()
jES
o vio [z; 0) S 2e
o Via (x;
$ 2CC((
Thus, as the loss function converges to 0, so does the importance attributed to features with no influence on the outcome.
C. Proof of Lemma 3.3
Since both V and V optimize their respective targets, they satisfy efficiency, i.e.:
SA S N, V(x9) = (iu 8"; V'(x = (2;;0
(9)
Then:
AS _ N (t),
V(x - V(23)= 2 V((55U(i)) - (2'(x>)
Σ (x; - 0² 2
Σ
Vio (z; (z;
Σ jES
o) vio (z; (x; 0r)
jESUf()
jES
jESU(S)
0 ( (x; 0*) 2 o via (x; 0³))
In the same way as for the Lemma 2, the proof assumes perfect minimization of the loss. Thus, we propose a relaxed variant, where the loss term (0) was minimized down to e² with e 2 0. Thus, following similar reasoning as in the proof of Lemma 2, we have that VS:
ViaSHAP((5)) Via SHAP (0)
1 via (z; 0) Se
We also have:
Via SHAP (x5) 1 Vio (z: ()
SHAP (2:5) (x; Via SHAP (0) + Via SHAP
(0)
By the triangle inequality on the right-hand side:
ViaSHAP (x - 1 Via (x; 0) < SHAP (HA - 15 Via (x;; 0) SHAP (0) + ViaSuAP SHAP (0)
But observe that all features in 0 are non-contributive since, AS N. 0 0 by definition of the masking operation. Thus, by the bound found in Lemma 2: Vi e N, (0, 0) s 2c. Thus ViaSHHP((0) (0) < 2nc.
Thus:
14

---

Prediction via Shapley Value Regression
Via SHAP (H:)) (i
Via (iASSAAPP) (0)
(0)
6 2nc
and we thus derive the following upper bound on the @i-wise error as:
- (z; Via SHAP (x0) 15
S c(2n 1).
D. Predictive Performance
We evaluated the performance of the four variants of ViaSHAP implementations mentioned in the experimental setup, i.e., KAN KAN MI.P'in, and MI.P3'n are compared the following algorithms for structured data: Random Forests, XGBoost, and TabNet, where Random Forests and XGBoost result in black-box models, while TabNet is explainable by visualizing feature selection masks that highlight important features. The predictive performance evaluation is conducted using 25 datasets. The results show that KANVi comes in first place as the best-performing classifier, followed by XGBoost and KAN based on AUC values.
The Friedman test confirmed that differences in predictive performance are statistically significant at the 0.05 level. subsequent post-hoc Nemenyi revealed that while the differences between KAN-based implementations and tree ensemble models (XGBoost and Random Forests) are statistically insignificant, the performance differences between KANDia and MLP variants are significant. Moreover, the differences between KANVia and TabNet are also statistically significant. The ranking of the seven models on the 25 datasets and the results of the post-hoe Nemenyi test are illustrated in Figure 3. The detailed results on the 25 datasets are shown in Table 1.
While the MLP implementations of ViaSHAP significantly underperformed compared to the KAN variants, their performance can still be enhanced by using, for instance, deeper and more expressive models, particularly for datasets with high dimensionality and large training sets. However, we defer the task of improving MLP-based ViaSHAP implementations future work, the core concept of ViaSHAP can be integrated with any deep learning model. More importantly, ViaSHAP is not limited to structured data and can be incorporated easily into the training loop of models in computer vision and natural language processing.
15

---

Prediction via Shapley Value Regression
Table 1. The AUC of KAN"", K MLPP and "M. TabNet, Random Forests, and XGBoost. The best- pest-performing model is
colored in light green and the second best pest-performing is colored in light blue
XGBoost
0.868
0.887
0.928
0.875
0.973
0.935
0.963
0.874
0.767
0.799
0.941
0.952
0.987
0.871
0.708
0.935
0.934
0.781
0.989
0.992
0.646
0.951
0.475
0.846
0.858
Random Forests
0.875 + 0.001
0.885 1: 0.001
0.913 4 0.001
+.000
0.875 0.855 + 0.003
0.97 1 0.0000
0.001
0.951 + 0.0000
0.855 H 0.00)
0.779 1 1.00
0.79 1 100
1.00002
0.935 T 1
0.0004
5.0000S
0.861 1746- H 0.004
F 0.0002
TT 0,0000
1945- # 0,02
+
7788- H 0,0000
8008
L000
+ 0.05
1t 0.002
0.485 = 0,00
0.847 1847747 + 0.002
0.855 + 0.00)
0979 E
988 T
0.909
0.935
0.979
0.746
1933
0.933
0.845
0,768
0.
1998
0.631
0.900
0
0.
0.873 = 0.01
0.82 = 0.062
0.911 = 0.001
0.862 0.005
0.88 10::00000 0.009
0.942 = 0.002
0.953 = 0.0008
0.883 - 0.002
0.774 = 0.007
0.801 + 0.003
0.942 = 0.0008
0.983 - 0.003
0.864 - 0.003
0.726 #.000
0.929 10099:000 1.9000000000 0.001
0.916 = 0.021
0.759 0.014
0.96 1066000000 0.004
0.991 + 0.002
0.631 0.16
0.918 1.18:00.000 0.014
0.493 - 0.012
= 0.004
0.727 + 0.01
0.94 0.94=0.000
TabNet
0.832 =
0.878 + 0.004
0.878 1: 0.005
0.912 0.006
0.872 + 0.005
0.87 1157:00000 0.008
0.938 0.003
0.952 + 0.002
0.881 + 0.002
0.773 + 0.004
0.786 + 0.005
0.932 +: 0.002
0.932 0.932 1 0.01
0.692 0.191
0.569 0
0.717
0:005
0.917 1 0.004
0.896 + 0.01
0.733 0.029 0.029
0.947 1: 0.001
0.996 + 0.001
0.662 66662-22 1 0.016
0.894 4: 0.032
0.5 0.014
0.839 + 0.004
0.64 +: 0.093
M
0.877 = 0.003
0.887 1: 0.005
0.91 = 0.007
0.873 0 = 0.002
0.864 1 0.003
0.922 0.031
0.953 1 0.001
0.88 1 0.002
0 0.006
0.786 0
0.001
0.929 #.005
0.918 0.014
0.014
0.571 0.159
0.719 0.01
0.917 0.917 0.004
0.909 1 0.014
0.746 1774=00000 0.005
0.948 1108800000 0.002
9.997 = 1.00001
0.685 6685-44 = - 888
0904 =t -.000
= -0000000
1000
0,00
14430000 +
0.737 =: 0.013
MLPVi
0.931
0.504
0.843
0,90
0.871 = 0.003
0.899 1: 0.002
0.916 0.001 0.001
0,876 0 + 0.005
0.004
0.939 0.939 0.001
0.955 + 0.003
+ 0.0005
0.002
0.003
H 0,0000
0.001
0.086
+ SWO0D
0.74 1 0.00
0.927 1 0.001
0.93 19013 0.013
0 # 0.003
0.962 112210000 1.000
0.002
+ 0.032
0.936 + 00000
0,007
7
0.852 + 0.003
+ 0.024
0.92 1
#
0.992 000200000000
KAN
0.784 #
0.805 1
1944#1
0.818 0
0.881
0.94 1942
0.944
0,885
0,861
0.765
0.496
0..65
0.87 = 0.003
0.89 0.005
0.914 + 0.003
0.878 + 0.001
0.93 1 0.004
0.933 0 0.002
0.96 + 0.0003
0,884 + 0.0000
0.788 1 0.002
0.801 + 0.001
0.0001
0.0007
0.000
0.864 + 0.001
0.003
0.732 + 0.003
0.929 1: 0.001
0.94 = 0.003
0.783 1..33000000 0.002
0.968 + 0.0008
0.996 + 0.001
0.827 + 0.009
0.946 1: 0.003
0.515 + 0.006
0.854 + 0.003
0.822 +: 0.002
KAN
0.944
0.949
0.985
1st order theorem proving
Telco Customer Chum
Micronggregation2
Dataset
Magic Telescope
LHC Identify
Indian Pines
Abalone
Ada Prior
Bank32nh
Electricity
Elevators
House 16H
Mozilla4
Satellite
Phonemes
Adult
Fars
Helena
Heloc
Higgs
Jannis
JM1
MC1
PC2
Pollen
16

---

Prediction via Shapley Value Regression
E. Explanations Accuracy Evaluation
The explainability of the four implementations of ViaSHAP, based on MLP and KAN, were evaluated by comparing their Shapley values (ø V (z; 0)) to the ground truth Shapley values (ø). As mentioned in the experimental set, the ground truth Shapley values were generated by KemelSHAP after convergence on each example in the test In the explainability evaluation, used the models trained with default hyperparameters in the predictive performance evaluation, which generally showed high similarity to the ground truth, as demonstrated by the cosine similarity measurements. The Friedman test found no significant differences in the cosine similarity between the compared algorithms over the 25 datasets. The detailed results are available in Table 2.
Table cosine similarity of the ground truth Shapley values to the Shapley values obtained from KAN KANG MLP MLP The best-performing model colored light green
Dataset
KAN
KAN
MI.PVia
MIPP
Abalone
0.969 + 0.0166
0.966 + 0.013
0.647 + 0.21
0.807 + 0.214
Ada Prior
0.935 + 0.046
0.982 + 0.006
0.663 0.142
0.908 + 0.045
Adult
0.931 + 0.049
0.992 + 0.011
0.574 0.16
0.947 = 0.032
Bank32nh
0,779 + 0.163
0.713 0.187
0.794:000 0.166
0.876 0.084
Electricity
0.970 + 0.02
0.971 + 0.017
0.912 + 0.131
0.913 + 0.09
Elevators
0.966 0.024
0.966 0.026
0.976 0.025
0.976 = 0.02
Fars
0,886 + 0.253
0.886 + 0.28
0.95 + 0.104
0.943 + 0.058
Helena
0.856 + 0.092
0.715 0.157
0.840 + 0.099
0.789 + 0.104
Heloc
0.844 + 0.111
0.671 0.182
0.759 0.176
0.832 + 0.125
Higgs
0.917 + 0.068
0.925 + 0.062
0.92 = 0.093
0.912 + 0.097
LHC Identify Jet
0.971 + 0.021
0.952 0.065
0.97 0.042
0.972 0.041
House 16H
0.919 + 0.048
0,922 + 0.043
0,927 + 0.06
0.944 + 0.048
Indian Pines
0.796 0.121
0.241 + 0.07
0.304 0.077
0.325 0.084
Jannis
0.852 + 0.141
0.546 + 0.189
0.675 + 0.13
0.439 + 0.1
JMI
0.88 + 0.044
0.667 0.217
0.795 + 0.203
0.839 + 0.159
Magic Telescope
0.922 + 0.067
0.935 0.058
0.973 + 0.035
0.962 + 0.058
MC1
0.466 + 0.268
0.794 0.084
0.777 = 0.127
0.887 = 0.055
Microaggregation2
0.938 + 0.049
0.610 0.149
0.840 0.099
0.81 0.096
Mozilla4
0.953 + 0.023
0.948 + 0.016
0.975 + 0.018
0.979 + 0.022
Satellite
0.841 0.116
0.870 0.077
0.766 = 0.159
0.861 0.093
PC2
0,534 + 0.183
0.905 + 0.053
0.786 0.137
0.827 + 0.098
Phonemes
0.811 + 0.162
0.868 + 0.082
0.873 + 0.126
0.916 + 0.083
Pollen
0.952 + 0.059
0.945 0.023
0.464 0.476
0.592 + 0.439
Telco Customer Churn
0.81 + 0.108
0.904 + 0.051
0.43 + 0.189
0.592 + 0.231
1st order theorem proving
0.725 0.179
0.464 0.517
0.387 0.182
0.539 0.144
We also measured similarity in ranking the important features between the computed Shapley values Vo 8)) and the ground truth Shapley values (ø) using the Spearman rank correlation coefficient. KAN is ranked first with respect to the correlation values across the 25 datasets, followed by both Kin o and MLPVing 0 in the second place, and MLPVia in the last place. The Spearman rank test revealed that the observed differences are significant. Subsequently, the post-hoc Nemenyi test confirmed that Vi significantly underperformed the compared algorithms, while the differences between the remaining algorithms are insignificant. Overall, if both the cosine similarity and the Spearman rank are considered, Via proved be more stable approximator, as detailed in Tables 2 and
17

---

Prediction via Shapley Value Regression
CD
KO KAND
MLP)
Figure 6. The average rank of KAN KAN MLP and MLP) the 25 datasets with respect Spearman correlation between the ground truth Shapley values and the values obtained from the compared models. A lower rank is better and the critical difference (CD) represents the largest difference that is not statistically significant.
Table The Spearman rank correlation between the ground truth Shapley values and the Shapley values obtained from KANVi, KAN and MLP The best-performing model is colored in light green
Dataset
KANVia
KAN
MLPVia
MI.PX
Abalone
0.663 0.234
0.879 0.14
0,529 + 0,246
0.649 + 0.236
Ada Prior
0.876 + 0.088
0.962 + 0.025
0.576 + 0.163
0.869 + 0.081
Adult
0.959 0.035
0.932 + 0.034
0.398 + 0.214
0.864 0.084
Bank32nh
0.432 + 0.151
0,433 + 0.139
0.349 0.15
0.486 + 0.129
Electricity
0.798 0.183
0.838 0.142
0.751 + 0.206
0.848 0.137
Flevators
0.920 + 0.064
0.888 + 0.072
0.883 + 0.07
0.902 + 0.06
Fars
0.347 0.328
0.106 0.133
0.512 + 0.164
0.491 0.115
Helena
0.669 + 0.152
0,475 + 0.188
0.656 + 0.159
0.660 + 0.168
Heloc
0.741 0.147
0.673 + 0.159
0.589 + 0.173
0.701 + 0.143
Higgs
0.674 + 0.12
0.718 + 0.112
0.535 + 0.143
0.568 + 0.139
LHC Identify Jet
0.857 0.119
0.726 + 0.184
0.737 + 0.164
0.724 + 0.146
House 16H
0.888 + 0.092
0,858 + 0.102
0,823 + 0.112
0.864 + 0.095
Indian Pines
0.699 0.1
0.057 + 0.054
0.099 0.07
0.181 + 0.056
Jannis
0.477 + 0.131
0.314 + 0.174
0.343 + 0.132
0.227 + 0.137
JMI
0.756 + 0.202
0.682 0.223
0.59 + 0.188
0.715 + 0.189
Magic Telescope
0.9 + 0.098
0.91 + 0.087
0.882 + 0.098
0.828 + 0.141
MC1
0.621 + 0.157
0.885 + 0.088
0.619 + 0.169
0.716 + 0.108
Microaggregation2
0.876 + 0.096
0.411 + 0.183
0.656 + 0.159
0.705 + 0.2
Mozilla4
0.942 + 0.092
0.971 + 0.063
0.909 + 0.161
0.913 + 0.137
Satellite
0.746 0.212
0.786 + 0.151
0.677 + 0.208
0.8 1 0.132
PC2
0.733 + 0.161
0.924 + 0.09
0.675 = 0.154
0.737 + 0.135
Phonemes
0.941 0.103
0.954 + 0.083
0.807 + 0.213
0.862 + 0.159
Pollen
0.285 + 0.442
0.171 + 0.484
0.297 + 0.498
0.407 + 0.545
Telco Customer Churn
0.848 0.098
0.938 + 0.043
0.262 + 0.297
0.471 + 0.211
1st order theorem proving
0.623 + 0.188
0.082 + 0.145
0.183 + 0.146
0.367 + 0.14
18

---

Prediction via Shapley Value Regression
F. Image Experiments
implemented ViaSHAP for image classification using three architectures: ResNe150 (He et al., 2016) (RexNe:50 ResNet18 (ResNer18 and U-Net (Ronneberger et al., 2015) (U-NerPla). The predictive performance these models was evaluated using Top-I Aceuracy, with the results summarized in Table 4. All models were trained the CIFAR-10 (Krizhevsky et al., 2014) dataset without transfer learning or pre-trained weights (i.e., trained from scratch) using four masks (samples) data instance. The training incorporated early stopping, terminating after ten epochs without improvement on a validation split (10% of the training data). The results of evaluating the performance of the trained models on the test set demonstrate that ViaSHAP can achieve high predictive performance on standard image classification tasks.
Table 4. A comparison of the predictive performance of ResNer50""
and U-Ner measured in AUC.
AUC
0.95 Confidence Interval
U-Net
0.983
(0.981, 0.986)
ResNet18
0.968
(0.964, 0.971)
ResNer50
0.96
(0.956, 0.964)
In order to assess the accuracy the Shapley values computed by ViaSHAP implementations, we followed a methodology similar to (Jethani et al., 2022). Specifically, we selected the top 50% most important features identified by the explainer and evaluated the predictive performance of the explained model under two conditions: using only the selected top features (Inclusion Accuracy) and excluding the top features (Exclusion Accuracy).
We compared the accuracy of Shapley value approximations of the three models (ResNer50 ResNer18 and Va), We also evaluated the accuracy of FastSHAP's approximations where the three ViaSHAP implementations for image classification are provided as black boxes to FastSHAP. The results indicate that the ViaSHAP implementations consistently provide more accurate Shapley value approximations than those generated by FastSHAP, as shown in Table 5. Figure 7 presents two examples illustrating the explanations generated by ViaSHAP models and FastSHAP, where the latter treats the ViaSHAP models black-box predictors. We also show the effects of using different percentages of the top features considered for inclusion and exclusion the top-1 accuracy in Figure 8.
Table 5. The accuracy of the Shapley values is evaluated using the top 50% of the most important features (according to their Shapley values). The Inclusion AUC (higher values are hetter) and the Exclusion AUC (lower values are better) are computed using the top accuracy.
Dataset
Exclusion AUC
Confidence Interval
Inclusion AUC 0.95 Confidence Interval
Uia FasSSHAP(U-NerPing
0.773
(0.747, 0.799)
0,988
(0.981, 0.995)
0,864
(0.843, 0.885)
0,978
(0.969, 0.987)
ResNer18
0.611
(0.581, 0.642)
0.99
(0.983, 0.996)
FastSHAPRRssNet/88kk)
0.755
(0.728, 0.782)
0,954
(0.941, 0.967)
Via
0.554
(0.523, 0.585)
0.997
(0.994, 1.0)
FasISHAP(ResNeetti
0.778
(0.753, 0.804)
0.978
(0.969, 0.987)
Ri
FastSHAP RrsNettss
ResNet50
FastSHAP ResNec50fie)
FastSHAP| U-Ne:fle)
Horsee
Figure 7. The explanations generated by ViaSHAP models and FastSHAP using randomly selected images from the CIFAR-10 dataset.
19

---

Prediction via Shapley Value Regression
Exclusion Curve
Inclusion Curve
1.0
0.8
Top 1 Accuracy
Top 1 Accuracy
0.6
ResNet18
FastSHAP(ResNet18)
ResNet50
0.4
0.4
FastSHAP(BesNett5)
U-Net
FastSHAP(U-Net)
0.2
0.2
0.0
40
20
40
80
100
Exclusion %
Inclusion
Figure 8. The inclusion and exclusion curves of ViaSHAP implementations as well as their FaPtSHAP explainers. We show how the top-1 accuracy of the predictive model changes as we exclude include an increasing share of the important features, where the important features are determined by each explainer in the comparison.
348
2:30
304
304
054
0.0
020
200
630
303
030
639
Figure 9. The explanations of RexNer18 Vio for 10 randomly selected predictions on the CIFAR- 10 dataset. Each column corresponds to a CIFAR- 10 class, and the predicted probability by ResNet18 displayed beneath each image.
20

---

Prediction via Shapley Value Regression
G. Relaxed Expected Prediction
ViaSHAP is optimized to minimize the loss function (7) and predict O if all features are marginalized out (ViasHAr = 0), which can be a limitation if the baseline removal approach (Sundararajan Najmi, 2020), for instance, is applied mask features using their mean values, i.c., Via (B(z)) O. some cases, particularly with heavily imbalanced data, the "average values" (x)) might strongly represent class over others. Moreover, in regression problems, the prediction of an accurate estimator using the expected values is unlikely to be O. Therefore, we propose relaxed variant of the optimization problem, where (iAS))A(()) (0) is not obliged to predict O in order to minimize the Shapley loss. We introduce bias term to
ViaSHAP, modifying the predictions such that = 1 T Φ Pia (x; 0) + 6. Accordingly, the relaxed loss function is formulated as follows:
(x( -000=EE
(
E p(S)
via (0;0) 00 (,))(((;;0) [1TT (x); +0) 00) (1) V
(z; (x::))))) (v,loo(1 (x;) + +6)) (10)
JEM
Consequently, ViaSHAP explains Via 6, where ò is optimized through the prediction loss function to minimize the overall prediction error. In the case of baseline removal using the expected values, SiA (B(x)) 8.
To measure the effect of the relaxed loss function on the predictive performance as well as the accuracy of the Shapley value approximations, we conduct an experiment where we compare the performance of two identical KAN implementations: one trained with the default loss function (7) and the other trained with the relaxed variant the loss function (10). The two implementations are trained using the default setting without link function applied to the predicted outcome. The detailed results of predictive performance are presented in Table 6, while the accuracy of the Shapley value approximations is provided in Table 7.
We test the null hypothesis that no significant difference exists in predictive performance, as measured by AUC, between ViaSHAP trained using the default architecture and its relaxed variant. Given that only two approaches are compared, the Wilcoxon signed-rank test (Wilcoxon, 1945) is employed. The results indicate that the null hypothesis cannot be rejected at the 0.05 significance level, i.e., there is no significant difference in the predictive performance of the compared approaches. Although the datasets used in the experiments include 19 imbalanced datasets, the results of both variants of ViaSHAP performing remarkably well.
The results regarding the similarity of the approximated Shapley values to the ground truth show no significant difference between the compared approaches, as measured by the three similarity metrics: cosine similarity, Spearman's 's rank, and R²
21

---

Prediction via Shapley Value Regression
Table The predictive performance of KAN optimized for Via ViasHiP (0) 0 the relaxed version of the optimization function
(Via
(0) d). The predictive performance measured by AUC. The best-performing model colored in light green
Dataset
KAN (Via SIAP (0) 0)
KANVia (Via
6)
Abalone
0.883 + 0.0002
0.883 + 0.0002
Ada Prior
0.898 + 0.003
0.900 + 0.003
Adult
0.919 = 0.0005
0.919 = 0.003
Bank32nh
0.883 = 0.003
0.887 + 0.002
Electricity
0.934 + 0.004
0.936 + 0.006
Elevators
0.936 + 0.003
0.937 + 0.001
Fars
0.958 + 0.002
0.958 + 0.001
Helena
0.868 = 0.006
0.873 = 0.003
Heloe
0.792 = 0.001
0.793 + 0.001
Higgs
0,801 = 0.001
0.801 = 0.001
LHC Identify Jet
0.939 + 0.0005
0.938 0.003
House 16H
0.949 + 0.001
0.951 1 0.001
Indian Pines
0.982 + 0.001
0.981 + 0.002
Jannis
0.861 + 0.001
0.858 + 0.003
JM1
0.686 + 0.025
0.703 0.025
Magic Telescope
0.921 + 0.002
0.925 + 0.002
MC1
0.952 + 0.011
0.942 + 0.013
Microaggregation22
0.764 = 0.008
0.766 + 0.011
Mozilla4
0.965 = 0.001
0.965 = 0.001
Satellite
0.944 = 0.010
0.964 + 0.022
PC2
0.659 + 0.060
0.689 + 0.031
Phonemes
0.923 + 0.003
0.922 + 0.002
Pollen
0.501 = 0.002
0.502 0.004
Telco Customer Churn
0.857 + 0.003
0.852 + 0.006
1st order theorem proving
0.810 + 0.006
0.761 + 0.003
22

---

Prediction via Shapley Value Regression
Table 7. A comparison to evaluate the similarity to the ground truth explanations between KAN optimized for Via (0) = 0 and the
relaxed version of the optimization function (Via (0) = 8). The similarity is measured using cosine similarity, Spearman's 's rank,
R². The berf-pnrn model is
colored light green.
Via SHAP (0) =
0.998 + 0.003
0.858 1 0.132
0.956 + 0.066
0.874 + 0.126
0.992 1 0.016
0.988 0.016
0.988 + 0.029
0.689 + 0.197
0.934 1 0.065
0.974 0.023
0.997 1: 0.012
0.965 1 0.04
0.266 :: 0.205
0.717 0.186
0.907 1 0.09
0.991 1: 0.012
0.894 + 0.236
0.956 - 0.057
0.9995 0.001
0.83 - 0.203
0.853 4 0.606
0.987 + 0.022
0.86 - 0.913
0.612 + 0.239
0.93 0.93 - 0.06
R²
0.9996 + 0.0007
0.999 + 0.002
0.9 1. 0.095
0.948 0.079
0.87 0.142
0.992 1 0.012
0.993 0.009
0.895 + 0.073
0.677 + 0.204
0.895 0.105
0.977 0 0.014
0.998 1: 0.005
0.961 0.057
0.333 0.192
0.722 + 0.183
0.901 1 0.094
0.98 0.02
0.873 + 0.332
0.929 + 0.114
0.814 + 0.296
0.895 4: 0.223
0.975 + 0.076
0.929 : 0.212
0.939 + 0.054
0.429 +: 0.479
Via SHAP
Via SHAP (0) = 8
0.972 = 0.054
0.886 + 0.088
0.941 - 0.056
0.646 = 0.113
0.965 - 0.043
0.963 0.031
0.724 0.119
0.701 + 0.143
0.885 0.078
0.833 0.833 0.07
0.961 1 0.047
0.948 - 0.051
0.511 0.163
0.607 0.125
0.917 0.917 0.077
0.968 1 0.039
0.891 0.121
0.959 0.056
0.968 - 0.071
0.856 0.131
0.857 + 0.133
0.972 + 0.058
0.933 0.933 0.11
0.913 + 0.068
0.771 + 0.142
Spearman's 's Rank
SHAP (0) = 0
0.971 - 0.052
0.909 + 0.068
0.931 + 0.074
0.648 0.
0.967 0.967 0.043
0.969 0.026
0.882 = 0.073
0.702 - 0.148
0.882 1 0.073
0.87 0.057
0.974 1: 0.032
0.952 + 0.044
0.553 0.18
0.18
0.624 0.113
0.916 0.085
0.959 1 0.042
0.881 0.881 0.139
0.957 0.957 0.049
0.967 + 0.074
0.894 + 0.102
0.875 0 0.127
0.951 = 0.094
0.959 + 0.076
0.934 + 0.052
0.66 4: 0.146
0.9998 + 0.0005
SHAP (0) =
0.999 + 0.001
0.945 0.053
0.984 + 0.023
0.948 = 0.043
0.997 0.005
0.995 + 0.006
0.995 + 0.012
0.873 1 0.101
0.973 = 0.029
0.989 0.01
0.999 + 0.005
0.987 + 0.018
0.648 - 0.147
0.884 4.84000.000 0.086
0.965 4
0.997 1: 0.004
0.957 = 0.089
0.985 + 0.021
0.961 + 0.052
0.945 + 0.096
0.996 + 0.005
0.988 + 0.02
0.973 + 0.027
0.814 +: 0.131
Cosine Similarity
SHiP (0) = 0
0.999 + 0.0008
0.963 0.037
0.981 + 0.03
0.948 0.045
0.998 1 0.004
0.997 1 0.004
0.962 # 0.036
0.874 - 0.095
0.962 = 0.036
0.991 0.006
0.999 1 0.002
0.988 + 0.015
0.683 0
0.898 =
0.965 0.042
0.994 119410000 0.006
0.951 =
0.982 + 0.021
0.9998 + 0.0003
0.976 0.976 0.033
0.956 0.087
0.993 + 0.013
0.994 0.013
0.978 - 0.025
0.778 + 0.123
1st order theorem proving
Telco Customer Chum
LHC Identify Jet
Magic Telescope
Microaggregationn
Abalone
Ada Prior
Bank32nh
Electricity
Elevators
House 16H
Indian Pines
Mozilla4
Satellite
Phonemes
Dataset
Adult
Fars
Helena
Heloc
Higgs
Jannis
JMI
MC1
PC2
Pollen
23

---

Prediction via Shapley Value Regression
H. ViaSHAP with Marginal Expectations
evaluate the predictive performance and the similarity of the approximate Shapley values to the ground truth values when ViaSHAP is trained using marginal expectations (Chen et al., 2020) as the strategy for marginalizing out features @ Viva (x X XMs; .00
Via
E
In the experiment, we compare ViaSHAP trained using baseline removal vs. marginal expectations. The compared models are trained using the default settings of ViaSHAP without link functions. The marginal expectations models employ 128 data instances background data selected from the validation set. The results of the experiment, presented in Tables 8 and indicate that ViaSHAP trained using the marginal expectations value function significantly underperforms the baseline removal approach. Furthermore, ViaSHAP trained with marginal expectations fails accurately approximate the solutions provided by the unbiased KernelSHAP that employs marginal expectations. The results indicate that employing the marginal expectations value function can hinder the learning of an accurate predictor.
Table The predictive performance of KAN optimized using marginal expectations for masking features using baseline removal. The predictive performance is measured by AUC. The best-performing model is colored in light green
Dataset
Marginal Expectations
Baseline Removal
Abalone
0.860 + 0.005
0.883 + 0.0002
Ada Prior
0.833 0.010
0.898 + 0.003
Adult
0.888 + 0.004
0.919 + 0.000
Bank32nh
0.833 0.007
0.883 + 0.003
Elevators
0.875 0.006
0.936 + 0.003
Helena
0.850 = 0.003
0.868 = 0.006
House 16H
0.929 0.003
0.949 = 0.001
Indian Pines
0.939 = 0.039
0.982 + 0.001
JM1
0.711 = 0.009
0.686 + 0.025
MC1
0.936 0.019
0.952 + 0.011
Microaggregation2
0.758 = 0.017
0.764 + 0.008
Mozilla4
0.924 0.006
0.965 + 0.001
Phonemes
0.907 = 0.006
0.923 = 0.003
Telco Customer Churn
0.824 + 0.010
0.857 + 0.003
24

---

Prediction via Shapley Value Regression
Table 9. The similarity to the ground truth explanations when KAN optimized using marginal expectations for masking features vs.
using baseline removal. The similarity is measured using cosine similarity, Spearman's 's rank, and R². The best-performing model colored in light green.
0.999 + 0.002
0.948 + 0.079
0.87 1 0.142
0.993 1 1033400000 0.009
0.677 1 0.204
0.961 1 0.057
0.333 115410000 0.192
0.901 + 0.094
0.873 1 0.332
0.929 1 0.114
0.9996 + 0.001
0.975 + 0.076
0.939 + 0.054
0.9 + 0.095
Baseline
R.
- + 13.19
-0.87 + 1.071
-0.778 + 1.261
0.544 + 0.301
1..998 + 1.427
-0.221 0.533
0.752 0.188
-0.6 1 0.328
-08009 + 1.666
-0.339 + 0.177
0.242 0.48
0.124 1 1.551
-8.204 + 14.423
-0.156 + 0.486
Marginal
0.971 + 0.052
0.909 + 0.068
0.931 + 0.074
0.648 0.114
0.969 1 0.026
0.702 0.702 0.148
0.952 +.044
0.553 + 0.18
0.916 1.088
0.881 0.139
0.957 0.049
0.967 + 0.074
0.951 + 0.094
0.934 + 0.052
Baseline
Spearman's 's Rank
0.697 0.253
0.526 + 0.223
0.665 0.176
0.426 0.143
0.559 0.224
-0.013 1 0.187
0.809 0.125
-0.0004 0.066
0.596 0.198
0.449 0.449 0.177
0.177 017740.22 0.231
0.909 + 0.141
0.845 + 0.188
0.607 + 0.154
Marginal
0.999 + 0.001
0.963 + 0.037
0.981 + 0.03
0.948 + 0.045
0.997 + 0.004
0.874 + 0.095
0.988 1 0.015
0.683 4 0.171
0.965 4 0.042
0.951 + 0.093
0.982 4 0.021
0.9998 + 0.0003
0.993 1 0.013
0.978 + 0.025
Baseline
Cosine Similarity
0.104 + 0.629
0.331 + 0.286
0.448 + 0.375
0.811 + 0.116
0.497 + 0.362
0.286 1 + 0.149
0.926 + 0.065
0.021 + 0.067
0.691 + 0.229
0.568 + 0.186
0.366 - 0.236
0.884 + 0.147
0.347 + 0.578
0.646 + 0.186
Marginal
Microaggregation2
Telco Cust. Churn
Indian Pines
House 16H
Dataset
Abalone
Ada Prior
Bank32nh
Elevators
Helena
Mozilla4
Phonemes
Adult
JM1
MC1
25

---

Prediction via Shapley Value Regression
I. A Comparison Between ViaSHAP and a KAN Model with the Same Architecture
conducted an experiment to assess the impact of incorporating Shapley loss in the optimization process on predictive performance of KAN model. Consequently, we compared KAN KAN model with am identical architecture that does compute Shapley values. As summarized in Table 10, the results indicate that KAN generally outperforms the KAN model with the same architecture. In order to determine the statistical significance of these results, the Wilcoxon signed-rank test (Wilcoxon, 1945) was employed to test the null hypothesis that difference exists in predictive performance, as measured by AUC, between KANY and the identical KAN model without Shapley values. The test results allowed for the rejection of the null hypothesis, indicating that KAN significantly outperforms the KAN architecture that is not optimized to compute Shapley values with respect to the predictive performance as measured by the AUC.
Table 10. A comparison between the predictive performance of KA and KAN model with identical architecture to KAN but does not compute Shapley values. The results are reported in AUC.
Dataset
K KAN
KAN 0.882 + 0.001
Abalone
0.87 + 0.003
Ada Prior
0.895 + 0.005
0.89 + 0.005
Adult
0.917 + 0.001
0.914 + 0.003
Bank32nh
0.886 + 0.001
0.878 + 0.001
Electricity
0.924 = 0.005
0.93 = 0.004
Elevators
0.935 0.003
0.935 1 0.002
Fars
0.957 0.001
0.96 + 0.0003
Helena
0.883 = 0.001
0.884 0.0001
Heloc
0.793 = 0.002
0.788 + 0.002
Higgs
0.801 + 0.002
0.801 + 0.001
LHC Identify Jet
0.944 + 0.0003
0.944 + 0.0001
House 16H
0.948 + 0.001
0.949 + 0.0007
Indian Pines
0.935 + 0.001
0.985 + 0.0004
Jannis
0.860 = 0.002
0.864 + 0.001
JMI
0.725 0.008
0.732 0.003
Magic Telescope
0.931 + 0.001
0.929 + 0.001
MC1
0.933 = 0.019
0.94 + 0.003
Microaggregation2
0.783 + 0.002
0.783 + 0.002
Mozilla4
0.967 = 0.001
0.968 + 0.0008
Satellite
0.987 + 0.003
0.996 + 0.001
PC2
0.458 = 0.049
0.827 + 0.009
Phonemes
0.945 0.002
0.946 0.003
Pollen
0.491 + 0.005
0.515 + 0.006
Telco Customer Churn
0.848 = 0.005
0.854 + 0.003
1st order theorem proving
0.805 + 0.005
0.822 + 0.002
26

---

Prediction via Shapley Value Regression
J. Ablation Study
this section, we explore the influence of key hyperparameters on the performance and behavior ViaSHAP. Specifically, we investigate the effects of the scaling hyperparameter and the number of sampled coalitions per data instance. We begin by analyzing how variations in 3 impact both predictive performance and the accuracy of the Shapley values generated by ViaSHAP. We then examine the role of the number of sampled coalitions in model performance, followed by an evaluation of how changes in 3 affect the progress of the computed loss values during training. The findings provide valuable insights into the robustness and efficiency of ViaSHAP under different hyperparameter settings.
J.1. The Impact of Scaling Hyperparameter / on the Performance of ViaSHAP
We evaluated the performance of the models trained with different 3 values (in equation 7), where exponentially increasing values are tested. The models were trained using the default hyperparameter settings described in the experimental setup, except for the values of B. The AUC of the trained models is measured on the test set, as well as the similarity of the predicted Shapley values to the ground truth. The results indicate that the predictive performance of ViaSHAP, as measured by the area under the ROC curve, remains largely unaffected by value of B3 even when 3 is increased exponentially. the other hand, the similarity between the computed Shapley values and the ground truth improves increases. However, the model struggles to learn effectively after /3 exceeds 200, shown in Figures 10 and 11.
First Order Theorem Proving
Bank32nh
Cooine Similrrty
Spearman Comvlation
A
0.8
0.6
0.2
0.0
Connne Sindarty Comslation
-0.2
AUC
128
256
2
16
32
128
256
512
1024
B
B
JM1
Phonemes
1.0
1.00
0.8
0.6
0.50
0.4
0.25
0.2
0.00
-025
-0.2
-0.50
-0.4
Couine Simlanty
-0775
8
128 256 512
2
4
8
16
32
64
128
256
512
1024
B
B
Figure 10. The effect of different values of 3 the predictive performance (AUC), alignment with the true Shapley values (cosine similarity), and the similarity in the order of features to the ground truth (Spearman rank).
27

---

Prediction via Shapley Value Regression
Heloc
House 16H
1.0
0.8
0.4
0.2
0.0
0.0
-022
-0.2
Conine Semdacty
Costne Simdarty
Cometation
Spearman Cormlation
AUC
-0.4
AUC
128 256
64
128
B
B
Figure 11. The effect of different values of 3 on the predictive performance (AUC), alignment with the true Shapley values (cosine similarity), and the similarity in the order features the ground truth (Spearman rank).
J.2. The Number of Samples
We assessed the impact of the number of sampled coalitions per data example on the performance of ViaSHAP, retraining the model using the default hyperparameters with the exception the sample size. We investigated an exponentially increasing range of sample sizes (2"), from 1 to 128. The findings suggest that the number of samples has smaller effect on the performance of the trained models compared to 8, which allows for effective training of ViaSHAP models with as few one sample per data instance. The results are illustrated in Figures 12 and 13.
Bank32nh
First Order Theorem Proving
0.9
0.8
0.8
0.7
0.6
0.5
0.6
0.4
0.3
0.2
0.4
1
2
8
16
32
64
128
2
4
8
16
32
64
128
# Samples
Samples
Figure 12. The effect of different number of samples the predictive performance (AUC), alignment with the Shapley values (cosine similarity), and the similarity in the order of features to the ground truth (Spearman rank).
28

---

Prediction via Shapley Value Regression
JM1
Phonemes
0.8
1.0
0,9
0.6
0.2
Cosine Simiarty Spearann Commiation
Collne Similartt
0.0
Spearman Comelation
AUG
ALC
2
16
32
128
32
# Samples
Samples
Heloc
House 16H
1.00
0.9
0.95
0.8
0.90
0.7
0.85
0.6
0.80
0.5
0.75
0.70
Cosine Simiaity
Simiarrty
Spearmn Corelation
Comelation
AUC
0.65
AUC
64
8
# Samples
# Samples
Figure The effect different number of samples on the predictive performance (AUC), alignment with the Shapley values (cosinc similarity), and the similarity in the order of fcatures to the ground truth (Spcarman rank).
29

---

Prediction via Shapley Value Regression
J.3. The Effect of Applying Link Function the Predicted Outcome
To examine the impact of employing link function on the predictive performance of ViaSHAP and the accuracy of its Shapley value approximations, trained KAN without applying link function at the output layer and compared the predictive performance to that of KAN with the default settings mentioned in the experimental setup. The results of the predictive comparison are summarized in Table 11 11. To evaluate the null hypothesis that there is no difference in predictive performance, measured by the AUC, between KANDia with and without a link function, the Wilcoxon signed-rank test was employed, given that only two methods were compared. The results indicate that the null hypothesis can be rejected at the 0.05 significance level. Therefore, the results indicate that the presence of a link function does not significantly influence predictive performance in general.
The similarity between the ground truth and the approximated Shapley values by KAND both with and without link functions, are reported in Table 12. The similarity of KAN approximations the ground truth is measured using the cosine similarity and the Spearman's 's Rank as described in the experimental setup, which allow for measuring the similarity even two explanations are not on the same scale, since ViaSHAP allows for applying link function to accommodate a valid range of outcomes which can lead ViaSHAP's approximations to be on a different scale than the ground truth obtained using the unbiased KemneISHA.. However, since we measure the effect of using the link function on the accuracy of Shapley values, we can also apply a metric that measures the similarity on the same scale for models without link function. Therefore, we also apply R functions. The results presented in Table 12 demonstrate that ViaSHAP without a link function significantly outperforms its counterpart with link function. In order to test the null hypothesis that no difference exists in the accuracy of Shapley value approximations by KAN2Pa with and without a link function, the Wilcoxon signed-rank test was applied. The test results confirm that the null hypothesis can be rejected both cases, whether Spearman's 's rank or cosine similarity is used as the similarity metric. Furthermore, the results show that Rt as a similarity metric consistent with both Spearman's rank and cosine similarity.
similarity metric to the ground truth Shapley values for models without link
30

---

Prediction via Shapley Value Regression
Table The effect the link function on the predictive performance KAN as measured by AUC. The hest-performing model
colored in light green
Dataset
KANVia (without a link function)
KANVia (default settings)
Abalone
0.883 + 0.0002
0.87 0.003
Ada Prior
0.898 0.003
0.89 + 0.005
Adult
0.919 + 0.0005
0.914 + 0.003
Bank32nh
0.883 + 0.003
0.878 + 0.001
Electricity
0.934 + 0.004
0.93 0.004
Elevators
0.936 0.002
0.935 + 0.002
Fars
0.958 + 0.001
0.96 + 0.0003
Helena
0.868 + 0.006
0.884 + 0.0001
Heloc
0.792 = 0.001
0.788 + 0.002
Higgs
0.801 = 0.001
0.801 + 0.001
LHC Identify Jet
0.939 = 0.0005
0.944 + 0.0001
House 16H
0.949 = 0.001
0.949 + 0.0007
Indian Pines
0.982 + 0.001
0.985 0.0004
Jannis
0.861 1 0.001
0.864 1: 0.001
JMI
0.686 + 0.024
0.732 +: 0.003
Magic Telescope
0,921 + 0.002
0.929 + 0.001
MC1
0.952 0.011
0.94 0.003
Microaggregation2
0.764 + 0.008
0.783 + 0.002
Mozilla4
0.965 + 0.001
0.968 + 0.0008
Satellite
0.944 + 0.01
0.996 1: 0.001
PC2
0.659 0.06
0.827 +: 0.009
Phonemes
0.923 + 0.003
0.946 + 0.003
Pollen
0.501 + 0.002
0.515 + 0.006
Telco Customer Churn
0.857 + 0.003
0.854 + 0.003
1st order theorem proving
0.810 + 0.006
0.822 + 0.002
31

---

Prediction via Shapley Value Regression
Table 12. The effect of the link function on the similarity of the approximated Shapley values by KANDiA
The best- best-performing model is
colored in light green
0.9996 + 0.0007
0.999 + 0.002
0.9 0.9 + 0.095
0.948 + 0.079
0.87 = 0.142
0.992 0.012
0.993 1 0.009
0.895 0.073
0.677 0.204
0.895 0.105
0.977 0.014
0.998 0.005
0.961 0.057
0.333 0.192
0.722 0.183
0.901 0.094
0.98 0.02
0.873 1 0.332
0.929 + 0.114
0 0.296
0.895 + 0.223
0.975 0.076
0.929 + 0.212
0.939 - 0.054
0.429 + 0.479
R²
ViaSHAP without a link function
Spearman's Rank
0.971 + 0.052
0.909 + 0.068
0.931 + 0.074
0.648 + 0.114
0.967 + 0.043
0.969 1 0.026
0 0.073
0.702 - 0.148
0.882 + 0.073
0.87 = 0.057
0.974 + 0.032
0.952 + 0.044
0.553 + 0.18
0 0.113
0.916 + 0.085
0.959 1 0.042
0.881 0.881 0.139
0.957 + 0.049
0.967 + 0.074
0.894 1 0.102
0.875 + 0.127
0.951 + 0.094
0.959 + 0.076
0.934 + 0.052
0.66 + 0.146
Cosine Similarity
0.999 + 0.0008
0.963 + 0.037
0.98 + 0.03
0.948 = 0.045
0.998 = 0.004
0.997 + 0.004
0 0.036
0.874 0.87440.00 0.095
0.962 - 0.036
0.991 0091=00000 0.006
0.999 - 0.002
0.988 = 0.015
0.683 = 0.171
0.898 = 0.072
0.965 - 0.042
0.994 + 0.006
0.951 1 0.093
0.982 - 0.021
0.9998 + 0.0003
0.976 + 0.033
0.956 + 0 0.087
0.993 0.993 0.013
0.994 + 0.013
0.978 - 0.025
0.778 = 0.123
Spearman's Rank
0.6635 1 0.234
0.8763 + 0.088
0.9594 + 0.035
0.432 0.151
0.7983 1 0.183
0.9203 1 0.064
0.347 0.327 0.328
0.669 0.152
0.7409 0.147
0.674 0.12
0.8575 0.119
0.8876 + 0.092
0.6991 + 0.116
0.4775 0 0.131
0.7561 0.202
0.9 0.9 0.098
4
0.8756 + 0.096
0.9423 4 0.092
0.746 0.212
0.7326 + 0.161
0.9407 4 0.103
0.372 +: 0.429
0.8476 + 0.098
0.6228 + 0.188
ViaSHAP with default settings
Cosine Similarity
0.969 + 0.017
0.935 + 0.046
0.931 + 0.049
0.779 0.163
1 0.02
0.966 1 0.024
0.886 1 0.253
0.856 + 0.092
0.844 0.844.0.111 0.111
0.917 0.068
0.971 1 0.021
0.919 + 0.048
0.796 + 0.121
0.852 + 0.141
0.88 0.044
0.922 1 1 0.067
0.466 + 0.268
0.938 + 0.049
0.953 4 0.023
0.841 0.116
0.534 + 0.183
0.811 + 0.162
0.952 + 0.059
0 0.81 +: 0.108
0.725 + 0.179
1
1st order theorem proving
Telco Customer Churn
Mieroaggregationz
LHC Identify Jet
Magic Telescope
Indian Pines
Dataset
Abalone
Ada Prior
Adult
Bank32nh
Electricity
Elevators
Helena
Heloc
Higgs
House 16H
Jannis
Mozilla4
Satellite
Phonemes
Pollen
Fars
JM1
MC1
PC2
32

---

Prediction via Shapley Value Regression
J.4. The Effect of the Efficiency Constraint
We investigate the impact of the efficiency constraint (4) on the predictive performance of ViaSHAP and the similarity of its approximate Shapley values to the ground truth. The experimental results, presented in Tables 13 and 14, indicate that imposing the efficiency constraint has significant effect either the predictive performance of ViaSHAP the accuracy of its explanations. To formally test this, we evaluate the null hypothesis that no significant difference exists in predictive performance, measured by AUC, between models trained with or without the efficiency constraint. Since only two approaches are compared, the Wilcoxon signed-rank test (Wilcoxon, 1945) is employed. The results confirm that the null hypothesis cannot be rejected at the 0.05 significance level, indicating no significant difference in predictive performance between the two approaches. Significance tests are also applied to evaluate the similarity of the approximated Shapley values the ground truth based on cosine similarity, Spearman's rank, and R2 The results indicate significant difference between ViaSHAP models trained with and without the efficiency constraint.
Table 13. The effect of the efficiency constraint on the predictive performance of KAN as measured by AUC. The best-performing model is colored in light green
Dataset
Unconstrained
Constrained
Abalone
0.883 + 0.0003
0.883 + 0.0002
Ada Prior
0.897 - 0.003
0.898 + 0.003
Adult
0.919 + 0.0007
0.919 = 0.0005
Bank32nh
0.884 + 0.002
0.883 + 0.003
Electricity
0.936 + 0.004
0,934 = 0.004
Elevators
0.933 + 0.002
0.936 + 0.003
Fars
0.959 + 0.001
0.958 + 0.002
Helena
0.870 + 0.005
0.868 = 0.006
Heloc
0.792 + 0.002
0.792 + 0.001
Higgs
0.800 + 0.002
0.801 + 0.001
LHC Identify Jet
0.939 + 0.0006
0.939 = 0.0005
House 16H
0.948 + 0.001
0.949 + 0.001
Indian Pines
0.982 = 0.002
0.982 = 0.001
Jannis
0.860 + 0.003
0.861 + 0.001
JM1
0.691 + 0.026
0.686 + 0.025
Magic Telescope
0.921 + 0.002
0.921 + 0.002
MCI
0.942 + 0.011
0.952 + 0.011
Microaggregation2
0.763 + 0.009
0.764 + 0.008
Mozilla4
0.965 + 0.001
0.965 + 0.001
Satellite
0.926 + 0.006
0.944 + 0.010
PC2
0.670 + 0.046
0.659 + 0.060
Phonemes
0.919 + 0.006
0.923 + 0.003
Pollen
0.499 + 0.002
0.501 = 0.002
Telco Customer Churn
0.853 + 0.004
0.857 - 0.003
1st order theorem proving
0.809 + + 0.007
0.810 0.006
33

---

Prediction via Shapley Value Regression
Table 14. The similarity to the ground truth explanations when KAN optimized with without efficiency constraint. The
similarity is measured using cosine similarity, Spearman's 's rank, and R².
The hest-performngg model is
colored in light green.
Constrained
0. 0.999 = 0.002
0.9 = 0.095
0.948 - 0.079
0.87 0.142
0.992 - 0.012
0.993 - 0.009
0.895 = 0.073
0.677 - 0.204
0.895 = 0.105
0.977 - 0.014
0.998 - 0.005
0.961 + 0.057
0.333 1 0.192
0.722 1 0.183
0.901 + 0.094
0.98 0.02
0.873 = 0.332
0.929 0.114
0.9996 + 0.001
0.814 1 0.296
0.895 1 0.223
0.975 0.076
0.929 + 0.212
0.939 - 0.054
0.429 + 0.479
Unconstrained
0. 0.999 + 0.002
0.8461 = 0.138
0.966 = 0.062
0.849 0.125
0.991 1 0.015
0.991 1 0.011
0.994 0.022
0.894 - 0.097
0.984 + 0.011
0. 0.998 - 0.006
0.969 0.032
0.297 0.178
0.727 1 0.191
0.851 1 0.123
0.985 0.017
0.869 - 0.239
0.956 + 0.061
0.9996 + 0.002
0.823 1 0.195
0.913 0.913 0.206
0.98 0.063
0.967 0.107
0.923 4 0.051
0.464 + 0.21
0.713
Constrained
0.971 = 0.052
0.909 = 0.068
0.931 0.074
0.648 = 0.114
0.967 = 0.043
0.969 = 0.026
0.882 = 0.073
0.702 = 0.148
0.882 = 0.073
0.87 0:77:0000 0.057
0.974 + 0.032
0.952 0.044
0.553 0.18
0.624 + 0.113
0.916 = 0.085
0.959 = 0.042
0.881 = 0.139
0.957 = 0.049
0.967 = 0.074
0.894 1 0.102
0.875 1 0.127
0.951 = 0.094
0.959 = 0.076
0.934 = 0.052
0.66 + 0.146
Spearman 's Rank
Unconstrained 0.972 + 0.056
0.909 = 0.067
0.955 = 0.048
0.654 1 0.117
0.964 1.04:.004
0.048
0.967 = 0.03
0,908 0.076
0.719 0.139
0.862 + 0.084
0.88 = 0.055
0.97 0.038
0.959 0.037
0.518 1 0.184
0.649 1 0.122
0.888 + 0.102
0.964 + 0.039
0.862 = 0.145
0.963 = 0.05
0.965 + 0.074
0.858 1 0.129
0.889 0.134
0.964 + 0.086
0.954 + 0.095
0.911 + 0.051
0.669 + 0.156
Constrained
0.999 = 0.001
0.963 0.037
0.981 + 0.03
0.948 = 0.043
0.998 + 0.004
0.997 1 0.004
0.962 = 0.036
0.874 0 0.095
0.962 006200005 0.036
0.991 1 0.006
0.999 + 0.002
0.988 + 0.015
0.683 1 0.171
0.898 1 0.072
0.965 1 0.042
0.994 0.006
0.951 = 0.093
0.982 1 0.021
0.9998 + 0.0003
0.976 0.033
0.956 0.087
0.993 0.013
0.994 + 0.013
0.978 0.025
0.778 + 0.123
Cosine Similarity
Unconstrained
0.999 + 0.001
0.949 0.047
0.987 + 0.023
0.945 + 0.045
0.997 + 0.005
0.996 + 0.004
0.998 = 0.008
0.888 + 0.085
0.96 = 0.037
0.993 + 0.004
0.999 + 0,002
0.99 0.012
0.663 + 0.164
0.903 1 0.071
0.95 = 0.041
0.995 = 0.005
0.941 + 0.103
0.986 + 0.021
0.9998 + 0.001
0.965 1 0.043
0.961 + 0.085
0.994 + 0.016
0.996 + 0.016
0.974 + 0.02
0.786 0,128
1st order theorem proving
Telco Customer Chum
Microaggregation2
LHC Identify Jet
Magic Télescope
Indian Pines
Dataset
Abalone
Ada Prior
Adult
Bank32nh
Electricity
Elevators
Helena
Heloc
Higgs
House 16H
Jannis
Mozilla4
Satellite
Phonemes
Pollen
Fars
JMI
MCI
PC2
34

---

Prediction via Shapley Value Regression
J.S. The Progress of Training and Validation Losses
In this subsection, report the progression of training and validation losses with different values of the hyperparameter using six datasets. common trend observed across models trained the six datasets is that, with different values of B, the Shapley loss (scaled by 3) consistently decreases quickly below the level of the classification loss, except for the First Order Theorem Proving dataset (Figure 15), which is multinomial classification dataset. For the First Order Theorem Proving dataset, the Shapley loss remains at a scale determined by the factor throughout the training time. However, the model for the First Order Theorem Proving dataset can still learn a function that estimates Shapley values with good accuracy, as shown in Tables 2 and 3. Moreover, benefits from larger 3 values to achieve accurate Shapley value approximations, as illustrated in Figure Additionally, the results indicate that ViaSHAP generally tends to take longer to converge as values increase.
Bank32nh (6=2)
0.8
0.6
0.2
0.0
2
3
10
11
12
13
14
#Epochs
A=16
2.00
Training Shapley
1.75
Validation Shapley Loss
Training Clessiiiation Loss
1.50
1.00
0.50
0.00
3
5
10
11
12
13
14
15
16
17
#Epochs
B=32
25
20
15
0.0
#Epochs
Figure 14. The effect value the progress of the training and the validation loss values.
35

---

Prediction via Shapley Value Regression
First Order Theorem Proving (B=2)
0.13
0.12
0.11
0.10
0.08
0.08
#Epochs
B=16
0.45
0.40
0.30
Training Shapley Loss
Validation Shapley Loss
0.25
Training Classification Loss
Validation Classification Loss
0.20
0.15
0.10
18
21
#Epochs
B=32
0.4
#Epochs
Figure 15. The effect of value the progress of the training and the validation loss values.
36

---

Prediction via Shapley Value Regression
JM1 (B=2)
1.0
0.8
0.6
0.4
0.2
0.0
1
3
4
6
7
9
10
11
12
13
14
15
16
#Epochs
B=16
1.50
1.25
1.00
0.75
0.50
0.25
0.00
2
3
5
6
7
9
10
11
12
13
14
15
16
17
18
#Epochs
B=32
3.0
Training Shapley Loss
Validation Shapley Loss
Training Classification Loss
Validation Classificatio Loss
0.5
#Epochs
Figure 16. The effect value the progress the training and the validation loss values.
37

---

Prediction via Shapley Value Regression
Phonemes (B=2)
0.6
0.5
0.4
0.3
0.2
0.1
22
24
#Epochs
B-16
Training Shapley Loss
2.0
Validation Shapley Loss
Training Classification Loss
Validation Classification Loss
1.5
1.0
0.5
0.0
6
12
15
24
27
30
33
36
39
42
45
48
#Epochs
B=32
4
3
2
10
12
24
26
#Epochs
Figure 17. The effect of /3 value on the progress of the training and the validation loss values.
38

---

Prediction via Shapley Value Regression
Heloc (B=2)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
17
#Epochs
B=16
2.00
Training Shapley Loss
1.75
Validation Shapley Loss
1.50
Training Classification Loss
Validation Classification Loss
1.25
1.00
0.75
0.50
0.25
0.00
3
8
13
#Epochs
B=32
3.5
1.5
1.0
0.5
0.0
16
20
35
68
#Epochs
Figure 18. The effect of / value on the progress of the training and the validation loss values.
39

---

Prediction via Shapley Value Regression
House 16H (B=2)
0.5
0.4
0.3
0.2
0.1
0.0
#Epochs
B=16
Training Shapley Loss
Validation Shapley Loss
Training Classification Loss
Validation Classification Loss
06
4
10
12
14
16
18
20
22
24
#Epochs
B=32
1.5
0.5
9
15
21
27
30
33
36
48
#Epochs
Figure 19, The effect value on the progress the training and the validation loss values.
40

---

Prediction via Shapley Value Regression
K. A Comparison Between ViaSHAP and FastSHAP
We compared the accuracy of ViaSHAP's Shapley value approximations to FastHHP,, using ViaSHAP as a black-box model within the FastSHAP framework. ViaSHAP implemented using KAN without link function, while FastSHAP is using the default settings. The evaluation employed metrics such as R², cosine similarity, and Spearman's 's rank correlation measure the similarity between the computed Shapley values and the ground truth. The results demonstrate that ViaSHAP achieves significantly higher similarity to the ground truth compared to FastSHAP. This conclusion is supported by the Wilcoxon signed-rank test, which enabled rejection of the null hypothesis that there is no difference in similarity to the ground truth Shapley values between ViaSHAP and FastSHAP. The test confirmed significant differences using all evaluated similarity metrics, including R², cosine similarity, and Spearman's 's rank correlation. The detailed results are available in Table 16.
L. A Comparison Between the Inference Time of ViaSHAP and KernelSHAP
In Table 15, we report the time required to explain 1000 instances using KernelSHAP and ViaSHAP (KANWi)) on six datasets using an NVIDIA Tesla V100f GPU and 16 cores of an Intel Xeon Gold 6338 processor.
Table 15. The time (in seconds) required to explain 1000 predictions from different datasets using KemeISHAP and ViaSHAP.
Dataset
KernelSHAP
KANV
Adult
56.92
0.0026
Elevators
54.22
0.0021
House 16
53. 12
0.0052
Indian Pines
43124.66
0.0023
Microaggregation 2
79.97
0.0022
First order proving theorem
436.25
0.0022
M. Limitations of ViaSHAP
ViaSHAP operates under the assumption that the selected base model can be optimized using backpropagation. Hence, models that employ other optimization algorithms, such as decision trees, not suitable for this approach. Nevertheless, ViaSHAP can be extended to work with methods that are not based on backpropagation. For example, can train one regressor per dimension of Vin (z))
The empirical results presented in Appendix H indicate that ViaSHAP does not yield accurate models when trained using the marginal expectations value function, which requires further investigation. Furthermore, as demonstrated Appendix D, ViaSHAP does not produce accurate predictors when trained using a small-sized MLP consisting of two hidden layers.

---

Prediction via Shapley Value Regression
Table 16. A comparison between ViaSHAP FastSHAP with respect the similarity the approximated Shapley values the ground
values. truth
The best-performing model
colored light green
0.996 1 0.008
0.042 : 1.359
0.853 0.298
0.728 0.728 0.29
0.914 0.306
0.983 0.023
0.991 1 0.028
0.532 #
0.824 0.177
0.986 + 0.01
0.997 1 0.016
0.89 - 0.107
-0.615 - 0.912
0.776 + 0.179
0.925 + 0.37
0.946 0.946 0.094
-0.024 + 9.964
0.966 1 0.054
0.984 0 0.049
0.126 0.793
0.274 + 1.616
0.925 0.165
0.855 0.23
0.899 0.109
0.367 + 2.832
FastSHAP
R
0.9996 + 0.0007
0.999 + 0.002
0.887 + 0.105
0.952 + 0.072
0.852 1 0.161
0.993 0.011
0.993 1 0.009
0.994 1 0.022
0.677 + 0.204
0.894 1: 0.098
0.998 = 0.005
0.964 1 0.039
0.333 - 0.192
0.722 + 0.183
0.887 + 0.206
0.881 = 0.346
0.944 1 1 0.061
0.873 + 0.151
0.891 - 0.272
0.971 0.071
0.933 : 0.276
0.924 + 0.085
0.429 + 0.479
ViaSHAP
+.977
0.98 39022.02
FastSHAP
0.966 + 0.055
0.64 0.24
0.893 0.115
0.527 0.133
0.921 0.101
0.941 0.047
0.124
0.193
10111
#.049
0.971 0.035
0.891 + 0.09
0.204 0.122
0.673 + 0.106
+.00800088
0.918 00084
0.638 0
0.97 1 0.041
0.921 0.141
0.55 0.25
0.619 0.249
0.946 0.0%
0.905 0.129
0.892 + 0.087
0.206
Spearman 's Rank
0.834
0.6
0.826
0.899
0.934
0.658
ViaSHAP
0.971 + 0.052
0.909 + 0.068
0.931 + 0.074
0.648 0.648 0.114
0.967 0.043
0.969 0.026
0.849 1 0.098
0.702 1702:0011 0.148
0.882 +:077..073
0.87 0.87 0.057
0.974 0.974 0.032
0.952 0.952:0.0 0.044
0.553 0.18
0.624 0.624+0.11
0.113
0.916 +
0.959 0.042
0.881 0.139
0.957
0.049
0.074
0.102
0.875
0.094
0.951 0
0.959 0.959
0.076
0.934 4.
0.052
0,146
0.967
0.894
0.66
0.999 + 0.002
0.703 0.25
0.956 + 0.072
0.897 - 0.079
0.978 0.06
0.006
0.997 0.021
0.822 0.139
0.935 + 0.064
0.994 1 0.004
0.999 - 0.003
0.964 0.035
0.423 0.154
0.92 + 0.064
0.98 + 0.042
0.984 0.023
0.023
0.789 0.789 0.254
0.99 0.017
0.994 + 0.017
0.858 0.114
0.786 + 0.234
0.981 + 0.036
0.984 +.98440002 0.024
0.963 0.963 #.045
0.776 0.776 0.174
FastSHAP
0.994 0.994
Cosine Similarity
ViaSHAP
0.999 + 0.0008
0.963 +: 0.037
0.991 + 0.03
0.948 0.045
0.998 + 0.004
0.997 0.997 0.004
0.997 1.008
0.874 + 0.095
0.962 + 0.036
0.991 = 0.006
0.999 + 0.002
0.988 + 0.015
0.683 + 0.171
0.898 + 0.072
0.965 = 0.042
0.994 = 0.006
0.951 + 0.093
1 0.021
0,9998 0.0003
0.976 +: 0.033
0.956 0
0.993 + 0.013
0.994 + 0.013
0.978 + 0.025
0.778 0.123
0.982 1
1st order theorem proving
Telco Customer Chur
Microaggregatton2
LHC Identify Jet
Magic Telescope
Indian Pines
Dataset
Abalone
Ada Prior
Adult
Bank32nh
Electricity
Elevators
Fars
Helena
Heloc
Higgs
House 16H
Jannis
JM1
MCI
Mozilla4
Satellite
PC2
Phonemes
Pollen
42

---

Prediction via Shapley Value Regression
N. Computational Cost
The experiments were conducted using an NVIDIA Tesla V100f GPU and 16 cores of an Intel Xeon Gold 6338 processor. The training time required for both KAN and MLPVia are recorded on 1,000 data examples with varying numbers of coalitions (Table 17). The inference time is also recorded 1,000 data example for both KAN and MLP as shown in Table 18. All the results reported the mean and standard deviation across five different runs. Generally, MLP is faster than KAN in both training and inference. Additionally, while the number of samples per data example increased exponentially, the computational cost during training did not rise at the same rate, as depicted in Figure 20.
Bank32nh
Adult
0.025
0.035
0.030
0.020
Seconds
0.015
0.020
0.015
0.010
0.005
0.000
0.000
2
16
32
Samples
Samples
House 16H
Heloc
0.025
Seconds
0.015
0.010
0.005
0.000
2
4
16
32
64
2
16
32
Samples
Samples
Phonemes
JM1
0.025
0.020
0.020
0.015
Seconds
0.015
0.010
0.005
0.000
Samples
# Samples
KA training time
ni prediction time MLP training time
MLP prediction
Figure 20. The training time and prediction time 1000 data instance of Via and ViP

---

Prediction via Shapley Value Regression
Table 17. The training time in seconds for 1000 data instances using KAN and MI..lia
0.0062 0.0000
0.0104 0.0007
0,0101 = 0.0004
0.007 + 0.0007
0.0061 0.000S
0.0004
0.0153 = 0.000
0.0073 = 0.0008
0.0065 = 0.0004
0,0055 = 0.000S
0.0003
000000
0 MUSSS = 110000
20090 = 010000
0,0004
10534 = 0.000
= 0,000
01006 = 0.000
= D.A PA4
32 Sammales
110 1AS5 +. 1-0000
0000 # 1000
EF 0
100300 =
00000 =
0.0053
0.0060
0.003
0000
04000
24000
0.00 180
0000
0000
16 Samples
0,0056 4: 0,0005 00005
0,0079 0,0005
0.0078 + 0,0004
0.0056 + 0.0005
0,0056 0,0005
0.000
0. 2011 1F 0.0000
MUS
E 0.000
0055 EH 10000
0000
257 H 1000
TOO MOND E + CUINN AODD
FL 0000
0.0033 L:
F1
H.
T H AAOI
H AOAA
F MODD
T F OAA
1000
MLP
0.0053 = 0,0007
0.0071 = 0.0003
0.0072 = 0.0003
0.0048 = 0.0009
0.0051 LL 0.0004
MADE 22
= 11XX
= - MMN
Samples
- 00000
1000
= 10000
=
1000 F -
- =
-
-
= - -
22000 -
-
=
00000
10000
0
0
00
00000
00
0
Sampling
0.002 = 0.0000
0.0003
0.003 1003- = 0.0055
0,002 + 0.0005
0,001 11 0.000
E 0000000000
5000
MIIL F 2000
2000 F 0000
C0001
10000
CHUSS F 00000
1O00 + Aoop
NX0D F 0000
10001 T 10000
connc F H Ax00
MODD F 10000
COUnI 1 MAMN
H H H 2000
+ 10000
0.0028
61000
10
000000
10000
000000
0000
00
0
32 Samples
0.0228 + 0.0016
0.0284 + 0.0021
0.0288 + 0.0027
0.0204 + 0.0012
0.023 = 0,0017
00093 1 0000000
01558 + 00000
H 10000000000
0000000000
10000 8
1000
00000
1000
000000000
10000 H 1000
1000 H 10000
1000
# # 0000
10030 H 10000
H 00000
= 1000
= 1000
- 1000
H. H
0,035
0.0200
0000
0000
0000
1000
0000
0000
1000
0.0224
Samples
0.0209 0,0009
0.0235 0.0017
0.0239 = 0.0018
0.019 + 0.0009
0.0013
1113
0,005
0.0187 = 0.000
0.0195 = 0.000
1014
1 0000
- 0,000
00013 = 0.000
= 10000
10.0000000000
02026 = 110000000
10020 #F 1000
1RS3 = 0000
1000
= 1000
MADE - 10000 SUS
0000 = CADNY
0.00 = 1000
0018 = 0.0013
0,0250 0,0016
0.0944 0.0197
0.0211
0.0274
001800
0.0995
000220
0.00
0.00
0.00
KAN
2 Samples
0.0013
0.024 0.0014
0.0227 + 0.0011
0.018 + 0,001
0,0209 1. 0.0012
2000
2001
TF COOD
1000000
20000
10000 I
H COOD
00000
00247 1F HOOD
+1000013
10117 1:
1
11
0.0208
00117
101
0.116
0000
0000
1000
1000
00053 H 1000 MAS 0000
0.00
1000
1000
1000
No Sampling
0.0058 +: 0,0005
0.0068 + 0.0005
0.0068 + 0,0008
0.0051 + 0.0005
0,0059 1. 0.0005
0.0051 T. 0.0005
0.009 # 0.0007
1:00000000000 0,0000
4005 H 0.000
1.000 1003
1. 01000
1003 H 1100
11000
+ 01000
0.000 1 H 1100 MAS
0.00S + 0100 2
0SISS 1 01000
1.000 M00
1 01080
0053 11 0000
F 01000
0.005 + 01000
0 0.0076 00000
+ 0.0066
0.0054
0,,00 11
0.0052
0
0.000
0.000
0
0.000
0.000
Tolco Customer Churn
LHC Identify Jet
MagicTelescope
Miemagreegaion
1st Ord. Theorem
Indian Pines
Dataset
Abalone
Ada Prior
Adult
Bank 32sh
Electricity
Elevators
Fars
Helena
Heloe
Higgs
House 16H1
Junnis
JMI
MC1
Movillad
Satellie
PC2
Phonemes
Pollen
44

---

Prediction via Shapley Value Regression
Table 18. The prediction running time in seconds for 1000 data instances using KAN and MLPVie
Dataset
KAN\Va
MI.PVia
Abalone
0.0024 + 0.0003
0.0004 + 0.00003
Ada Prior
0.003 + 0.0008
0.0006 +: 0.000005
Adult
0,0026 + 0.0004
0.0006 + 0,000005
Bank32nh
0.0021 = 0.0002
0.0004 + 0.0001
Electricity
0.0024 + 0.0003
0.0005 + 0.0002
Elevators
0.0021 = 0.0002
0.0005 = 0.0003
Fars
0.0031 + 0.0005
0.0009 0.0001
Helena
0.0023 +: 0.0004
0.0004 = 0.0001
Heloc
0.0022 +: 0.0002
0.0003 + 0.000005
Higgs
0.0022 + 0.0002
0.0003 "0.00001
LHC Identify Jet
0.0023 + 0.0004
0.0004 + 0.00001
House 16H
0.0052 + 0.0005
0.0004 + 0.0001
Indian Pines
0.0023 + 0.0003
0.0004 + 0.0001
Jannis
0.0023 + 0.0003
0.0004 + 0.0000
JMI
0.0026 + 0.0012
0.0003 + 0.00001
MagicTelescope
0.0022 + 0.0002
0.0003 + 0.00001
MC1
0.0023 + 0.0003
0.0004 + 0.0001
Microaggregation 2
0.0022 + 0.0002
0.0004 + 0.00001
Mozilla 4
0.0022 + 0.0002
0.0004 0.0001
Satellite
0.0022 + 0.0003
0.0004 0.0001
PC2
0.0021 + 0.0003
0.0003 + 0.00001
Phonemes
0.0021 + 0.0001
0.0003 + 0.000005
Pollen
0.0022 + 0.0003
0.0004 + 0.0001
Telco Customer Churn
0.003 + 0.0005
0.0009 + 0.0001
1st Order Theorem Proving
0.0022 + 0.0003
0.0004 + 0.000004
45

---

Prediction via Shapley Value Regression
O. Dataset Details
Table 19 presents an overview of the datasets used in the experiments. The table includes the number of classes, number features, dataset size, training, validation, and test split sizes. Additionally, the table provides the corresponding dataset ID from OpenML.
Table 19.
The dataset information.
OpenML ID
720
1037
1590
833
151
846
40672
41
45023
23512
42468
821
41972
41168
1053
1120
1056
41671
1046
40900
1069
1489
871
42178
1475
Test Set
833
912
2442
1.229
4.531
2,490
10.097
13,040
1,250
4,902
41 41,500
2,278
1.829
16,747
1.088
1.902
994
4,000
1,554
1.147
1,118
811
770
1.056
1,224
Val. Set
836
91.
2443
1.229
4.532
2,490
10,097
10,432
1,250
4,903
39,425
2,279
1,463
13,398
1,089
1.902
994
3,200
1.555
1.148
1.118
811
770
1.057
979
Train. Set
2506
2737
43957
5,734
36,249
11.619
80,774
41,724
7.500
38,245
749.075
18, (8,227
5.852
53,588
8,708
15,216
7,478
12,800
12,436
2,805
3.353
3,782
2,308
4,930
3,915
Dataset Size
4177
4562
48842
8,192
45,312
16,599
100.968
65, 15,196
10,000
98,050
830,000
22, 78.784
9,144
93,733
10,885
19,020
9,466
20,000
15,545
5,100
5,589
5,404
3,848
7,043
6,118
# Classes
100
2
2
2
8
2
5
8
2
2
2
5
2
2
2
2
2
2
6
# Features
14
14
32
18
29
27
22
28
16
16
220
54
21
10
38
20
5
36
36
19
51
1st Order Theorem Proving
Telco Customer Churn
Dataset
LHC Identify Jet
Microaggreeattoon
Magic Telescpe
Indian Pines
House 16H
Ada Prior
Bnh
Electricity
Elevators
Mozilla 4
Phonemes
Abalone
Satellite
Adult
Fars
Helena
Heloc
Higgs
Jannis
JM1
MC1
PC2
Pollen
46


Reviews:
[{"review_id": "KsC1A04rkw", "reviewer": "Reviewer_TGJg", "summary": "The paper proposes a new method, ViaSHAP, which learns a function that computes Shapley values alongside the model’s prediction. This method works by training a machine learning model that minimizes a weighed least squares lost similar to that of KernelSHAP and FastSHAP. The authors present many experiments showing that the predictions of the model on tabular data are on-par with state-of-the-art tabular models (XGBoost, MLP, Kolmogorov-Arnold Networks) and that the explanations are more accurate than FastSHAP.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and valuable feedback on our manuscript. We provide answers and clarifications below.\n\n> The authors argue that one major drawback of existing methods is that Shapley values are computed post-hoc....\n\nWe do not intend to suggest that computing Shapley values in a post-hoc setup is inherently a drawback. Rather, our work builds on existing post-hoc methods and aims to further reduce computational costs by eliminating the need for training a separate explainer model and running two models at inference time. In particular, it is true that in cases where a pre-trained model already exists or when users cannot access or modify the black-box model, post-hoc methods might be more suitable. Nonetheless, this is not always the case. ViaSHAP offers an alternative for users who prioritize developing an inherently explainable model, as well as for scenarios where computational efficiency and resources are important considerations.\n\n> There are already several works on computing Shapley's explanation faster. Again, even though this work is new from the angel learning to explain rather than postdoc explanation, still the methologies is not very distinct from FastSHAP.\n\nOur proposed method is indeed inspired by FastSHAP. However, a key difference is that we do not start with a pre-trained black-box model that requires post-hoc explanations. This difference is fundamental in the settings, i.e., building a by-design explainable model vs. explaining a pre-trained model. In the adapted setting, ViaSHAP thus reduces the computational cost needed from training two models to a single one. The use of a by-design explainable model also ensures that the model explains itself, thus providing explanations that align with the prediction. On the other hand, explaining a black-box model through post-hoc methods leaves the explanations open to the inherent randomness issues of statistical learning [1].\n\n> It would be good to discuss the recent papers on amortized SHAP...\n\nWe thank the reviewer for pointing out the missing recent paper. We will indeed add it to the discussion.\n\n> In what problem settings should one decide to use ViaSHAP instead of training a regular model and then apply FastSHAP?\n\nViaSHAP is particularly beneficial in environments with limited time or computational resources, as it eliminates the need to train and run separate models for prediction and explanation. Additionally, it is well-suited for scenarios where high-fidelity explanations are a central concern, as the model’s predictions are inherently tied to its explanations.\n\nOn the other hand, post-hoc explanation methods remain necessary when users cannot access or modify the black-box model or when the model does not support optimization via gradients and backpropagation. We do not propose to replace or discard post-hoc methods, but rather to offer an alternative approach that promotes explainability while addressing computational efficiency.\n\n> how much computation time will be saved using ViaSHAP...\n\nWhen comparing ViaSHAP to KernelSHAP, ViaSHAP significantly reduces computational cost at inference time by avoiding the need to solve a separate optimization problem for each prediction. Therefore, computational cost is reduced by an order of magnitude, as demonstrated in Appendix L and Table 15.\n\nWhen compared to real-time explanation methods such as FastSHAP, ViaSHAP further reduces computation at training time (by avoiding the training of a separate explainer model) and at inference time (by running inference on a single model instead of two). \n\nMore importantly, the computational cost of ViaSHAP at inference time remains the same as a similar model that does not compute Shapley values. Detailed computational cost analyses for ViaSHAP can be found in Appendix N.\n\n> Can ViaSHAP be used while training LLMs? What are the tradeoffs and fundamental computational limits?\n\nIn principle, ViaSHAP could be adapted to train LLMs. However, there are significant challenges. For instance, LLM training is exceptionally computationally intensive, and incorporating Shapley-based explanations during training would add more computational burden at training time, as ViaSHAP requires sampling from the input and propagating gradients. From the previously evoked results (Appendix N), the inference time should not be harmed by this procedure.\n\nAnother possible solution is to use ViaSHAP on pre-trained LLMs, where all the learned parameters of the model are fixed, and one or two output layers are modified and optimized to approximate the explanation.\n\nThat being said, we cannot provide an accurate answer without a proper study similar to what we did with tabular data and images.\n\n[1] Rudin et al., Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead, 2019", "claims_and_evidence": "Strengths\n\nThe authors perform experiments across a large number of datasets, showing that the predictive power of ViaSHAP is on-par with other methods. Additionally, they show that ViaSHAP generates more accurate explanations than FastSHAP. The main selling point of ViaSHAP is to train a model that simultaneously provides accurate predictions and Shapley values.\n\nWeaknesses\n\nThe authors argue that one major drawback of existing methods is that Shapley values are computed post-hoc, that is, after a model has already been trained. More specifically, “generating instance-base explanations or learning a pre-trained explainer always demands\nfurther data, time, and resources” (page 1). The authors’ main selling point of ViaSHAP, then, is the ability to train a single model that generates both accurate predictions and Shapley values—training a model plus training a FastSHAP explainer model unnecessarily increases computational resources. However, the importance of computing Shapley values efficiently lies in explaining already existing machine learning models. Therefore, one might argue that the authors’ claim that generating Shapley explanations post-hoc is a drawback is a fundamentally biased view of existing Shapley literature.\n\nThe methodologies are highly similar and inspired by FashSHAP, even though FastSHAP is a posthoc method and ViaSHAP is not.", "methods_and_evaluation_criteria": "The method is explained clearly, and the evaluation criteria make sense.", "theoretical_claims": "No theoretical claims.", "experimental_designs_or_analyses": "Extensive empirical evaluations have been performed. Several of these are supplementary, perhaps due to space constraints. But it would be good to have some summary tables of these results in the main text.", "relation_to_broader_scientific_literature": "There are already several works on computing Shapley's explanation faster. Again, even though this work is new from the angel learning to explain rather than postdoc explanation, still the methologies is not very distinct from FastSHAP.", "essential_references_not_discussed": "It would be good to discuss the recent papers on amortized SHAP, which directly related to the fast computation of SHAP, for example, \n\n[1] Stochastic Amortization: A Unified Approach to Accelerate Feature and Data Attribution https://arxiv.org/pdf/2401.15866\n[2] SHAP zero Explains Genomic Models with Near-zero Marginal Cost for Future Queried Sequences https://arxiv.org/abs/2410.19236", "other_strengths_and_weaknesses": "Discussed before.", "other_comments_or_suggestions": "Discussed before.", "questions_for_authors": "1) In what problem settings should one decide to use ViaSHAP instead of training a regular model and then apply FastSHAP?\n2) In total, how much computation time will be saved using ViaSHAP in the previous question?\n3) Can ViaSHAP be used while training LLMs? What are the tradeoffs and fundamental computational limits?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Jlv8U8FwKU", "reviewer": "Reviewer_L8aK", "summary": "This paper proposes a method that simultaneously computes both the Shapley values and the predicted output, where the predicted output is equal to the sum of the Shapley values. To achieve this, the authors train the network to learn and approximate Shapley values by minimizing the weighted least squares in Eq. (6), while simultaneously optimizing the predicted output in Eq. (7). The authors employ MLP and KAN as the model architectures. Experimental results on several tabular datasets demonstrate the effectiveness of the proposed explanation method.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's feedback. We provide answers in the following part.\n\n> My major concern is its positioning...\n\nThe proposed method does not fall strictly into the categories of model-specific or model-agnostic approaches. Instead, we introduce a framework for training by-design explainable models, where predictions and their Shapley value explanations are learned simultaneously. This distinguishes our approach from existing methods that compute Shapley values post-hoc or in a separate forward pass.\n\n> ... authors should sufficiently compare it with prior works...\n\nWe appreciate the reviewer’s suggestion. Our primary objective, however, is to demonstrate that the proposed method is inherently explainable and that the generated explanations are accurate. To this end, we compared our approach to the unbiased KernelSHAP, as well as FastSHAP, which directly inspired our method and represents the most closely related work. Unfortunately, due to the constraints of the rebuttal phase, we are unable to add new results at this stage. However, we acknowledge the value of broader comparisons.\n\n> ...on complex datasets such as CIFAR-10...\n\nWe have indeed reported the model’s predictive performance on complex datasets such as CIFAR-10, along with the approximation error of the explanation, and compared the results to FastSHAP. We kindly refer the reviewer to Appendix F, which provides detailed results for CIFAR-10.\nThe evaluation of Shapley value estimates on images using the root mean square error requires access to ground truth Shapley values, which is computationally infeasible for image datasets. Instead, we assess explanation quality using inclusion and exclusion curves, which measure an explanation’s ability to identify informative image regions. This evaluation strategy is commonly used for image datasets, as seen in [1,2,3].\n\n> Is the second categorical loss in Eq. (7) incorrectly written?...\n\nWe thank the reviewer for pointing out that the notations in Eq.7 are confusing. Indeed, j in Eq.7 represents the true probability to be estimated for each category. We will correct and clarify this in the revised paper.\n\n> ...explicitly explain the physical meaning of Eq. (6)...\n\nWe appreciate the reviewer’s suggestion. We explained the intuition behind Eq.6 in lines [173,186] and illustrated the main idea in Figure 2 and Algorithm 1. However, as recommended by the reviewer, we will further refine our explanation to enhance clarity.\n\n> ...using KernelSHAP as the ground-truth...\n\nWe agree with the reviewer that KernelSHAP is unreliable, particularly for high-dimensional data. Therefore, we employed an unbiased version of KernelSHAP [4], which aligns with the reviewer’s suggested approach [5], as mentioned in lines 358 and 359 \" the ground truth Shapley values ($\\phi$), computed by the unbiased KernelSHAP\". The same evaluation setup, using unbiased KernelSHAP, has been employed in [3]. For the tabular datasets, we allowed the unbiased KernelSHAP to keep sampling until convergence. The bias of KernelSHAP shrinks to zero as the sample size grows [4,6]. It has been shown that the unbiased KernelSHAP converges to the true values if provided with a large enough number of samples [3,4], which we did for the tabular data in the experiments. \n\n> Using cosine similarity as the metric...\n\nWe completely agree with the reviewer that cosine similarity alone is insufficient for comparing explanations. Therefore, we employed three metrics: Spearman rank correlation, $R^2$, and cosine similarity. The results are reported using the three metrics in Tables 7, 9, 12, 14, and 16.\n\n> ... include a broader set of model-agnostic approaches...\n\nWe acknowledge that incorporating additional experiments can enhance the quality of the paper. We want to clarify that the method in reference [5] provided by the reviewer (Covert et al. Improving KernelSHAP...) is the unbiased KernelSHAP method, which we have already compared our proposed method with in our evaluation.\n\n> lacks a discussion of related works...\n\nWe appreciate the reviewer’s suggestions for additional related work to discuss. We kindly clarify that the papers (Mitchell et al. Sampling Permutations for Shapley Value Estimation) and (Covert et al. Improving KernelSHAP) are already cited multiple times in our manuscript.\n\n[1] Hooker, S., et al. A benchmark for interpretability methods in deep neural networks.\n\n[2] Jethani, N., et al. Have we learned to explain?: How interpretability methods can learn to encode predictions in their interpretations. AISTATS 2021\n\n[3] Jethani, N., et al. FastSHAP: Real-time shapley value estimation. ICLR, 2022\n\n[4] Covert, I. et al. Improving kernelshap: Practical shapley value estimation using linear regression. AISTATS 2021\n\n[5] Castro, J., et al. Polynomial calculation of the shapley value based on sampling. Computers & Operations Research\n\n[6] Covert, I., et al. Stochastic Amortization: A Unified Approach to Accelerate Feature and Data Attribution.", "claims_and_evidence": "1.\tMy major concern is its positioning within the existing literature. If the proposed method is model-specific, prior works such as [1][2] have already demonstrated approaches to compute exact Shapley values in a single forward pass. If the method is intended to be model-agnostic, various approaches [3][4][5] already exist to compute Shapley values either in an unbiased or biased manner.\n\nSince the proposed method does not apply to arbitrary black-box models, but rather trains an inherently interpretable model that also predicts outputs, the authors should sufficiently compare it with prior works (not limited to the listed interpretable models), and highlight the advantages of the proposed method.\n\n2.\tThe main challenge of jointly training a model to predict Shapley values while performing task predictions is that, it may be difficult to achieve great model performance on complex tasks with relatively accurate Shapley values. For example, on complex datasets such as CIFAR-10, CIFAR-100, and Tiny ImageNet, the authors should report both the classification accuracy and the approximation error of the Shapley values (e.g., root mean square error). Besides, the choice of the trade-off parameter $\\beta$ in Algorithm 1 should be reported in different tasks.\n\n\n\n[1] Chen et al. HarsanyiNet: Computing Accurate Shapley Values in a Single Forward Propagation. ICML, 2023.\n\n[2] Wang et al. Shapley explanation networks. ICLR, 2021.\n\n[3] Castro et al. Polynomial calculation of the shapley value based on sampling. Computers & Operations Research, 36(5):1726–1730, 2009.\n\n[4] Mitchell et al. Sampling permutations for shapley value estimation. Journal of Machine Learning Research, 23:1–46, 2022.\n\n[5] Covert et al. Improving kernelshap: Practical shapley value estimation using linear regression. International Conference on Artificial Intelligence and Statistics, 2021.", "methods_and_evaluation_criteria": "Is the second categorical loss in Eq. (7) incorrectly written? Why is it $-j log (\\hat{j})$, where $j$ is the category, $j\\in\\\\{1, \\cdots, d\\\\}$, and $\\hat{j}$ is the probability of the $j$-th category. \n\nIn addition, it would be helpful to explicitly explain the physical meaning of Eq. (6) to improve clarity.", "theoretical_claims": "Theorem 1 is not carefully checked.", "experimental_designs_or_analyses": "1.\tIn Lines 278-283, using KernelSHAP as the ground-truth Shapley value may be inappropriate. One concern is that the number of samples required for KernelSHAP to produce reliable estimates depends on the complexity of the task [1]. A more rigorous approach would be to compute exact Shapley values using the definition for datasets with a small number of input variables. For larger image datasets, the authors should compare their method with unbiased sampling-based Shapley value estimation [3].\n\n2.\tUsing cosine similarity as the metric for evaluating the difference between approximated and ground-truth Shapley values may be insufficient. The authors could consider incorporating additional metrics, such as root mean square error, to provide a more comprehensive assessment.\n\n\n3.\tIn Section 4.3, when comparing interpretability methods, it is necessary to include a broader set of model-agnostic approaches, such as those proposed in [3], [4], and [5].", "relation_to_broader_scientific_literature": "See \"Claims And Evidence\"", "essential_references_not_discussed": "The paper lacks a discussion of related works in closely related directions.\n\n1.\tMethods that compute exact Shapley values and predict model outputs in a single forward pass:\n\n[1] Chen et al. HarsanyiNet: Computing Accurate Shapley Values in a Single Forward Propagation. ICML, 2023.\n\n[2] Wang et al. Shapley explanation networks. ICLR, 2021.\n\n\n2.\tApproximation algorithms for Shapley values:\n\n[3] Castro et al. Polynomial calculation of the shapley value based on sampling. Computers & Operations Research, 36(5):1726–1730, 2009.\n\n[4] Mitchell et al. Sampling permutations for shapley value estimation. Journal of Machine Learning Research, 23:1–46, 2022.\n\n[5] Covert et al. Improving kernelshap: Practical shapley value estimation using linear regression. International Conference on Artificial Intelligence and Statistics, 2021.\n\n3. Other methods for attribution computation in a single forward pass", "other_strengths_and_weaknesses": "Strengths:\n\n1.\tThe paper is well written and easy to follow.\n\n2.\tThe authors explored the possibility of training both predicted shapley values and predicted outcomes.\n\nWeaknesses:\n\nSee “Claims And Evidence”,“Methods And Evaluation Criteria” and “Experimental Designs Or Analyses.”", "other_comments_or_suggestions": "N/A", "questions_for_authors": "See “Claims And Evidence”,“Methods And Evaluation Criteria” and “Experimental Designs Or Analyses.”", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PVu4gjmK5u", "reviewer": "Reviewer_86f8", "summary": "The paper presents a method called ViaSHAP which aims to learn a function to compute the Shapley Values as the model trains. This function predicts the Shapley Values from inputs, directly uses those values to form the model output and bypasses the need for post-hoc computation (i.e. to fit a KernelSHAP to the model). ViaSHAP is implemented as an constraint on a network for an MLP and KAN architecture. The Shapley results are shown to perform comparably to KernelSHAP and FastSHAP across several experiments without a significant loss in model performance.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's time and feedback on our paper. Below, we provide our responses and clarifications.\n\n> I'm a bit concerned with how the comparisons between ViaSHAP and the ground truth are made...\n\nWe appreciate the reviewer’s concern and would like to clarify our experimental setup. In the explainability evaluation, we treated ViaSHAP models as standard black boxes and computed the ground truth Shapley values using the unbiased KernelSHAP [1], which involves solving an optimization problem for each prediction. We then compared the explanations generated by ViaSHAP to the ground truth obtained from the unbiased KernelSHAP.\n\nAs established in previous work, the bias of KernelSHAP shrinks to zero as the sample size grows [1, 2], and with a sufficiently large number of samples, it converges to the true Shapley values [1, 3]. For the tabular datasets, we allowed the unbiased KernelSHAP to continue sampling and updating the learned values until convergence. We followed the same evaluation setup using the unbiased KernelSHAP that has been employed in [3].\n\n> The paper focuses on the classification case with regression being briefly discussed in the appendices...\n\nWe appreciate the reviewer’s feedback and acknowledge that incorporating additional regression datasets could further strengthen the experiments. However, due to the constraints of the rebuttal phase, we are unable to add new results at this stage.  \n\nWe would also like to clarify that the term *regression* in the paper title, *Prediction via Shapley Value Regression*, specifically refers to the fact that we solve the regression task of Shapley values and use Shapley values in the prediction task itself. In other words, the title is intended to convey *\"(Classification and Regression) via Shapley Value Regression.\"*\n\n> incorporating more discussion about when this method would be best used...\n\nWe sincerely appreciate the reviewer’s suggestion and will update the manuscript accordingly.\nViaSHAP is particularly advantageous in settings where computational resources or time are limited, as it removes the need to train and run separate models for prediction and explanation. Moreover, it is well-suited for scenarios where high-fidelity explanations are crucial, as the model’s predictions are inherently tied to the explanations.\n\nPost-hoc methods might be more suitable in cases where a pre-trained black-box model is already employed or when users cannot access or modify the black-box model.\n\n> Why is it the case that the results are particularly weak over certain datasets (such as pollen)?\n\nSimilar to other machine learning models, ViaSHAP's performance can vary depending on the characteristics of the dataset. Certain datasets pose inherent challenges, such as a limited number of training examples, a high-dimensional feature space, complex data distributions, or noisy data, all of which can impact model performance.\nWhile ViaSHAP's performance on the Pollen dataset is relatively weaker compared to other datasets, it is noteworthy that it also outperforms XGBoost, Random Forests, and TabNet on this dataset, which suggests that the Pollen dataset is inherently challenging, rather than an issue specific to ViaSHAP.\n\n> ...why is it the case that the AUC of XGBoost does not have a +- value...\n\nAll compared models are trained and evaluated using the same data splits to ensure a fair comparison. The $\\pm$ values indicate variations in performance due to different random initializations, as stated in the experimental setup (Lines [312, 315]):\n*\"If the model’s performance varies with different random seeds, it will be trained using five different seeds, and the average result will be reported alongside the standard deviation.\"*\nSince XGBoost with the default settings is deterministic and its performance does not vary across different random seeds, its AUC is reported without a $\\pm$ value.\n\n> What exactly does Figure 5 show? I'm not sure how much this illustration contributes to the main paper.\n\nFigure 5 illustrates that the explanations generated by ViaSHAP are more precise compared to those from FastSHAP when applied to the same image. Specifically, FastSHAP tends to highlight broader regions, whereas ViaSHAP primarily focuses on the shape of the classified instance within the image, providing a more precise and sparse explanation.\n\n[1] Covert, I. and Lee, S.-I. Improving kernelshap: Practical shapley value estimation using linear regression. In Proceedings of The 24th International Conference on Artificial Intelligence and Statistics.\n\n[2] Covert, I., Kim, C., Lee, S., Zou, J., Hashimoto, T. Stochastic Amortization: A Unified Approach to Accelerate Feature and Data Attribution. In The Thirty-eighth Annual Conference on Neural Information Processing Systems 2024.\n\n[3] Jethani, N., Sudarshan, M., Covert, I. C., Lee, S.-I., and Ranganath, R. FastSHAP: Real-time shapley value estimation. In the International Conference on Learning Representations, 2022", "claims_and_evidence": "The claims are quite clear and the experiments are quite exhaustive.", "methods_and_evaluation_criteria": "Comparisons with KernelSHAP and FastSHAP across the tested datasets makes sense for this application. The performance of ViaSHAP is then compared with XGBoost, TabNet, and Random Forests to validate the model performance.", "theoretical_claims": "I don't see any particular issues with the proofs presented.", "experimental_designs_or_analyses": "I'm a bit concerned with how the comparisons between ViaSHAP and the ground truth are made, as noted in lines 358-360, the ground truth KernelSHAP values are computed using the ViaSHAP network as a black box. Is the comparison then how different the output of a model are to a (weighted) least squares approximation of that model itself?\n\nOther than that, the experimental design all made sense to me.", "relation_to_broader_scientific_literature": "This paper contributes a method of significantly reducing the cost of computing Shapley Values by removing the need for post-hoc fitting.", "essential_references_not_discussed": "The citations are quite exhaustive", "other_strengths_and_weaknesses": "Posed in questions section.", "other_comments_or_suggestions": "Overall I quite like the idea of the paper, some minor issues in below sections.\n\nMinor Comments:\n- The paper focuses on the classification case with regression being briefly discussed in the appendices, which doesn't seem to reflect the paper title very well", "questions_for_authors": "Questions:\n- I would suggest incorporating more discussion about when this method would be best used, and why one would choose it over other \nmethods.\n- Why is it the case that the results are particularly weak over certain datasets (such as pollen)?\n- In Table 1, why is it the case that the AUC of XGBoost does not have a +- value associated? Additionally, what exactly do the +- values indicate, are they variations over random (seed) initialization of the model, or over different train/test splits?\n- What exactly does Figure 5 show? I'm not sure how much this illustration contributes to the main paper.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2tCu7ylI8F", "reviewer": "Reviewer_1ExE", "summary": "This paper introduces a framework for estimating Shapley values in explainable AI. Typically, a single observation $\\mathbf{x}$ is considered for attribution and gives rise to a set function $v: 2^{[n]} \\to \\mathbb{R}$. Then we compute the Shapley values for just the set function $v$ i.e., $\\phi_i = \\frac1{n} \\sum_{S \\subseteq [n] \\setminus \\{i\\}} \\frac{v(S \\cup \\{i\\}) - v(S)}{\\binom{n-1}{|S|}}$. If we have many observations $\\mathbf{x}$ that we want to explain, this is clearly a computational issue. In prior work (FastSHAP) and this work (ViaSHAP), the goal is to learn one function that simultaneously outputs the Shapley values for many observations $\\mathbf{x}$ and the induced set functions $v$.\n\nThe authors propose a training method with two loss functions:\n• the prediction loss which is the standard measure of how accurately the model can recover the label $y$ for a given input $\\mathbf{x}$\n• the loss on the Shapley estimates produced by the model, the goal is for the Shapley values to sum to the correct value on a given coalition $S$\n\nThey evaluate their method on 25 (impressive!) datasets and four different networks. However, they don't seem to use the ground truth Shapley values and instead estimate them with approximation methods like Kernel SHAP.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's time and feedback. Please find our responses below.\n\n> A) It's unclear to me what it means to run KernelSHAP until convergence...\n> B) KernelSHAP produces estimates which are slightly biased...\n\nWe agree with the reviewer that KernelSHAP can be biased, therefore, we employed the unbiased KernelSHAP [1]. For the tabular datasets, we allowed unbiased KernelSHAP to continue sampling and updating the learned values iteratively until convergence, meaning that the sampling is not restricted to a fixed number of coalitions but continues until the estimates are stable.\n\nThe bias of KernelSHAP shrinks to zero as the sample size grows [1,2]. Furthermore, it has been shown that the unbiased KernelSHAP converges to the true Shapley values given a sufficiently large number of samples [1,3], which we provided for the tabular data. We employ the same evaluation setup that has been employed in [3].\n\n> I'm concerned about how they compute ground truth...Use a small value of $2^n$...\n\nWe included datasets with a small number of features $n$, such as Phonemes (5 $n$), Pollen (5 $n$), Mozilla 4 (5 $n$), Abalone (8 $n$), Electricity (8 $n$), and MagicTelescope (10 $n$). For these datasets, the total number of possible coalitions is small enough that the unbiased KernelSHAP solution provides the exact Shapley values, as all possible coalitions are generated.\n\n> ... Use a linear model.... Use a decision tree\n\nWe appreciate the reviewer's perspective. However, training a linear model or TreeSHAP to obtain the exact Shapley values may not be applicable to evaluating the explainability of ViaSHAP, as ViaSHAP explains its own predictions rather than the predictions of a separate model, e.g., a decision tree where exact Shapley values can be obtained. Also, ViaSHAP, in the current form, cannot be implemented using decision trees.\n\n> They state three lemmas and a theorem, but I would describe these as \"decorative theory\"...\n\nWe respectfully disagree that the lemmas and the theorem are of marginal value for the following reasons:\n\n1-The proofs of Lemmas 3.2 and 3.3 **provide upper bounds on the error associated with satisfying the missingness and consistency** properties under realistic optimization assumptions. \n\n2-Without the theoretical results, there would be no formal guarantee that ViaSHAP computes Shapley values, given it differs from post-hoc explainers and computes the Shapley values before making predictions.\n\n> simple baseline is missing: Train a decision tree model...\n\nWe want to clarify again that ViaSHAP explains itself rather than acting as a post-hoc explainer. Since ViaSHAP, as currently proposed, can only be implemented with algorithms that can be optimized using backpropagation (i.e., cannot be implemented using decision trees), we cannot see how training a decision tree-based model and explaining it could help evaluate ViaSHAP's explainability.\n\n> \"There are $2^n - 1$ possible coalitions\", I guess this is semantic but doesn't the empty set trivially count as a coalition?\n\nWe mean that there are $2^n - 1$ possible coalitions that can be used to compute the exact Shapley values, since we cannot add a new player $i$ to the grand coalition $S$ of all players as follows: $\\frac{|S|! (n - |S| - 1)!}{n!} (v(S \\cup \\\\{i\\\\}) - v(S))$. We will update the sentence to be clear.\n\n> Notation of $p(S)$ and $p(x)$ in 2.4 is confusing..\n\nWe agree with the reviewer and will update the notation to eliminate confusion.\n\n> Figure 3 is very strange...\n\nFigure 3 is a standard visualization for summarizing the results of Friedman-Nemenyi statistical significance tests. Similar plots have been used in [4,5,6]. For a detailed breakdown of the results, please refer to Table 1.\n\n> are you computing the Shapley values for each pixel...\n\nWe computed Shapley values using super-pixels of size 2×2. However, this is a hyperparameter, and users can choose between pixel-wise explanation or larger super-pixel sizes.\n\n> title \"Prediction via Shapley Value Regression\" is very broad...\n\nMethods like KernelSHAP indeed involve Shapley value regression. However, a key difference is that they do not formulate predictions using the regressed Shapley values, which is central to our approach. We appreciate the reviewer’s suggestion for an alternative title and are open to modifying the title to be more specific.\n\n[1]-Covert, I., et al. Improving kernelshap: Practical shapley value estimation using linear regression. AISTATS 2021.\n\n[2]-Covert, I., et al. Stochastic Amortization: A Unified Approach to Accelerate Feature and Data Attribution. NeurIPS 2024.\n\n[3]-Jethani, N., et al. FastSHAP: Real-time shapley value estimation. ICLR, 2022\n\n[4]-Dedja, K., et al. BELLATREX: Building Explanations through a LocaLly AccuraTe Rule EXtractor.\n\n[5]-Werner, H., et al. Evaluating Different Approaches to Calibrating Conformal Predictive Systems.\n\n[6]-Pugnana, A., et al. Deep Neural Network Benchmarks for Selective Classification", "claims_and_evidence": "They claim \"we generate ground truth Shapley values by running KernelSHAP until it converges since it has been demonstrated that KernelSHAP will converge to the true Shapley values when given a sufficiently large number of data samples\". But, KernelSHAP produces estimates which are slightly biased so the \"sufficiently large number of data samples\" required to recover the true Shapley values is $2^n$ since KernelSHAP samples without replacement.", "methods_and_evaluation_criteria": "As mentioned above, I'm concerned about how they compute ground truth Shapley values. I would recommend running experiments using all of the following approaches:\n• Use a small value of $n$ where the Shapley values can be computed exactly\n• Use a linear model as the model to be explained where the Shapley values are simply the coefficients of the model\n• Use a decision tree or forest model as the model to be explained where the Shapley values can be exactly computed using Tree SHAP", "theoretical_claims": "They state three lemmas and a theorem, but I would describe these as \"decorative theory\" that are straightforward and of marginal value.", "experimental_designs_or_analyses": "I think an obvious and simple baseline is missing: Train a decision tree model (e.g., XGBoost). Then for a given input $\\mathbf{x}$, exactly (and efficiently) compute the Shapley values using Tree SHAP. This approach is far simpler than creating their model to produce Shapley values, and the computation of the Shapley values is exact; the only approximation is from the original explainer model to the decision tree model.", "relation_to_broader_scientific_literature": "Inspired by and a follow up work to Fast SHAP.", "essential_references_not_discussed": "I don't think so.", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "Miscellaneous comments:\n• Third paragraph in 2.2 looks like it was supposed to be commented out?\n• 2.3 \"There are $2^n-1$ possible coalitions\", I guess this is semantic but doesn't the empty set trivially count as a coalition?\n• Notation of $p(S)$ and $p(x)$ in 2.4 is confusing because two different distributions but written like one function that takes both input data $x$ and coalitions $S$.\n• 3.3 second paragraph \"bounded domain can [be] represented by a finite sum\"\n• Figure 3 is very strange to me. Could you represent as a table? Or bar plots?", "questions_for_authors": "• In the image experiments, are you computing the Shapley values for each pixel in the image input?\n• The title \"Prediction via Shapley Value Regression\" is very broad, and could include e.g., Kernel SHAP. Can you make it more specific and relevant to your work e.g., \"A Framework for Simultaneously Predicting Shapley Values on Many Observations\"?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes a method dubbed ViaShap, which is an interesting variation of Shapley-based methods that jointly learns to predict outputs and compute Shapley value explanations in a single model. This sits in-between learning to explain and post-hoc methods; Unlike the latter, ViaSHAP trains a network (via MLPs and KANs) such that the output is the sum of its learned Shapley values. The method optimizes two losses: one for prediction accuracy and another to ensure Shapley consistency via a weighted least squares loss inspired by KernelSHAP and FastSHAP. Experimental results across 25 tabular datasets show that ViaSHAP achieves predictive performance comparable to state-of-the-art models (e.g., XGBoost, MLP) and produces more accurate Shapley explanations than FastSHAP, without significant performance degradation. While exact Shapley values are not used, the authors benchmark their estimates against approximations from KernelSHAP.

There is broad general support for this papers. All reviewers praised the extensive numerical validation across a large number of diverse datasets. Reviewers TGJg and L8aK had some concerns on similarities and differences with FastShap, Kernel Shap, the potential limited utility (not being applicable to off-the-shelf models), and concerns on their notions of "ground-truth" Shapley values. All/most concerns were addressed successfully. 

Reviewer 86f8 had some concerns on the empirical evaluation. These were clarified by the authors, and I agree with the authors in that the score provided by 86f8 is disproportionately low compared to their written evaluation. Finally, reviewer 1ExE had some concerns on the empirical validation and methods of comparison, and suggested experiments based on tree-based models. The authors addressed these concerns explaining why they are not relevant here. While I agree that the experiment suggested by the reviewer is interesting (and the authors should consider adding it to the paper), it is not severe to prevent my recommendation for acceptance.
Decision: Accept (poster)