dataesr/scientific-paragraphs-categorization huggingface dataset:
, a m } with each new attribute being a linear composition of the previous attributes : ∀i, a i ∈ A , ∃ {α 1 ,. , α n } ∈ R n , a i = α 1 * a 1 +. + α n + a n. Each new instance is associated with an instance of the previous dataset. ∀x = {x 1 ,. , x m } ∈ X , ∃! x ∈ X, ∀i ∈ [1,. , m]∃ α 1 ,. , α n ∈ R n , x i = α 1 * x 1 +. + α n + x n. Given this set of factors α 1 ,. , α n , for each attribute, we consider each factor as an evaluation of the importance of the attributes in the group. We can then constitute a coalition of attributes by exploiting the groups formed by the most important factors. This gives us the algorithm 2. For the sake of simplicity, we consider each a ∈ A as a vector of its α i factors. 4. We have two principal components : A = 0. 65D -0. 4B + 0. 25C + 0. 1A and B = 0.
VIF and revVIF based coalition The variance inflation factor (VIF) is an estimation of the multicollinearity of the attributes of the dataset regarding a given target attribute. Given a dataset D = (A, X), the VIF value of a ∈ A is calculated by running a standard linear regression with a as the target for the prediction. Then, given R the coefficient of determination of the linear regression, we have: V IF (a) = 1 1 -R 2 (7) It is commonly accepted that a variance inflation factor superior to 10 indicates strong multicollinearity of the attribute with other attributes of the dataset. This threshold of 10 is arbitrary but considered as a standard in numerous publications (e. g. [35] ). Moreover, when an attribute is removed from the dataset, the VIF of the attributes multicollinear with it decreases.
Then, we can automatically detect groups of attributes by calculating the VIF of each attribute (considered as a target) of the dataset, and then comparing them with a new VIF calculation with an attribute removed. For this purpose, we consider two possible approaches: • Considering as a priority the calculation of strongly multicollinear groups of attributes: Those are groups of attributes with a dependency on one another. In the context of this approach, attributes whose VIF varies strongly when an attribute is removed from the dataset is considered as part of the group. • Considering as a priority the calculation of weakly or non-multicollinear groups of attributes: Given the fact that correlated attributes tend to bring the same information to the model, it may be preferable to prioritize groups for which the addition or removal of an attribute changes greatly the information brought by the group.
These two approaches are named VIF coalition and reverse VIF coalition, respectively. This gives us the algorithm 3, for the VIF coalition. The reverse VIF coalition can be obtained simply by replacing the condition for adding an attribute to a group by if newvif s(a ) > oldvif s(a ) * (1 -t * 0. 05). This supplementary ratio of 0. 05 has been obtained by preliminary experiments, which showed that just keeping the 1 -t factor led to a generation of all the possible subgroups, which defeat the principle of an approximation. Example 3. Given our dataset of 4 attributes, we calculate the VIFs of each attribute. Then, we calculate the VIFs again each time with one of the attributes removed. The results are depicted in Figure  5.
Given a dataset D = (A, X), with A = {a 1 ,. , a n } the correlation matrix C is obtained by computing the Spearman correlation coefficient of each attribute couple : C(1, 2) = corr(a 1 , a 2 ). Thus C is symmetrical and have 1 as the value of its whole diagonal. For each line i of the matrix C, we consider as grouped with a i the attributes strongly (or weakly) correlated with a i , for the Spearman coalition (or the reverse Spearman coalition).
Algorithm 4 Spearman-based coalition extraction. Input: a threshold t, the set of attributes of the dataset A, and a function spearman(A) calculating the matrix of all the absolute Spearman correlation coefficient of all the subsets of a set of attributes. a max and min functions which returns the maximum and minimum of a matrix line. Output: σ a coalition of attributes σ ← {} corrmat ← spearman(A) calculating the correlation matrix for all a ∈ A do g ← {} for all a ∈ A do if corrmat(a, a ) > max(corrmat(a)) * (1 -t) and max(corrmat(a)) > 0. 1 then If the most correlated attribute have a coefficient less than 0. 1, we consider a as a singleton add a to g end if end for add g to σ end for return σ The algorithm 4 details the Spearman coalition method. The reverse Spearman coalition method can be obtained by replacing the condition for adding an attribute to a group by corrmat(a, a ) < min(corrmat(a))+max(corrmat(a)) * t and min(corrmat(a)) < 0.
5. This adds the least correlated attributes up to a threshold : if the attribute least correlated to a have its Spearman correlation to a superior to 0. 5, we consider the attribute a as a singleton. Example 4. Given our previous dataset of 4 attributes, we calculate the matrix of the spearman correlation coefficients as depicted in Figure  6. In this matrix, we iterate on each row of the matrix, in order to create groups based on the most correlated attributes. In the first line, we see that the attribute most correlated to A is B, and the two other attributes are very weakly correlated to A. Thus, we have a first group: {A, B}. The second line tells us that A and C are both strongly correlated to B. So, we have a second group: {A, B, C}. Similarly, the third line indicates that B and D are correlated to C, and so we add a third group: {B, C, D}.
Experimental protocol Our experiments are run on an AMD Ryzen 3700 processor with 8 x 3. 6 GHz cores and 32 GB of RAM. Our tests are realized from the data available on the Openml platform  [36]. We select the biggest collection of datasets 2  on which classification tasks have been run. We also consider two classification tasks: Random Forest and Support Vector Machine (SVM) with the non-linear Radial-Basis-Function (RBF) kernel. Experiments are conducted using Python 3. 7. 9 with the Scikit-Learn package on both models  3. Due to the heavy computational cost of the complete influenceconsidered as the reference of our experiments-we select the datasets having at most nine attributes. Thus, a collection of 243 datasets is obtained. Considering the two types of workflows, we have a total of 486 runs.
For each of those Figure  6 : Depiction of the groups calculated by the spearman based coalition method for a 4 attributes dataset. The spearman correlation matrix is calculated. For each line, the attributes most correlated with the line's attribute are considered as part of a group. runs, we generate each type of influence described in this paper, for each instance of the 243 datasets: the complete influence for the baseline, along with the coalitional influences, k-depth influence, and the Kernel and Tree SHAP influences. The coalitional influences are generated using the different group generation methods described in Section 4, which are based on an α ∈]0, 0. 5[ parameter (small values of α resulting in smaller subgroups, and high values in bigger ones). We generate the possible subgroups with 5 different values of α to study the influence of subgroup size.
To compare the different explanation methods, we consider the explanation results as a vector of attribute influences noted I(x) = [i 1 ,. , i n ] with n the number of attributes in the dataset. Thus, each of the attributes a k is given an influence i k ∈ [0, 1] by the method I : ∀k ∈ [1. n], i k = I ai (x), with x an instance of the dataset. We then define a difference between two vectors of influences i, j as the normalised Euclidean distance: d(i, j) = 1 2 √ n n k=1 (i k -j k ) 2 (8) Considering this formula, we define an error score based on the difference between an explanation method and the complete influence method. Given an instance x, an explanation method I(x), and the complete influence method I C (x): err(I, x) = d(I(x), I C (x)) (9) For each instance of each dataset, we generate the error score of every method, allowing us to compare their performances across the different collected datasets.
Each error score is the distance of one of the coalitional methods from the complete method. Thus, lesser error is indicative of a more precise estimation of the complete method. To compare methods, we also consider the time needed to explain a set of data, called computation time. This includes the time to determine the subgroups of interest and to compute the influences of all instances in the coalitional and kdepth methods. For the SHAP methods, the calculation time is the one taken to calculate the influences of all instances. 1  details the number of datasets and mean number of instances for each number of attributes. Since the number of instances impacts the total computation time for a dataset, each computational time is normalized by dividing by the number of instances in the dataset to compare times per instance.
Evaluation of the literature methods In this section, we focus on the k-depth and SHAP methods and evaluate them with the protocol described previously. Figure  7  indicates the mean error with respect to the complete method, and with the distinction of the datasets based on their number of attributes. For both models, the linear (or 1-depth) method gives the worst results, particularly as the number of attributes grows. A larger k in k-depth methods results in more precise influence attribution. This is fully expected since a higher k generates larger groups closer to the Complete method. Interestingly, SHAP methods -KernelSHAP for both models and TreeSHAP only for the Random Forest-give rather accurate results, without really losing accuracy as the number of attributes grows. Figure  8  represents the mean computation time per instance for all methods.
The y-axis is log scaled in order to have a clear view of all results, even for datasets with a low number of attributes, as the growth is exponential. The k-depth methods take more and more time as the k grows -from linear to complete. The TreeSHAP method, only usable with the Random Forest, gives rather good computation time, being between those of linear and 2-depth methods. On the contrary, the KernelSHAP methods give poor results, being slower than the complete, especially with the SVM model. This is a major inconvenience since it suggests that interpretability with models that are not tree-based would often be intractable in practice. We then present, in the next section, the evaluations of our coalitional methods, including the first comparison with literature approaches.
On the contrary, Model-based takes mostly more time than the complete method. Indeed, the time taken by the Model-based method to generate a set of groups is particularly long compared to other coalition-based methods, thus canceling out the effects of group selection. This automatically disqualifies this method for practical use. In particular, the computation time for RevV IF generally seems higher than the other methods. A possible reason is that it should generate larger groups, closer to the Complete one. To have a clearer view of the methods' performances, we average the error with respect to the Complete method and the computation time globally, thus independently of the number of attributes in the datasets. Therefore, it gives us a single representation, called Performance Map (Figures  11  and 12 ), with the computation time normalized by one of the complete on the horizontal axis, and the error for the complete on the vertical axis.
The complete method is thus placed on the point with coordinates (1, 0). All the methods are thus placed above the complete since they can not have a null error with respect to the complete, and methods placed at the left of the complete have lower computation time than the complete, while those placed at the complete right are slower to compute. We also retained only the most promising grouping methods from previous results -PCA, Spearman, Reverse Spearman and Reverse VIFfor α ∈ [0. 2, 0. 4] so as to avoid a figure with too much noisy information. The k-depth and SHAP methods are also shown. Figure  7 : Error score between each explanation method and the complete influence depending on the number of attributes in the dataset. Figure  11  shows the Performance Map for the Random Forest model.
KernelSHAP is not included since its computation time is too high, thus flattening the rest of the graph. Nevertheless, TreeSHAP method is still on average slower to compute than the complete, despite being specially designed for tree-based models. As for the coalitional methods, PCA is the fastest, but not the most accurate, while Reverse VIF is the most accurate but also the slowest method between all coalitional methods. Spearman and Reverse Spearman seems to be the best balance between precision and computation time. Figure  12  shows the Performance Map for the SVM model. The subfigure on the left includes the KernelSHAP method, but as this method is on average 40 times slower than the Complete method for our datasets, it flattens the other ones. Thus the subfigure on the right does not include KernelSHAP method.
In a similar fashion to Random Forest, PCA is the fastest coalitional method but the least accurate, while Reverse VIF is the most accurate. Again, Spearman-based grouping methods seem to be the most balanced ones. The only advantage of SHAP methods is that one does not need to retrain any model. SHAP simulates missing attributes through a heavy number of perturbations of the to-be-explained instance, implying a substantial cost. While our methods, only retraining once the model for every coalitional group, seems to be beneficial over SHAP. That is Figure  8 : Time per instance (in seconds and log scaled) taken by each method depending on the number of attributes in the dataset. why the complete method, which has the highest number of coalitional groups, is often faster to compute than SHAP methods.
Performance between literature and coalitional methods over different types of datasets So as to have a clearer view of the impact of all methods in one figure, Performance Map are displayed for each model. But as the number of attributes and instances in a dataset has a strong impact on the performances, we split the datasets into two parts. The first one includes datasets that have relatively few instances (strictly less than 500) or attributes (strictly less than 6), whereas the second part only includes datasets with a higher number of instances (at least 500) and attributes (at least 6). There are 213 datasets in the first set and 30 in the second one. Figure  17  shows the results for the Random Forest model for the coalitional methods -PCA, Spearman, Reverse Spearman and Reverse VIFfor three complexity proportion -10%, 25% and 50%-, the k-depth methods -from linear to completeand TreeSHAP.
The left subgraph shows the results for the first set containing datasets with few attributes or instances and the subgraph on the right indicates results for more complex datasets as described previously. TreeSHAP has on average a higher computation time than the complete method with an error between the 4-depth and 5-depth for both sets of datasets. Coalitional methods show strong results, PCA -50% has a computation time equivalent to a 3-depth but has an error closer to the 5-depth for both sets. Similarly, all coalitional methods with a threshold of 25% are in terms of computation time closer to the 2-depth while being more accurate than 3-depth. This is explained by the fact that coalitional methods generates smarter groups with less useless or redundant information than k-depth, thus being more efficient.
How to interpret the coalitional explanations In this section, we show an example of the use of our coalitional methods on a real use case dataset and for specific instances from this dataset. For this experiment, we only consider the Spearman method, since it produces one of the best results as discussed in the previous sections. We also include the Kernel SHAP method for a comparison with the explanations found with the Spearman approach. The quality of an interpretability method is a subjective concept and it would be difficult to theorize measures to assess what constitutes good interpretability. Nevertheless, some criteria exist in the literature to evaluate individual explanations  [37]. Properties such as fidelity and comprehensibility can help non-experts to evaluate and compare individual explanations, thus explanations methods.
All these behaviors from our model are coherent with the clinical literature about Covid-19  [39]. In contrast, with the KernelSHAP method, the near-zero average influences for some attributes are inconsistent with known risk factors. Another important point of the explanations is the fidelity and ease of understanding and interpreting them. Although very subjective, these parameters are essential to take into account in the medical field, since a lack of fidelity to the model and understanding of the explanations can lead to wrong decision making and consequences for the health of patients. To evaluate this, one instance of each class from the Covid-19 dataset was randomly drawn to describe and evaluate the explanation of the KernelSHAP and the Spearman coalitional method. Figures  21  and 22  show the influence of each attribute for these patients, whose descriptions are given below.
Spearman's explanations are slightly more contrasted than Kernel SHAP's ones. For this use case, the two methods are easy to understand as there are based on the same additive strategy. For both methods and both examples, influences approximate closely model predictions, and therefore have a high fidelity. However, this fidelity is only local, as methods only explain individual data instances. Moreover, and based on the clinical literature about Covid-19  [39] , the explanations from Spearman method seem more consistent for comorbidities. Finally, the complete dataset was computed in 51 seconds with the Spearman method when it took more than 18 minutes for the Kernel SHAP method, for similar results.
Tests were conducted on 243 datasets with a different number of attributes and instances. Coalitional methods show the most promising results, especially those based on PCA, Spearman and Reverse VIF. These new methods notably outperform existing ones, whether k-depth or SHAP-based, for more complex datasets where attribute interdependence is more likely to be present. Although our results open up encouraging perspectives of practical application of individual prediction interpretability, either in terms of accuracy or in computation time, the main problem is that computing the complete influences, which is the comparison baseline for our study, becomes near impossible with larger attribute numbers. Thus, it is very difficult to monitor the performance of our different methods with this baseline.
A possible way to address this problem could be first to run a global attribute importance study for large datasets using methods such as Permutation Importance that is model agnostic or Gini Importance for tree-based models. Then use this information to compute influences only for the most important attributes during the individual explanation generation. A longer-term perspective is also to take into account the context where the predictive analysis is conducted. Indeed, the explanations provided for particular instances cannot be totally satisfactory for any users in any situations. The degree of user expertise in the data seems very important to consider: an expert user will certainly be more interested in very precise explanations than a novice one. Moreover, the analysis process may have an impact on the type of explanations to consider.
Algorithm 2 PCA-based coalition extraction. Input: a threshold t and the set of attributes A of the PCA Output: σ a coalition of attributes σ ← {} for all a ∈ A do for each attribute generated by the PCA g ← {} g, a new possible group αmax ← max(a = α 1 ,. , α n ) find the most important factor for all α i ∈ a do if α i ≥ α max * (1 -t) then add a i to g the attribute is included in the group if close to the max Example 2. Given our previous dataset of 4 attributes, we run a PCA on it. The new attributes generated are as in Figure.
The use case dataset concerns the SARS-COV2 -also called Covid-19 -epidemic outbreak in France during spring 2020. Data collection complied with the European GDPR rules and consists of anonymized medical information of 409 patients with Covid-19 virus hospitalized at the Centre Hospitalier Intercommunal de Créteil 4 between March and May 2020. The primary binary outcome consists of the deterioration of the patient's state of health during their stay, also called aggravation. Deterioration was defined as the requirement for mechanical ventilation, presence of septic shock, acute respiratory distress syndrome, a requirement for resuscitation maneuvers during hospitalization or hospital mortality. Out of the total number of patients, 176 of them had a deterioration in their health state, i. e.
Algorithm 2 PCA-based coalition extraction. Input: a threshold t and the set of attributes A of the PCA Output: σ a coalition of attributes σ ← {} for all a ∈ A do for each attribute generated by the PCA g ← {} g, a new possible group αmax ← max(a = α 1 ,. , α n ) find the most important factor for all α i ∈ a do if α i ≥ α max * (1 -t) then add a i to g the attribute is included in the group if close to the max Example 2. Given our previous dataset of 4 attributes, we run a PCA on it. The new attributes generated are as in Figure end if end for add g to σ end for return σ.
Numerical experiments In this section, we provide numerical experiments for different gradient methods by two types of problems. The first one is generated randomly by MATLAB and the second one is a two-point boundary value problem. In both examples, the right-hand side b in system (1) is computed by b = Ax * where x * is a random vector such that x * ∈ (-10, 10). The tests are started from zero vectors and the stopping criterion is fixed with g n < 10 -6 g 0. All experiments are performed using MATLAB R2018b on a machine with Double Intel Core i7 2. 8 GHz CPU. In the first example, we consider the random problem generated by the MATLAB builtin function sprandsym, which has appeared in  [14]. We would like to know the impact of parameters on the convergence behavior of alternate gradient methods.
In one direction, the MG-based method minimizes indeed the residual value. On the other hand, it is known that stability generally favors short steplengths. Along with  (23) , the desired conclusion follows. Finally, we compare our new methods with the conjugate gradient (CG) method  [21]. Two examples are used to show the robustness and efficiency of the proposed methods. The first example concerns the random problems with perturbation generated by MATLAB, which have the following form Ãx = b, Ã = A + δV, where δ is a small positive value and V is a nonsymmetric matrix. Still, let κ be the condition number of A. We choose δ = 10 -4. V is generated by the MATLAB function sprand. We compare also our methods with the generalized minimum residual (GMRES) method  [32]  in view of the perturbation.
In order to avoid duplicates of N 1 networks, i. e. keep only one network instance per equivalence class, we must exploit the abstract representation of their functions presented above (see Section 2. 1). Full details concerning the implementation of this computation involving a Java processing pipeline orchestrating ASP modules are given in appendices A and B. ∀w ni→nj ∈ N 1 , ∃!w ′ ni→nj ∈ N ′ 1 , w ni→nj ≤ w ′ ni→nj. i, j ∈ [1; d].