In this paper, we consider the problem of learning a (first-order) theorem prover where we use a representation of beliefs in mathematical claims instead of a proof system to search for proofs. The inspiration for doing so comes from the practices of human mathematicians where a proof system is typically used after the fact to justify a sequence of intuitive steps obtained by "plausible reasoning" rather than to discover them. Towards this end, we introduce a probabilistic representation of beliefs in first-order statements based on first-order distributive normal forms (dnfs) devised by the philosopher Jaakko Hintikka. Notably, the representation supports Bayesian update and does not enforce that logically equivalent statements are assigned the same probability---otherwise, we would end up in a circular situation where we require a prover in order to assign beliefs. We then examine (1) conjecturing as (statistical) model selection and (2) an alternating-turn proving game amenable (in principle) to self-play training to learn a prover that is both complete in the limit and sound provided that players maintain "reasonable" beliefs. Dnfs have super-exponential space requirements so the ideas in this paper should be taken as conducting a thought experiment on "learning to prove". As a step towards making the ideas practical, we will comment on how abstractions can be used to control the space requirements at the cost of completeness.

We consider worker skill estimation for the single-coin Dawid-Skene crowdsourcing model. In practice, skill-estimation is challenging because worker assignments are sparse and irregular due to the arbitrary and uncontrolled availability of workers. We formulate skill estimation as a rank-one correlation-matrix completion problem, where the observed components correspond to observed label correlations between workers. We show that the correlation matrix can be successfully recovered and skills are identifiable if and only if the sampling matrix (observed components) does not have a bipartite connected component. We then propose a projected gradient descent scheme and show that skill estimates converge to the desired global optima for such sampling matrices. Our proof is original and the results are surprising in light of the fact that even the weighted rank-one matrix factorization problem is NP-hard in general. Next, we derive sample complexity bounds in terms of spectral properties of the signless Laplacian of the sampling matrix. Our proposed scheme achieves state-of-art performance on a number of real-world datasets.

Many expensive black-box optimisation problems are sensitive to their inputs. In these problems it makes more sense to locate a region of good designs, than a single, possible fragile, optimal design. Expensive black-box functions can be optimised effectively with Bayesian optimisation, where a Gaussian process is a popular choice as a prior over the expensive function. We propose a method for robust optimisation using Bayesian optimisation to find a region of design space in which the expensive function's performance is insensitive to the inputs whilst retaining a good quality. This is achieved by sampling realisations from a Gaussian process modelling the expensive function and evaluating the improvement for each realisation. The expectation of these improvements can be optimised cheaply with an evolutionary algorithm to determine the next location at which to evaluate the expensive function. We describe an efficient process to locate the optimum expected improvement. We show empirically that evaluating the expensive function at the location in the candidate sweet spot about which the model is most uncertain or at random yield the best convergence in contrast to exploitative schemes. We illustrate our method on six test functions in two, five, and ten dimensions, and demonstrate that it is able to outperform a state-of-the-art approach from the literature.

In recent years, there has been an exponential growth in the number of complex documents and texts that require a deeper understanding of machine learning methods to be able to accurately classify texts in many applications. Many machine learning approaches have achieved surpassing results in natural language processing. The success of these learning algorithms relies on their capacity to understand complex models and non-linear relationships within data. However, finding suitable structures, architectures, and techniques for text classification is a challenge for researchers. In this paper, a brief overview of text classification algorithms is discussed. This overview covers different text feature extractions, dimensionality reduction methods, existing algorithms and techniques, and evaluations methods. Finally, the limitations of each technique and their application in the real-world problem are discussed.

Data driven soft sensor design has recently gained immense popularity, due to advances in sensory devices, and a growing interest in data mining. While partial least squares (PLS) is traditionally used in the process literature for designing soft sensors, the statistical literature has focused on sparse learners, such as Lasso and relevance vector machine (RVM), to solve the high dimensional data problem. In the current study, predictive performances of three regression techniques, PLS, Lasso and RVM were assessed and compared under various offline and online soft sensing scenarios applied on datasets from five real industrial plants, and a simulated process. In offline learning, predictions of RVM and Lasso were found to be superior to those of PLS when a large number of time-lagged predictors were used. Online prediction results gave a slightly more complicated picture. It was found that the minimum prediction error achieved by PLS under moving window (MW), or just-in-time learning scheme was decreased up to ~5-10% using Lasso, or RVM. However, when a small MW size was used, or the optimum number of PLS components was as low as ~1, prediction performance of PLS surpassed RVM, which was found to yield occasional unstable predictions. PLS and Lasso models constructed via online parameter tuning generally did not yield better predictions compared to those constructed via offline tuning. We present evidence to suggest that retaining a large portion of the available process measurement data in the predictor matrix, instead of preselecting variables, would be more advantageous for sparse learners in increasing prediction accuracy. As a result, Lasso is recommended as a better substitute for PLS in soft sensors; while performance of RVM should be validated before online application.

Since the advent of the horseshoe priors for regularization, global-local shrinkage methods have proved to be a fertile ground for the development of Bayesian methodology in machine learning, specifically for high-dimensional regression and classification problems. They have achieved remarkable success in computation, and enjoy strong theoretical support. Most of the existing literature has focused on the linear Gaussian case; see Bhadra et al. (2019) for a systematic survey. The purpose of the current article is to demonstrate that the horseshoe regularization is useful far more broadly, by reviewing both methodological and computational developments in complex models that are more relevant to machine learning applications. Specifically, we focus on methodological challenges in horseshoe regularization in nonlinear and non-Gaussian models; multivariate models; and deep neural networks. We also outline the recent computational developments in horseshoe shrinkage for complex models along with a list of available software implementations that allows one to venture out beyond the comfort zone of the canonical linear regression problems.

Structure learning of Bayesian networks has always been a challenging problem. Nowadays, massive-size networks with thousands or more of nodes but fewer samples frequently appear in many areas. We develop a divide-and-conquer framework, called partition-estimation-fusion (PEF), for structure learning of such big networks. The proposed method first partitions nodes into clusters, then learns a subgraph on each cluster of nodes, and finally fuses all learned subgraphs into one Bayesian network. The PEF method is designed in a flexible way so that any structure learning method may be used in the second step to learn a subgraph structure as either a DAG or a CPDAG. In the clustering step, we adapt the hierarchical clustering method to automatically choose a proper number of clusters. In the fusion step, we propose a novel hybrid method that sequentially add edges between subgraphs. Extensive numerical experiments demonstrate the competitive performance of our PEF method, in terms of both speed and accuracy compared to existing methods. Our method can improve the accuracy of structure learning by 20% or more, while reducing running time up to two orders-of-magnitude.

Modeling strategic conflict from a game theoretical perspective involves dealing with epistemic uncertainty. Payoff uncertainty models are typically restricted to simple probability models due to computational restrictions. Recent breakthroughs Artificial Intelligence (AI) research applied to Poker have resulted in novel approximation approaches such as counterfactual regret minimization, that can successfully deal with large-scale imperfect games. By drawing from these ideas, this work addresses the problem of arbitrary continuous payoff distributions. We propose a method, Harsanyi-Counterfactual Regret Minimization, to solve two-player zero-sum extensive-form games with arbitrary payoff distribution models. Given a game $\Gamma$, using a Harsanyi transformation we generate a new game $\Gamma^\#$ to which we later apply Counterfactual Regret Minimization to obtain $\varepsilon$-Nash equilibria. We include numerical experiments showing how the method can be applied to a previously published problem.

A great deal of attention has been recently given to Machine Learning (ML) techniques in many different application fields. This paper provides a vision of what ML can do in Power Line Communications (PLC). We firstly and briefly describe classical formulations of ML, and distinguish deterministic problems from statistical problems with relevance to communications. We then discuss ML applications in PLC for each layer, namely, for characterization and modeling, for physical layer algorithms, for media access control and networking algorithms. Finally, other applications of PLC that can benefit from the usage of ML, as grid diagnostics, are analyzed. Illustrative numerical examples are reported to serve the purpose of validating the ideas and motivate future research endeavors in this stimulating signal/data processing field.

There are three quite distinct ways to train a machine learning model on recommender system logs. The first method is to model the reward prediction for each possible recommendation to the user, at the scoring time the best recommendation is found by computing an argmax over the personalized recommendations. This method obeys principles such as the conditionality principle and the likelihood principle. A second method is useful when the model does not fit reality and underfits. In this case, we can use the fact that we know the distribution of historical recommendations (concentrated on previously identified good actions with some exploration) to adjust the errors in the fit to be evenly distributed over all actions. Finally, the inverse propensity score can be used to produce an estimate of the decision rules expected performance. The latter two methods violate the conditionality and likelihood principle but are shown to have good performance in certain settings. In this paper we review the literature around this fundamental, yet often overlooked choice and do some experiments using the RecoGym simulation environment.