From SHAP Scores to Feature Importance Scores

A central goal of eXplainable Artificial Intelligence (XAI) is to assign relative importance to the features of a Machine Learning (ML) model given some prediction. The importance of this task of explainability by feature attribution is illustrated by the ubiquitous recent use of tools such as SHAP and LIME. Unfortunately, the exact computation of feature attributions, using the game-theoretical foundation underlying SHAP and LIME, can yield manifestly unsatisfactory results, that tantamount to reporting misleading relative feature importance. Recent work targeted rigorous feature attribution, by studying axiomatic aggregations of features based on logic-based definitions of explanations by feature selection. This paper shows that there is an essential relationship between feature attribution and a priori voting power, and that those recently proposed axiomatic aggregations represent a few instantiations of the range of power indices studied in the past. Furthermore, it remains unclear how some of the most widely used power indices might be exploited as feature importance scores (FISs), i.e. the use of power indices in XAI, and which of these indices would be the best suited for the purposes of XAI by feature attribution, namely in terms of not producing results that could be deemed as unsatisfactory. This paper proposes novel desirable properties that FISs should exhibit. In addition, the paper also proposes novel FISs exhibiting the proposed properties. Finally, the paper conducts a rigorous analysis of the best-known power indices in terms of the proposed properties.

Distance-Restricted Explanations: Theoretical Underpinnings & Efficient Implementation

The uses of machine learning (ML) have snowballed in recent years. In many cases, ML models are highly complex, and their operation is beyond the understanding of human decision-makers. Nevertheless, some uses of ML models involve high-stakes and safety-critical applications. Explainable artificial intelligence (XAI) aims to help human decision-makers in understanding the operation of such complex ML models, thus eliciting trust in their operation. Unfortunately, the majority of past XAI work is based on informal approaches, that offer no guarantees of rigor. Unsurprisingly, there exists comprehensive experimental and theoretical evidence confirming that informal methods of XAI can provide human-decision makers with erroneous information. Logic-based XAI represents a rigorous approach to explainability; it is model-based and offers the strongest guarantees of rigor of computed explanations. However, a well-known drawback of logic-based XAI is the complexity of logic reasoning, especially for highly complex ML models. Recent work proposed distance-restricted explanations, i.e. explanations that are rigorous provided the distance to a given input is small enough. Distance-restricted explainability is tightly related with adversarial robustness, and it has been shown to scale for moderately complex ML models, but the number of inputs still represents a key limiting factor. This paper investigates novel algorithms for scaling up the performance of logic-based explainers when computing and enumerating ML model explanations with a large number of inputs.

On Correcting SHAP Scores

Recent work uncovered examples of classifiers for which SHAP scores yield misleading feature attributions. While such examples might be perceived as suggesting the inadequacy of Shapley values for explainability, this paper shows that the source of the identified shortcomings of SHAP scores resides elsewhere. Concretely, the paper makes the case that the failings of SHAP scores result from the characteristic functions used in earlier works. Furthermore, the paper identifies a number of properties that characteristic functions ought to respect, and proposes several novel characteristic functions, each exhibiting one or more of the desired properties. More importantly, some of the characteristic functions proposed in this paper are guaranteed not to exhibit any of the shortcomings uncovered by earlier work. The paper also investigates the impact of the new characteristic functions on the complexity of computing SHAP scores. Finally, the paper proposes modifications to the tool SHAP to use instead one of our novel characteristic functions, thereby eliminating some of the limitations reported for SHAP scores.

Refutation of Shapley Values for XAI -- Additional Evidence

Recent work demonstrated the inadequacy of Shapley values for explainable artificial intelligence (XAI). Although to disprove a theory a single counterexample suffices, a possible criticism of earlier work is that the focus was solely on Boolean classifiers. To address such possible criticism, this paper demonstrates the inadequacy of Shapley values for families of classifiers where features are not boolean, but also for families of classifiers for which multiple classes can be picked. Furthermore, the paper shows that the features changed in any minimal $l_0$ distance adversarial examples do not include irrelevant features, thus offering further arguments regarding the inadequacy of Shapley values for XAI.

A Refutation of Shapley Values for Explainability

Recent work demonstrated the existence of Boolean functions for which Shapley values provide misleading information about the relative importance of features in rule-based explanations. Such misleading information was broadly categorized into a number of possible issues. Each of those issues relates with features being relevant or irrelevant for a prediction, and all are significant regarding the inadequacy of Shapley values for rule-based explainability. This earlier work devised a brute-force approach to identify Boolean functions, defined on small numbers of features, and also associated instances, which displayed such inadequacy-revealing issues, and so served as evidence to the inadequacy of Shapley values for rule-based explainability. However, an outstanding question is how frequently such inadequacy-revealing issues can occur for Boolean functions with arbitrary large numbers of features. It is plain that a brute-force approach would be unlikely to provide insights on how to tackle this question. This paper answers the above question by proving that, for any number of features, there exist Boolean functions that exhibit one or more inadequacy-revealing issues, thereby contributing decisive arguments against the use of Shapley values as the theoretical underpinning of feature-attribution methods in explainability.

Explainability is NOT a Game

Explainable artificial intelligence (XAI) aims to help human decision-makers in understanding complex machine learning (ML) models. One of the hallmarks of XAI are measures of relative feature importance, which are theoretically justified through the use of Shapley values. This paper builds on recent work and offers a simple argument for why Shapley values can provide misleading measures of relative feature importance, by assigning more importance to features that are irrelevant for a prediction, and assigning less importance to features that are relevant for a prediction. The significance of these results is that they effectively challenge the many proposed uses of measures of relative feature importance in a fast-growing range of high-stakes application domains.

From Robustness to Explainability and Back Again

In contrast with ad-hoc methods for eXplainable Artificial Intelligence (XAI), formal explainability offers important guarantees of rigor. However, formal explainability is hindered by poor scalability for some families of classifiers, the most significant being neural networks. As a result, there are concerns as to whether formal explainability might serve to complement other approaches in delivering trustworthy AI. This paper addresses the limitation of scalability of formal explainability, and proposes novel algorithms for computing formal explanations. The novel algorithm computes explanations by answering instead a number of robustness queries, and such that the number of such queries is at most linear on the number of features. Consequently, the proposed algorithm establishes a direct relationship between the practical complexity of formal explainability and that of robustness. More importantly, the paper generalizes the definition of formal explanation, thereby allowing the use of robustness tools that are based on different distance norms, and also by reasoning in terms of some target degree of robustness. The experiments validate the practical efficiency of the proposed approach.

The Inadequacy of Shapley Values for Explainability

This paper develops a rigorous argument for why the use of Shapley values in explainable AI (XAI) will necessarily yield provably misleading information about the relative importance of features for predictions. Concretely, this paper demonstrates that there exist classifiers, and associated predictions, for which the relative importance of features determined by the Shapley values will incorrectly assign more importance to features that are provably irrelevant for the prediction, and less importance to features that are provably relevant for the prediction. The paper also argues that, given recent complexity results, the existence of efficient algorithms for the computation of rigorous feature attribution values in the case of some restricted classes of classifiers should be deemed unlikely at best.

On Computing Probabilistic Abductive Explanations

The most widely studied explainable AI (XAI) approaches are unsound. This is the case with well-known model-agnostic explanation approaches, and it is also the case with approaches based on saliency maps. One solution is to consider intrinsic interpretability, which does not exhibit the drawback of unsoundness. Unfortunately, intrinsic interpretability can display unwieldy explanation redundancy. Formal explainability represents the alternative to these non-rigorous approaches, with one example being PI-explanations. Unfortunately, PI-explanations also exhibit important drawbacks, the most visible of which is arguably their size. Recently, it has been observed that the (absolute) rigor of PI-explanations can be traded off for a smaller explanation size, by computing the so-called relevant sets. Given some positive {\delta}, a set S of features is {\delta}-relevant if, when the features in S are fixed, the probability of getting the target class exceeds {\delta}. However, even for very simple classifiers, the complexity of computing relevant sets of features is prohibitive, with the decision problem being NPPP-complete for circuit-based classifiers. In contrast with earlier negative results, this paper investigates practical approaches for computing relevant sets for a number of widely used classifiers that include Decision Trees (DTs), Naive Bayes Classifiers (NBCs), and several families of classifiers obtained from propositional languages. Moreover, the paper shows that, in practice, and for these families of classifiers, relevant sets are easy to compute. Furthermore, the experiments confirm that succinct sets of relevant features can be obtained for the families of classifiers considered.

Feature Necessity & Relevancy in ML Classifier Explanations

Given a machine learning (ML) model and a prediction, explanations can be defined as sets of features which are sufficient for the prediction. In some applications, and besides asking for an explanation, it is also critical to understand whether sensitive features can occur in some explanation, or whether a non-interesting feature must occur in all explanations. This paper starts by relating such queries respectively with the problems of relevancy and necessity in logic-based abduction. The paper then proves membership and hardness results for several families of ML classifiers. Afterwards the paper proposes concrete algorithms for two classes of classifiers. The experimental results confirm the scalability of the proposed algorithms.