Multi-agent reinforcement learning (MARL) is well-suited for runtime decision-making in optimizing the performance of systems where multiple agents coexist and compete for shared resources. However, applying common deep learning-based MARL solutions to real-world problems suffers from issues of interpretability, sample efficiency, partial observability, etc. To address these challenges, we present an event-driven formulation, where decision-making is handled by distributed co-operative MARL agents using neuro-symbolic methods. The recently introduced neuro-symbolic Logical Neural Networks (LNN) framework serves as a function approximator for the RL, to train a rules-based policy that is both logical and interpretable by construction. To enable decision-making under uncertainty and partial observability, we developed a novel probabilistic neuro-symbolic framework, Probabilistic Logical Neural Networks (PLNN), which combines the capabilities of logical reasoning with probabilistic graphical models. In PLNN, the upward/downward inference strategy, inherited from LNN, is coupled with belief bounds by setting the activation function for the logical operator associated with each neural network node to a probability-respecting generalization of the Fr\'echet inequalities. These PLNN nodes form the unifying element that combines probabilistic logic and Bayes Nets, permitting inference for variables with unobserved states. We demonstrate our contributions by addressing key MARL challenges for power sharing in a system-on-chip application.

The typical phases of Bayesian network (BN) structured development include specification of purpose and scope, structure development, parameterisation and validation. Structure development is typically focused on qualitative issues and parameterisation quantitative issues, however there are qualitative and quantitative issues that arise in both phases. A common step that occurs after the initial structure has been developed is to perform a rough parameterisation that only captures and illustrates the intended qualitative behaviour of the model. This is done prior to a more rigorous parameterisation, ensuring that the structure is fit for purpose, as well as supporting later development and validation. In our collective experience and in discussions with other modellers, this step is an important part of the development process, but is under-reported in the literature. Since the practice focuses on qualitative issues, despite being quantitative in nature, we call this step qualitative parameterisation and provide an outline of its role in the BN development process.

Functional linear discriminant analysis (FLDA) is a powerful tool that extends LDA-mediated multiclass classification and dimension reduction to univariate time-series functions. However, in the age of large multivariate and incomplete data, statistical dependencies between features must be estimated in a computationally tractable way, while also dealing with missing data. There is a need for a computationally tractable approach that considers the statistical dependencies between features and can handle missing values. We here develop a multivariate version of FLDA (MUDRA) to tackle this issue and describe an efficient expectation/conditional-maximization (ECM) algorithm to infer its parameters. We assess its predictive power on the "Articulary Word Recognition" data set and show its improvement over the state-of-the-art, especially in the case of missing data. MUDRA allows interpretable classification of data sets with large proportions of missing data, which will be particularly useful for medical or psychological data sets.

Bayesian neural networks (BNNs) have recently gained popularity due to their ability to quantify model uncertainty. However, specifying a prior for BNNs that captures relevant domain knowledge is often extremely challenging. In this work, we propose a framework for integrating general forms of domain knowledge (i.e., any knowledge that can be represented by a loss function) into a BNN prior through variational inference, while enabling computationally efficient posterior inference and sampling. Specifically, our approach results in a prior over neural network weights that assigns high probability mass to models that better align with our domain knowledge, leading to posterior samples that also exhibit this behavior. We show that BNNs using our proposed domain knowledge priors outperform those with standard priors (e.g., isotropic Gaussian, Gaussian process), successfully incorporating diverse types of prior information such as fairness, physics rules, and healthcare knowledge and achieving better predictive performance. We also present techniques for transferring the learned priors across different model architectures, demonstrating their broad utility across various settings.

Diffusion models are a remarkably effective way of learning and sampling from a distribution $p(x)$. In posterior sampling, one is also given a measurement model $p(y \mid x)$ and a measurement $y$, and would like to sample from $p(x \mid y)$. Posterior sampling is useful for tasks such as inpainting, super-resolution, and MRI reconstruction, so a number of recent works have given algorithms to heuristically approximate it; but none are known to converge to the correct distribution in polynomial time. In this paper we show that posterior sampling is \emph{computationally intractable}: under the most basic assumption in cryptography -- that one-way functions exist -- there are instances for which \emph{every} algorithm takes superpolynomial time, even though \emph{unconditional} sampling is provably fast. We also show that the exponential-time rejection sampling algorithm is essentially optimal under the stronger plausible assumption that there are one-way functions that take exponential time to invert.

Neuro-Symbolic (NeSy) predictors that conform to symbolic knowledge - encoding, e.g., safety constraints - can be affected by Reasoning Shortcuts (RSs): They learn concepts consistent with the symbolic knowledge by exploiting unintended semantics. RSs compromise reliability and generalization and, as we show in this paper, they are linked to NeSy models being overconfident about the predicted concepts. Unfortunately, the only trustworthy mitigation strategy requires collecting costly dense supervision over the concepts. Rather than attempting to avoid RSs altogether, we propose to ensure NeSy models are aware of the semantic ambiguity of the concepts they learn, thus enabling their users to identify and distrust low-quality concepts. Starting from three simple desiderata, we derive bears (BE Aware of Reasoning Shortcuts), an ensembling technique that calibrates the model's concept-level confidence without compromising prediction accuracy, thus encouraging NeSy architectures to be uncertain about concepts affected by RSs. We show empirically that bears improves RS-awareness of several state-of-the-art NeSy models, and also facilitates acquiring informative dense annotations for mitigation purposes.

Our aim of this paper is to develop new map matching algorithms and to improve upon previous work. We address two key approaches: Analytic Hierarchy Process (AHP) map matching and fuzzy logic map matching. AHP is a decision-making method that combines mathematical analysis with human judgment, and fuzzy logic is an approach to computing based on the degree of truth and aims at modeling the imprecise modes of reasoning from 0 to 1 rather than the usual boolean logic. Of these algorithms, the way of our applying AHP to map matching is newly developed in this paper, meanwhile, our application of fuzzy logic to map matching is mostly the same as existing research except for some small changes. Because of the common characteristic that both methods are designed to handle imprecise information and simplicity for implementation, we decided to use these methods.

We investigate the concept of Best Approximation for Feedforward Neural Networks (FNN) and explore their convergence properties through the lens of Random Projection (RPNNs). RPNNs have predetermined and fixed, once and for all, internal weights and biases, offering computational efficiency. We demonstrate that there exists a choice of external weights, for any family of such RPNNs, with non-polynomial infinitely differentiable activation functions, that exhibit an exponential convergence rate when approximating any infinitely differentiable function. For illustration purposes, we test the proposed RPNN-based function approximation, with parsimoniously chosen basis functions, across five benchmark function approximation problems. Results show that RPNNs achieve comparable performance to established methods such as Legendre Polynomials, highlighting their potential for efficient and accurate function approximation. Widely used in Machine Learning (ML) tasks, including image classification, signal analysis, speech recognition, text-toimage synthesis and language processing [84, 68, 83, 37, 18, 39], the celebrated universal approximation theorem arXiv:2402.11397v1 Since the 1990s, there has been a notable increase in research focusing on solving numerical analysis problems. Lee and Kang (1990) [59] employed a Hopfield Neural Network to address the numerical solution of differential equations. Rico-Martinez et al. (1992) [76] introduced a recursive multilayer ANNs architecture, emulating the implementation of the 4th-order Runge-Kutta scheme, for the identification of continuous-time ODEs and the construction of their bifurcation diagram from experimental data. Yes ildirek and Lewis (1995) [85] employed a neural network-based controller for the feedback linearization of dynamical systems. Lagaris et al. (1998) [58] systematically investigate the use of FNNs for the solution of linear and nonlinear differential equations, addressing a range of scenarios from initial and boundary value problems. Random Projection Neural Networks of Best Approximation: Convergence theory and practical applications and increased computational power.

Abstract--Federated learning allows us to run machine learning algorithms on decentralized data when data sharing is not permitted due to privacy concerns. Ensemble-based learning works by training multiple (weak) classifiers whose output is aggregated. Federated ensembles are ensembles applied to a federated setting, where each classifier in the ensemble is trained on one data location. In this article, we explore the use of federated ensembles of Bayesian networks (FBNE) in a range of experiments and compare their performance with locally trained models and models trained with VertiBayes, a federated learning algorithm to train Bayesian networks from decentralized data. Our results show that FBNE outperforms local models and provides a significant increase in training speed compared with VertiBayes while maintaining a similar performance in most settings, among other advantages. We show that FBNE is a potentially useful tool within the federated learning toolbox, especially when local populations are heavily biased, or there is a strong imbalance in population size across parties. We discuss the advantages and disadvantages of this approach in terms of time complexity, model accuracy, privacy protection, and model interpretability.

Bayesian networks model relationships between random variables under uncertainty and can be used to predict the likelihood of events and outcomes while incorporating observed evidence. From an eXplainable AI (XAI) perspective, such models are interesting as they tend to be compact. Moreover, captured relations can be directly inspected by domain experts. In practice, data is often real-valued. Unless assumptions of normality can be made, discretization is often required. The optimal discretization, however, depends on the relations modelled between the variables. This complicates learning Bayesian networks from data. For this reason, most literature focuses on learning conditional dependencies between sets of variables, called structure learning. In this work, we extend an existing state-of-the-art structure learning approach based on the Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) to jointly learn variable discretizations. The proposed Discretized Bayesian Network GOMEA (DBN-GOMEA) obtains similar or better results than the current state-of-the-art when tasked to retrieve randomly generated ground-truth networks. Moreover, leveraging a key strength of evolutionary algorithms, we can straightforwardly perform DBN learning multi-objectively. We show how this enables incorporating expert knowledge in a uniquely insightful fashion, finding multiple DBNs that trade-off complexity, accuracy, and the difference with a pre-determined expert network.