# Fidex and FidexGlo: From Local Explanations to Global Explanations of Deep Models +
## 2 Explainability Techniques
### Rule-Based Explanations  
The formalism of propositional rules is natural with respect to the logic of human reasoning. Specifically, rules are given as follows: \"if \((x_{1}<t_{1})\) and \((x_{2}>t_{2})\) and \(\ldots\)\((x_{n}<t_{n})\) then DECISION\"; with \(x_{i}\) corresponding to input variables \((i=1\ldots n)\), \(t_{i}\) being real values, and DECISION corresponding to a decision, such as \"patient healthy\". For instance, we can say the following: \"this patient is sick, because input variable \(i\) is greater than threshold \(t_{i}\) and input variable \(j\) is great than threshold \(t_{j}\)\". The first taxonomy describing the general characteristics of all techniques for extracting rules from connectionist models was introduced in [6]. Specifically, they distinguished three main categories of methods: pedagogical; decompositional; and eclectic. Pedagogical techniques use a transparent model, such as decision trees (DTs) [20] to learn input/output associations, without considering weight values of neural networks. Decompositional methods generate propositional rules by analyzing the weight values. However, most algorithms in this category present exponential algorithmic complexity. To alleviate this difficulty, a pruning step is helpful to reduce the number of weights. Eclectic techniques are both pedagogical and decompositional. [PERSON] proved that the problem of rule extraction from multilayer perceptrons (MLPs) is NP-hard [21]. Finally, for support vector machines (SVMs), which are functionally equivalent to MLPs with a single hidden layer, a review of many explainability techniques were proposed in [22]. Note that the majority of decompositional techniques suffers from exponential algorithmic complexity [12]. Examples of pedagogical techniques are reported in [23; 24]. Deep architectures are even more complex, so their opacity with respect to MLPs has increased. Unfortunately, the simplest pedagogical approach of adapting transparent models, such as decision trees, to deep neural networks generally yields unsatisfactory results in terms of accuracy and fidelity [25]. Although new rule extraction algorithms are always being proposed based on a single MLP, little work has been achieved for ensembles of connectionist models. For example, in REFNE, a trained ensemble generated additional samples and then extracted propositional rules [26]. Here, the input variables were discretized during rule extraction. The rules were limited to only three antecedents. Using genetic programming, [PERSON] presented a technique for generating rules from ensembles of 20 neural networks [27]. Specifically, the problem of rule extraction has been considered as an optimization problem, where a trade-off between accuracy and comprehensibility has to be considered. Moreover, [PERSON] and [PERSON] introduced a rule extraction technique for a limited number of MLPs in an ensemble [28; 29]. [PERSON] et al. trained DIMLP ensembles by optimizing their diversity [30]. Rule extraction was then performed on each individual network, and for each sample, the rule with the highest confidence score was selected. Finally, in RE-E-NNES (rule extraction with ensemble of NN resembles) the C4.5 decision tree algorithm was used to extract rulesets from MLP ensembles trained by a boosting algorithm (NNBOOST); the number of MLPs in an ensemble was limited to ten [31]. RE-E-NNES generated rules from the first MLP, then refined them with the second MLP and so on with the other MLPs. Ensembles of decision trees are very good models for data classification. Furthermore, a single tree can be explained by the rules that are generated from it, but many trees in an ensemble lose their transparency and require special explanation techniques. In particular, it is always possible to generate all the rules from all the trees in an ensemble, but since there can be hundreds of trees, the number of rules is too large. Therefore, several techniques try to remove as many rules as possible.