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The techniques used to generate pseudo-random numbers for Monte Carlo (MC) applications bear many implications on the quality and speed of that programs work. As a random number generator (RNG) slows, the production of random numbers begins to dominate runtime. As RNG output grows in correlation, the final product becomes less reliable. These difficulties are further compounded by the need for reproducibility and parallelism. For reproducibility, the numbers generated to determine any outcome must be the same each time a simulation is run. However, the concurrency that comes with most parallelism introduces race conditions. To have both reproducibility and concurrency, separate RNG states must be tracked for each independently schedulable unit of simulation, forming independent random number streams. We propose an alternative to the stride-based parallel LCG seeding approach that scales more practically with increased concurrency and workload by generating seeds through hashing and allowing for repeated outputs. Data gathered from normality tests of tally results from simple MC transport benchmark calculations indicates that the proposed hash-based RNG does not significantly affect the tally result normality property as compared to the conventional stride-based RNG.

Although there is mounting empirical evidence for the increase in affective polarization, few mechanistic models can explain its emergence at the population level. The question of how such a phenomenon can emerge from divergent opinions of a population on an ideological issue is still an open issue. In this paper, we establish that human normativity, that is, individual expression of normative opinions based on beliefs about the population, can lead to population-level polarization when ideological institutions distort beliefs in accordance with their objective of moving expressed opinion to one extreme. Using a game-theoretic model, we establish that individuals with more extreme opinions will have more extreme rhetoric and higher misperceptions about their outgroup members. Our model also shows that when social recommendation systems mediate institutional signals, we can observe the formation of different institutional communities, each with its unique community structure and characteristics. Using the model, we identify practical strategies platforms can implement, such as reducing exposure to signals from ideological institutions and a tailored approach to content moderation, both of which can rectify the affective polarization problem within its purview.

A recent development in Bayesian optimization is the use of local optimization strategies, which can deliver strong empirical performance on high-dimensional problems compared to traditional global strategies. The "folk wisdom" in the literature is that the focus on local optimization sidesteps the curse of dimensionality; however, little is known concretely about the expected behavior or convergence of Bayesian local optimization routines. We first study the behavior of the local approach, and find that the statistics of individual local solutions of Gaussian process sample paths are surprisingly good compared to what we would expect to recover from global methods. We then present the first rigorous analysis of such a Bayesian local optimization algorithm recently proposed by M\"uller et al. (2021), and derive convergence rates in both the noisy and noiseless settings.

Instead of testing solely a precise hypothesis, it is often useful to enlarge it with alternatives that are deemed to differ from it negligibly. For instance, in a bioequivalence study one might consider the hypothesis that the concentration of an ingredient is exactly the same in two drugs. In such a context, it might be more relevant to test the enlarged hypothesis that the difference in concentration between the drugs is of no practical significance. While this concept is not alien to Bayesian statistics, applications remain confined to parametric settings and strategies on how to effectively harness experts' intuitions are often scarce or nonexistent. To resolve both issues, we introduce PROTEST, an accessible nonparametric testing framework that seamlessly integrates with Markov Chain Monte Carlo (MCMC) methods. We develop expanded versions of the model adherence, goodness-of-fit, quantile and two-sample tests. To demonstrate how PROTEST operates, we make use of examples, simulated studies - such as testing link functions in a binary regression setting, as well as a comparison between the performance of PROTEST and the PTtest (Holmes et al., 2015) - and an application with data on neuron spikes. Furthermore, we address the crucial issue of selecting the threshold - which controls how much a hypothesis is to be expanded - even when intuitions are limited or challenging to quantify.

A Kochen-Specker (KS) set is a finite collection of vectors on the two-sphere containing no antipodal pairs for which it is impossible to assign 0s and 1s such that no two orthogonal vectors are assigned 1 and exactly one vector in every triplet of mutually orthogonal vectors is assigned 1. The existence of KS sets lies at the heart of Kochen and Specker's argument against non-contextual hidden variable theories and the Conway-Kochen free will theorem. Identifying small KS sets can simplify these arguments and may contribute to the understanding of the role played by contextuality in quantum protocols. In this paper we derive a weak lower bound of 10 vectors for the size of any KS set by studying the opposite notion of large non-KS sets and using a probability argument that is independent of the graph structure of KS sets. We also point out an interesting connection with a generalisation of the moving sofa problem around a right-angled hallway on the two-sphere.

In 1991 Ramshaw and Mesina introduced a clever synthesis of penalty methods and artificial compression methods. Its form makes it an interesting option to replace the pressure update in the Uzawa iteration. The result, for the Stokes problem, is \begin{equation} \left\{ \begin{array} [c]{cc} Step\ 1: & -\triangle u^{n+1}+\nabla p^{n}=f(x),\ {\rm in}\ \Omega,\ u^{n+1}|_{\partial\Omega}=0,\\ Step\ 2: & p^{n+1}-p^{n}+\beta\nabla\cdot(u^{n+1}-u^{n})+\alpha ^{2}\nabla\cdot u^{n+1}=0. \end{array} \right. \end{equation} For saddle point problems, including Stokes, this iteration converges under a condition similar to the one required for Uzawa iteration.

The paper presents an evaluation of popular audio inpainting methods based on autoregressive modelling, namely, extrapolation-based and Janssen methods. A novel variant of the Janssen method suitable for gap inpainting is also proposed. The main differences between the particular popular approaches are pointed out, and a mid-scale computational experiment is presented. The results demonstrate the importance of the choice of the AR model estimator and the suitability of the new gap-wise Janssen method.

Large Language Models (LLMs) have shown excellent generalization capabilities that have led to the development of numerous models. These models propose various new architectures, tweaking existing architectures with refined training strategies, increasing context length, using high-quality training data, and increasing training time to outperform baselines. Analyzing new developments is crucial for identifying changes that enhance training stability and improve generalization in LLMs. This survey paper comprehensively analyses the LLMs architectures and their categorization, training strategies, training datasets, and performance evaluations and discusses future research directions. Moreover, the paper also discusses the basic building blocks and concepts behind LLMs, followed by a complete overview of LLMs, including their important features and functions. Finally, the paper summarizes significant findings from LLM research and consolidates essential architectural and training strategies for developing advanced LLMs. Given the continuous advancements in LLMs, we intend to regularly update this paper by incorporating new sections and featuring the latest LLM models.

AI is undergoing a paradigm shift with the rise of models (e.g., BERT, DALL-E, GPT-3) that are trained on broad data at scale and are adaptable to a wide range of downstream tasks. We call these models foundation models to underscore their critically central yet incomplete character. This report provides a thorough account of the opportunities and risks of foundation models, ranging from their capabilities (e.g., language, vision, robotics, reasoning, human interaction) and technical principles(e.g., model architectures, training procedures, data, systems, security, evaluation, theory) to their applications (e.g., law, healthcare, education) and societal impact (e.g., inequity, misuse, economic and environmental impact, legal and ethical considerations). Though foundation models are based on standard deep learning and transfer learning, their scale results in new emergent capabilities,and their effectiveness across so many tasks incentivizes homogenization. Homogenization provides powerful leverage but demands caution, as the defects of the foundation model are inherited by all the adapted models downstream. Despite the impending widespread deployment of foundation models, we currently lack a clear understanding of how they work, when they fail, and what they are even capable of due to their emergent properties. To tackle these questions, we believe much of the critical research on foundation models will require deep interdisciplinary collaboration commensurate with their fundamentally sociotechnical nature.

The recent proliferation of knowledge graphs (KGs) coupled with incomplete or partial information, in the form of missing relations (links) between entities, has fueled a lot of research on knowledge base completion (also known as relation prediction). Several recent works suggest that convolutional neural network (CNN) based models generate richer and more expressive feature embeddings and hence also perform well on relation prediction. However, we observe that these KG embeddings treat triples independently and thus fail to cover the complex and hidden information that is inherently implicit in the local neighborhood surrounding a triple. To this effect, our paper proposes a novel attention based feature embedding that captures both entity and relation features in any given entity's neighborhood. Additionally, we also encapsulate relation clusters and multihop relations in our model. Our empirical study offers insights into the efficacy of our attention based model and we show marked performance gains in comparison to state of the art methods on all datasets.