This paper proposes a new perspective for analyzing the generalization power of deep neural networks (DNNs), i.e., directly disentangling and analyzing the dynamics of generalizable and non-generalizable interaction encoded by a DNN through the training process. Specifically, this work builds upon the recent theoretical achievement in explainble AI, which proves that the detailed inference logic of DNNs can be can be strictly rewritten as a small number of AND-OR interaction patterns. Based on this, we propose an efficient method to quantify the generalization power of each interaction, and we discover a distinct three-phase dynamics of the generalization power of interactions during training. In particular, the early phase of training typically removes noisy and non-generalizable interactions and learns simple and generalizable ones. The second and the third phases tend to capture increasingly complex interactions that are harder to generalize. Experimental results verify that the learning of non-generalizable interactions is the the direct cause for the gap between the training and testing losses.

Integrating symbolic techniques with statistical ones is a long-standing problem in artificial intelligence. The motivation is that the strengths of either area match the weaknesses of the other, and $\unicode{x2013}$ by combining the two $\unicode{x2013}$ the weaknesses of either method can be limited. Neuro-symbolic AI focuses on this integration where the statistical methods are in particular neural networks. In recent years, there has been significant progress in this research field, where neuro-symbolic systems outperformed logical or neural models alone. Yet, neuro-symbolic AI is, comparatively speaking, still in its infancy and has not been widely adopted by machine learning practitioners. In this survey, we present the first mapping of neuro-symbolic techniques into families of frameworks based on their architectures, with several benefits: Firstly, it allows us to link different strengths of frameworks to their respective architectures. Secondly, it allows us to illustrate how engineers can augment their neural networks while treating the symbolic methods as black-boxes. Thirdly, it allows us to map most of the field so that future researchers can identify closely related frameworks.

Probabilistic logical models are a core component of neurosymbolic AI and are important models in their own right for tasks that require high explainability. Unlike neural networks, logical models are often handcrafted using domain expertise, making their development costly and prone to errors. While there are algorithms that learn logical models from data, they are generally prohibitively expensive, limiting their applicability in real-world settings. In this work, we introduce precision and recall for logical rules and define their composition as rule utility -- a cost-effective measure to evaluate the predictive power of logical models. Further, we introduce SPECTRUM, a scalable framework for learning logical models from relational data. Its scalability derives from a linear-time algorithm that mines recurrent structures in the data along with a second algorithm that, using the cheap utility measure, efficiently ranks rules built from these structures. Moreover, we derive theoretical guarantees on the utility of the learnt logical model. As a result, SPECTRUM learns more accurate logical models orders of magnitude faster than previous methods on real-world datasets.

Link prediction is an important task in addressing the incompleteness problem of knowledge graphs (KG). Previous link prediction models suffer from issues related to either performance or explanatory capability. Furthermore, models that are capable of generating explanations, often struggle with erroneous paths or reasoning leading to the correct answer. To address these challenges, we introduce a novel Neural-Symbolic model named FaSt-FLiP (stands for Fast and Slow Thinking with Filtered rules for Link Prediction task), inspired by two distinct aspects of human cognition: "commonsense reasoning" and "thinking, fast and slow." Our objective is to combine a logical and neural model for enhanced link prediction. To tackle the challenge of dealing with incorrect paths or rules generated by the logical model, we propose a semi-supervised method to convert rules into sentences. These sentences are then subjected to assessment and removal of incorrect rules using an NLI (Natural Language Inference) model. Our approach to combining logical and neural models involves first obtaining answers from both the logical and neural models. These answers are subsequently unified using an Inference Engine module, which has been realized through both algorithmic implementation and a novel neural model architecture. To validate the efficacy of our model, we conducted a series of experiments. The results demonstrate the superior performance of our model in both link prediction metrics and the generation of more reliable explanations.

In September 2018, I published a blog about my forthcoming book on The Mathematical Foundations of Data Science. The central question we address is: How can we bridge the gap between mathematics needed for Artificial Intelligence (Deep Learning and Machine learning) with that taught in high schools (up to ages 17/18)? In this post, we present a chapter from this book called "A Taxonomy of Machine Learning Models." The book is now available for an early bird discount released as chapters. If you are interested in getting early discounted copies, please contact ajit.jaokar at feynlabs.ai.

The recent success of Bayesian methods in neuroscience and artificial intelligence gives rise to the hypothesis that the brain is a Bayesian machine. Since logic and learning are both practices of the human brain, it leads to another hypothesis that there is a Bayesian interpretation underlying both logical reasoning and machine learning. In this paper, we introduce a generative model of logical consequence relations. It formalises the process of how the truth value of a sentence is probabilistically generated from the probability distribution over states of the world. We show that the generative model characterises a classical consequence relation, paraconsistent consequence relation and nonmonotonic consequence relation. In particular, the generative model gives a new consequence relation that outperforms them in reasoning with inconsistent knowledge. We also show that the generative model gives a new classification algorithm that outperforms several representative algorithms in predictive accuracy and complexity on the Kaggle Titanic dataset.

In September 2018, I published a blog about my forthcoming book on The Mathematical Foundations of Data Science. The central question we address is: How can we bridge the gap between mathematics needed for Artificial Intelligence (Deep Learning and Machine learning) with that taught in high schools (up to ages 17/18)? In this post, we present a chapter from this book called "A Taxonomy of Machine Learning Models." The book is now available for an early bird discount released as chapters. If you are interested in getting early discounted copies, please contact ajit.jaokar at feynlabs.ai.

Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead Rudin et al., arXiv 2019 It's pretty clear from the title alone what Cynthia Rudin would like us to do! The paper is a mix of technical and philosophical arguments and comes with two main takeaways for me: firstly, a sharpening of my understanding of the difference between explainability and interpretability, and why the former may be problematic; and secondly some great pointers to techniques for creating truly interpretable models. A model can be a black box for one of two reasons: (a) the function that the model computes is far too complicated for any human to comprehend, or (b) the model may in actual fact be simple, but its details are proprietary and not available for inspection. In explainable ML we make predictions using a complicated black box model (e.g., a DNN), and use a second (posthoc) model created to explain what the first model is doing. A classic example here is LIME, which explores a local area of a complex model to uncover decision boundaries.

Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead Rudin et al., arXiv 2019 It's pretty clear from the title alone what Cynthia Rudin would like us to do! The paper is a mix of technical and philosophical arguments and comes with two main takeaways for me: firstly, a sharpening of my understanding of the difference between explainability and interpretability, and why the former may be problematic; and secondly some great pointers to techniques for creating truly interpretable models. A model can be a black box for one of two reasons: (a) the function that the model computes is far too complicated for any human to comprehend, or (b) the model may in actual fact be simple, but its details are proprietary and not available for inspection. In explainable ML we make predictions using a complicated black box model (e.g., a DNN), and use a second (posthoc) model created to explain what the first model is doing. A classic example here is LIME, which explores a local area of a complex model to uncover decision boundaries.

Although the notion of diagnostic problem has been extensively investigated in the context of static systems, in most practical applications the behavior of the modeled system is significantly variable during time. The goal of the paper is to propose a novel approach to the modeling of uncertainty about temporal evolutions of time-varying systems and a characterization of model-based temporal diagnosis. Since in most real world cases knowledge about the temporal evolution of the system to be diagnosed is uncertain, we consider the case when probabilistic temporal knowledge is available for each component of the system and we choose to model it by means of Markov chains. In fact, we aim at exploiting the statistical assumptions underlying reliability theory in the context of the diagnosis of timevarying systems. We finally show how to exploit Markov chain theory in order to discard, in the diagnostic process, very unlikely diagnoses.