Mixture models are traditionally represented and learned by adding several distributions as components. Allowing mixtures to subtract probability mass or density can drastically reduce the number of components needed to model complex distributions. However, learning such subtractive mixtures while ensuring they still encode a non-negative function is challenging. We investigate how to learn and perform inference on deep subtractive mixtures by squaring them. We do this in the framework of probabilistic circuits, which enable us to represent tensorized mixtures and generalize several other subtractive models. We theoretically prove that the class of squared circuits allowing subtractions can be exponentially more expressive than traditional additive mixtures; and, we empirically show this increased expressiveness on a series of real-world distribution estimation tasks.

This paper studies the problem of learning message propagation strategies for graph neural networks (GNNs). One of the challenges for graph neural networks is that of defining the propagation strategy. For instance, the choices of propagation steps are often specialized to a single graph and are not personalized to different nodes. To compensate for this, in this paper, we present learning to propagate, a general learning framework that not only learns the GNN parameters for prediction but more importantly, can explicitly learn the interpretable and personalized propagate strategies for different nodes and various types of graphs. We introduce the optimal propagation steps as latent variables to help find the maximum-likelihood estimation of the GNN parameters in a variational Expectation-Maximization (VEM) framework. Extensive experiments on various types of graph benchmarks demonstrate that our proposed framework can significantly achieve better performance compared with the state-of-the-art methods, and can effectively learn personalized and interpretable propagate strategies of messages in GNNs.

The problem of how to take the right actions to make profits in sequential process continues to be difficult due to the quick dynamics and a significant amount of uncertainty in many application scenarios. In such complicated environments, reinforcement learning (RL), a reward-oriented strategy for optimum control, has emerged as a potential technique to address this strategic decision-making issue. However, reinforcement learning also has some shortcomings that make it unsuitable for solving many financial problems, excessive resource consumption, and inability to quickly obtain optimal solutions, making it unsuitable for quantitative trading markets. In this study, we use two methods to overcome the issue with contextual information: contextual Thompson sampling and reinforcement learning under supervision which can accelerate the iterations in search of the best answer. In order to investigate strategic trading in quantitative markets, we merged the earlier financial trading strategy known as constant proportion portfolio insurance (CPPI) into deep deterministic policy gradient (DDPG). The experimental results show that both methods can accelerate the progress of reinforcement learning to obtain the optimal solution.

Machine learning models have undeniably achieved impressive performance across a range of applications. However, their often perceived black-box nature, and lack of transparency in decision-making, have raised concerns about understanding their predictions. To tackle this challenge, researchers have developed methods to provide explanations for machine learning models. In this paper, we introduce LaPLACE-explainer, designed to provide probabilistic cause-and-effect explanations for any classifier operating on tabular data, in a human-understandable manner. The LaPLACE-Explainer component leverages the concept of a Markov blanket to establish statistical boundaries between relevant and non-relevant features automatically. This approach results in the automatic generation of optimal feature subsets, serving as explanations for predictions. Importantly, this eliminates the need to predetermine a fixed number N of top features as explanations, enhancing the flexibility and adaptability of our methodology. Through the incorporation of conditional probabilities, our approach offers probabilistic causal explanations and outperforms LIME and SHAP (well-known model-agnostic explainers) in terms of local accuracy and consistency of explained features. LaPLACE's soundness, consistency, local accuracy, and adaptability are rigorously validated across various classification models. Furthermore, we demonstrate the practical utility of these explanations via experiments with both simulated and real-world datasets. This encompasses addressing trust-related issues, such as evaluating prediction reliability, facilitating model selection, enhancing trustworthiness, and identifying fairness-related concerns within classifiers.

Solving program induction problems requires searching through an enormous space of possibilities. DreamCoder is an inductive program synthesis system that, whilst solving problems, learns to simplify search in an iterative wake-sleep procedure. The cost of search is amortised by training a neural search policy, reducing search breadth and effectively "compiling" useful information to compose program solutions across tasks. Additionally, a library of program components is learnt to express discovered solutions in fewer components, reducing search depth. In DreamCoder, the neural search policy has only an indirect effect on the library learnt through the program solutions it helps discover. We present an approach for library learning that directly leverages the neural search policy, effectively "decompiling" its amortised knowledge to extract relevant program components. This provides stronger amortised inference: the amortised knowledge learnt to reduce search breadth is now also used to reduce search depth. We integrate our approach with DreamCoder and demonstrate faster domain proficiency with improved generalisation on a range of domains, particularly when fewer example solutions are available.

The maximum likelihood method is the best-known method for estimating the probabilities behind the data. However, the conventional method obtains the probability model closest to the empirical distribution, resulting in overfitting. Then regularization methods prevent the model from being excessively close to the wrong probability, but little is known systematically about their performance. The idea of regularization is similar to error-correcting codes, which obtain optimal decoding by mixing suboptimal solutions with an incorrectly received code. The optimal decoding in error-correcting codes is achieved based on gauge symmetry. We propose a theoretically guaranteed regularization in the maximum likelihood method by focusing on a gauge symmetry in Kullback -- Leibler divergence. In our approach, we obtain the optimal model without the need to search for hyperparameters frequently appearing in regularization.

Most existing federated learning methods are unable to estimate model/predictive uncertainty since the client models are trained using the standard loss function minimization approach which ignores such uncertainties. In many situations, however, especially in limited data settings, it is beneficial to take into account the uncertainty in the model parameters at each client as it leads to more accurate predictions and also because reliable estimates of uncertainty can be used for tasks, such as out-of-distribution (OOD) detection, and sequential decision-making tasks, such as active learning. We present a framework for federated learning with uncertainty where, in each round, each client infers the posterior distribution over its parameters as well as the posterior predictive distribution (PPD), distills the PPD into a single deep neural network, and sends this network to the server. Unlike some of the recent Bayesian approaches to federated learning, our approach does not require sending the whole posterior distribution of the parameters from each client to the server but only the PPD in the distilled form as a deep neural network. In addition, when making predictions at test time, it does not require computationally expensive Monte-Carlo averaging over the posterior distribution because our approach always maintains the PPD in the form of a single deep neural network. Moreover, our approach does not make any restrictive assumptions, such as the form of the clients' posterior distributions, or of their PPDs. We evaluate our approach on classification in federated setting, as well as active learning and OOD detection in federated settings, on which our approach outperforms various existing federated learning baselines.

Active learning provides a framework to adaptively query the most informative experiments towards learning an unknown black-box function. Various approaches of active learning have been proposed in the literature, however, they either focus on exploration or exploitation in the design space. Methods that do consider exploration-exploitation simultaneously employ fixed or ad-hoc measures to control the trade-off that may not be optimal. In this paper, we develop a Bayesian hierarchical approach, referred as BHEEM, to dynamically balance the exploration-exploitation trade-off as more data points are queried. To sample from the posterior distribution of the trade-off parameter, We subsequently formulate an approximate Bayesian computation approach based on the linear dependence of queried data in the feature space. Simulated and real-world examples show the proposed approach achieves at least 21% and 11% average improvement when compared to pure exploration and exploitation strategies respectively. More importantly, we note that by optimally balancing the trade-off between exploration and exploitation, BHEEM performs better or at least as well as either pure exploration or pure exploitation.

The lighting requirements are subjective and one light setting cannot work for all. However, there is little work on developing smart lighting algorithms that can adapt to user preferences. To address this gap, this paper uses fuzzy logic and reinforcement learning to develop an adaptive lighting algorithm. In particular, we develop a baseline fuzzy inference system (FIS) using the domain knowledge. We use the existing literature to create a FIS that generates lighting setting recommendations based on environmental conditions i.e. daily glare index, and user information including age, activity, and chronotype. Through a feedback mechanism, the user interacts with the algorithm, correcting the algorithm output to their preferences. We interpret these corrections as rewards to a Q-learning agent, which tunes the FIS parameters online to match the user preferences. We implement the algorithm in an aircraft cabin mockup and conduct an extensive user study to evaluate the effectiveness of the algorithm and understand its learning behavior. Our implementation results demonstrate that the developed algorithm possesses the capability to learn user preferences while successfully adapting to a wide range of environmental conditions and user characteristics. and can deal with a diverse spectrum of environmental conditions and user characteristics. This underscores its viability as a potent solution for intelligent light management, featuring advanced learning capabilities.

For a broad class of choice and ranking models based on Luce's choice axiom, including the Bradley--Terry--Luce and Plackett--Luce models, we show that the associated maximum likelihood estimation problems are equivalent to a classic matrix balancing problem with target row and column sums. This perspective opens doors between two seemingly unrelated research areas, and allows us to unify existing algorithms in the choice modeling literature as special instances or analogs of Sinkhorn's celebrated algorithm for matrix balancing. We draw inspirations from these connections and resolve important open problems on the study of Sinkhorn's algorithm. We first prove the global linear convergence of Sinkhorn's algorithm for non-negative matrices whenever finite solutions to the matrix balancing problem exist. We characterize this global rate of convergence in terms of the algebraic connectivity of the bipartite graph constructed from data. Next, we also derive the sharp asymptotic rate of linear convergence, which generalizes a classic result of Knight (2008), but with a more explicit analysis that exploits an intrinsic orthogonality structure. To our knowledge, these are the first quantitative linear convergence results for Sinkhorn's algorithm for general non-negative matrices and positive marginals. The connections we establish in this paper between matrix balancing and choice modeling could help motivate further transmission of ideas and interesting results in both directions.