Decision trees are widely used in many settings where interpretable models are preferred orrequired.

Decision trees are widely used in many settings where interpretable models are preferred or required.

C.1 Proof of Lemma 4 The lemma will be a direct consequence of the following result.

Machine learning models and algorithms appear more and more frequently in systems that make decisions with an impact in our lives.

Despite significant advancements in post-hoc explainability techniques for neural networks, many current methods rely on heuristics and do not provide formally provable guarantees over the explanations provided. Recent work has shown that it is possible to obtain explanations with formal guarantees by identifying subsets of input features that are sufficient to determine that predictions remain unchanged using neural network verification techniques. Despite the appeal of these explanations, their computation faces significant scalability challenges. In this work, we address this gap by proposing a novel abstraction-refinement technique for efficiently computing provably sufficient explanations of neural network predictions. Our method abstracts the original large neural network by constructing a substantially reduced network, where a sufficient explanation of the reduced network is also provably sufficient for the original network, hence significantly speeding up the verification process. If the explanation is in sufficient on the reduced network, we iteratively refine the network size by gradually increasing it until convergence. Our experiments demonstrate that our approach enhances the efficiency of obtaining provably sufficient explanations for neural network predictions while additionally providing a fine-grained interpretation of the network's predictions across different abstraction levels.

Recent improvements in the expressive power of spatio-temporal models have led to performance gains in many real-world applications, such as traffic forecasting and social network modelling. However, understanding the predictions from a model is crucial to ensure reliability and trustworthiness, particularly for high-risk applications, such as healthcare and transport. Few existing methods are able to generate explanations for models trained on continuous-time dynamic graph data and, of these, the computational complexity and lack of suitable explanation objectives pose challenges. In this paper, we propose $\textbf{S}$patio-$\textbf{T}$emporal E$\textbf{X}$planation $\textbf{Search}$ (STX-Search), a novel method for generating instance-level explanations that is applicable to static and dynamic temporal graph structures. We introduce a novel search strategy and objective function, to find explanations that are highly faithful and interpretable. When compared with existing methods, STX-Search produces explanations of higher fidelity whilst optimising explanation size to maintain interpretability.

Despite the practical success of Artificial Intelligence (AI), current neural AI algorithms face two significant issues. First, the decisions made by neural architectures are often prone to bias and brittleness. Second, when a chain of reasoning is required, neural systems often perform poorly. Neuro-symbolic artificial intelligence is a promising approach that tackles these (and other) weaknesses by combining the power of neural perception and symbolic reasoning. Meanwhile, the success of AI has made it critical to understand its behaviour, leading to the development of explainable artificial intelligence (XAI). While neuro-symbolic AI systems have important advantages over purely neural AI, we still need to explain their actions, which are obscured by the interactions of the neural and symbolic components. To address the issue, this paper proposes a formal approach to explaining the decisions of neuro-symbolic systems. The approach hinges on the use of formal abductive explanations and on solving the neuro-symbolic explainability problem hierarchically. Namely, it first computes a formal explanation for the symbolic component of the system, which serves to identify a subset of the individual parts of neural information that needs to be explained. This is followed by explaining only those individual neural inputs, independently of each other, which facilitates succinctness of hierarchical formal explanations and helps to increase the overall performance of the approach. Experimental results for a few complex reasoning tasks demonstrate practical efficiency of the proposed approach, in comparison to purely neural systems, from the perspective of explanation size, explanation time, training time, model sizes, and the quality of explanations reported.

Decision trees are widely used in many settings where interpretable models are preferred or required. There is indeed a vast literature on the design and analysis of decision tree algorithms that aim at optimizing these parameters.This paper contributes to this important line of research: we propose as a novel criterion of measuring the interpretability of a decision tree, the sparsity of the set of attributes that are (on average) required to explain the classification of the examples. We give a tight characterization of the best possible guarantees achievable by a decision tree built to optimize both our newmeasure (which we call the {\em explanation size}) and the more classical measures of worst-case and average depth. In particular, we give an algorithm that guarantees O(\ln n) -approximation (hence optimal if P eq NP) for the minimization of both the average/worst-case explanation size and the average/worst-case depth. In addition to our theoretical contributions, experiments with 20 real datasets show that our algorithm has accuracy competitive with CART while producing trees that allow for much simpler explanations.

Recent research has developed a number of eXplainable AI (XAI) techniques, such as gradient-based approaches, input perturbation-base methods, and blackbox explanation methods. Although extracting meaningful insights from deep learning models, how to properly evaluate these XAI methods remains an open problem. The most widely used approach is to perturb or even remove what the XAI method considers to be the most important features in an input and observe the changes in the output prediction. This approach although efficient suffers the Out-of-Distribution (OOD) problem as the perturbed samples may no longer follow the original data distribution. A recent method RemOve And Retrain (ROAR) solves the OOD issue by retraining the model with perturbed samples guided by explanations. However, the training may not always converge given the distribution difference. Furthermore, using the model retrained based on XAI methods to evaluate these explainers may cause information leakage and thus lead to unfair comparisons. We propose Fine-tuned Fidelity (F-Fidelity), a robust evaluation framework for XAI, which utilizes i) an explanation-agnostic fine-tuning strategy, thus mitigating the information leakage issue and ii) a random masking operation that ensures that the removal step does not generate an OOD input. We designed controlled experiments with state-of-the-art (SOTA) explainers and their degraded version to verify the correctness of our framework. We conducted experiments on multiple data structures, such as images, time series, and natural language. The results demonstrate that F-Fidelity significantly improves upon prior evaluation metrics in recovering the ground-truth ranking of the explainers. Furthermore, we show both theoretically and empirically that, given a faithful explainer, F-Fidelity metric can be used to compute the sparsity of influential input components, i.e., to extract the true explanation size.

Building on VeriX (Verified eXplainability, arXiv:2212.01051), a system for producing optimal verified explanations for machine learning model outputs, we present VeriX+, which significantly improves both the size and the generation time of verified explanations. We introduce a bound propagation-based sensitivity technique to improve the size, and a binary search-based traversal with confidence ranking for improving time -- the two techniques are orthogonal and can be used independently or together. We also show how to adapt the QuickXplain (Junker 2004) algorithm to our setting to provide a trade-off between size and time. Experimental evaluations on standard benchmarks demonstrate significant improvements on both metrics, e.g., a size reduction of 38% on the GTSRB dataset and a time reduction of 90% on MNIST. We also explore applications of our verified explanations and show that explanation size is a useful proxy for both incorrectness detection and out-of-distribution detection.