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This paper investigates the use of probabilistic neural networks (PNNs) to model aleatoric uncertainty, which refers to the inherent variability in the input-output relationships of a system, often characterized by unequal variance or heteroscedasticity. Unlike traditional neural networks that produce deterministic outputs, PNNs generate probability distributions for the target variable, allowing the determination of both predicted means and intervals in regression scenarios. Contributions of this paper include the development of a probabilistic distance metric to optimize PNN architecture, and the deployment of PNNs in controlled data sets as well as a practical material science case involving fiber-reinforced composites. The findings confirm that PNNs effectively model aleatoric uncertainty, proving to be more appropriate than the commonly employed Gaussian process regression for this purpose. Specifically, in a real-world scientific machine learning context, PNNs yield remarkably accurate output mean estimates with R-squared scores approaching 0.97, and their predicted intervals exhibit a high correlation coefficient of nearly 0.80, closely matching observed data intervals. Hence, this research contributes to the ongoing exploration of leveraging the sophisticated representational capacity of neural networks to delineate complex input-output relationships in scientific problems.

This paper analyzes a popular computational framework to solve infinite-dimensional Bayesian inverse problems, discretizing the prior and the forward model in a finite-dimensional weighted inner product space. We demonstrate the benefit of working on a weighted space by establishing operator-norm bounds for finite element and graph-based discretizations of Mat\'ern-type priors and deconvolution forward models. For linear-Gaussian inverse problems, we develop a general theory to characterize the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and MAP estimators for nonlinear inverse problems. Our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh refinement.

As cloud computing usage grows, cloud data centers play an increasingly important role. To maximize resource utilization, ensure service quality, and enhance system performance, it is crucial to allocate tasks and manage performance effectively. The purpose of this study is to provide an extensive analysis of task allocation and performance management techniques employed in cloud data centers. The aim is to systematically categorize and organize previous research by identifying the cloud computing methodologies, categories, and gaps. A literature review was conducted, which included the analysis of 463 task allocations and 480 performance management papers. The review revealed three task allocation research topics and seven performance management methods. Task allocation research areas are resource allocation, load-Balancing, and scheduling. Performance management includes monitoring and control, power and energy management, resource utilization optimization, quality of service management, fault management, virtual machine management, and network management. The study proposes new techniques to enhance cloud computing work allocation and performance management. Short-comings in each approach can guide future research. The research's findings on cloud data center task allocation and performance management can assist academics, practitioners, and cloud service providers in optimizing their systems for dependability, cost-effectiveness, and scalability. Innovative methodologies can steer future research to fill gaps in the literature.

In this paper, we explore online convex optimization (OCO) and introduce a new analysis that provides fast rates by exploiting the curvature of feasible sets. In online linear optimization, it is known that if the average gradient of loss functions is larger than a certain value, the curvature of feasible sets can be exploited by the follow-the-leader (FTL) algorithm to achieve a logarithmic regret. This paper reveals that algorithms adaptive to the curvature of loss functions can also leverage the curvature of feasible sets. We first prove that if an optimal decision is on the boundary of a feasible set and the gradient of an underlying loss function is non-zero, then the algorithm achieves a regret upper bound of $O(\rho \log T)$ in stochastic environments. Here, $\rho > 0$ is the radius of the smallest sphere that includes the optimal decision and encloses the feasible set. Our approach, unlike existing ones, can work directly with convex loss functions, exploiting the curvature of loss functions simultaneously, and can achieve the logarithmic regret only with a local property of feasible sets. Additionally, it achieves an $O(\sqrt{T})$ regret even in adversarial environments where FTL suffers an $\Omega(T)$ regret, and attains an $O(\rho \log T + \sqrt{C \rho \log T})$ regret bound in corrupted stochastic environments with corruption level $C$. Furthermore, by extending our analysis, we establish a regret upper bound of $O\Big(T^{\frac{q-2}{2(q-1)}} (\log T)^{\frac{q}{2(q-1)}}\Big)$ for $q$-uniformly convex feasible sets, where uniformly convex sets include strongly convex sets and $\ell_p$-balls for $p \in [1,\infty)$. This bound bridges the gap between the $O(\log T)$ regret bound for strongly convex sets ($q=2$) and the $O(\sqrt{T})$ regret bound for non-curved sets ($q\to\infty$).

We study the computational complexity of computing Bayes-Nash equilibria in first-price auctions with discrete value distributions and discrete bidding space, under general subjective beliefs. It is known that such auctions do not always have pure equilibria. In this paper we prove that the problem of deciding their existence is NP-complete, even for approximate equilibria. On the other hand, it can be shown that mixed equilibria are guaranteed to exist; however, their computational complexity has not been studied before. We establish the PPAD-completeness of computing a mixed equilibrium and we complement this by an efficient algorithm for finding symmetric approximate equilibria in the special case of iid priors. En route to these results, we develop a computational equivalence framework between continuous and discrete first-price auctions, which can be of independent interest, and which allows us to transfer existing positive and negative results from one setting to the other. Finally, we show that correlated equilibria of the auction can be computed in polynomial time.

To fully exploit the benefits of the fog environment, efficient management of data locality is crucial. Blind or reactive data replication falls short in harnessing the potential of fog computing, necessitating more advanced techniques for predicting where and when clients will connect. While spatial prediction has received considerable attention, temporal prediction remains understudied. Our paper addresses this gap by examining the advantages of incorporating temporal prediction into existing spatial prediction models. We also provide a comprehensive analysis of spatio-temporal prediction models, such as Deep Neural Networks and Markov models, in the context of predictive replication. We propose a novel model using Holt-Winter's Exponential Smoothing for temporal prediction, leveraging sequential and periodical user movement patterns. In a fog network simulation with real user trajectories our model achieves a 15% reduction in excess data with a marginal 1% decrease in data availability.

This paper studies the case of possibly high-dimensional covariates in the regression discontinuity design (RDD) analysis. In particular, we propose estimation and inference methods for the RDD models with covariate selection which perform stably regardless of the number of covariates. The proposed methods combine the local approach using kernel weights with $\ell_{1}$-penalization to handle high-dimensional covariates. We provide theoretical and numerical results which illustrate the usefulness of the proposed methods. Theoretically, we present risk and coverage properties for our point estimation and inference methods, respectively. Under certain special case, the proposed estimator becomes more efficient than the conventional covariate adjusted estimator at the cost of an additional sparsity condition. Numerically, our simulation experiments and empirical example show the robust behaviors of the proposed methods to the number of covariates in terms of bias and variance for point estimation and coverage probability and interval length for inference.

This paper identifies a cultural dominance issue within large language models (LLMs) due to the predominant use of English data in model training (e.g., ChatGPT). LLMs often provide inappropriate English-culture-related answers that are not relevant to the expected culture when users ask in non-English languages. To systematically evaluate the cultural dominance issue, we build a benchmark of concrete (e.g., holidays and songs) and abstract (e.g., values and opinions) cultural objects. Empirical results show that the representative GPT models suffer from the culture dominance problem, where GPT-4 is the most affected while text-davinci-003 suffers the least from this problem. Our study emphasizes the need to critically examine cultural dominance and ethical consideration in their development and deployment. We show that two straightforward methods in model development (i.e., pretraining on more diverse data) and deployment (e.g., culture-aware prompting) can significantly mitigate the cultural dominance issue in LLMs.

We ask whether multilingual language models trained on unbalanced, English-dominated corpora use English as an internal pivot language -- a question of key importance for understanding how language models function and the origins of linguistic bias. Focusing on the Llama-2 family of transformer models, our study uses carefully constructed non-English prompts with a unique correct single-token continuation. From layer to layer, transformers gradually map an input embedding of the final prompt token to an output embedding from which next-token probabilities are computed. Tracking intermediate embeddings through their high-dimensional space reveals three distinct phases, whereby intermediate embeddings (1) start far away from output token embeddings; (2) already allow for decoding a semantically correct next token in the middle layers, but give higher probability to its version in English than in the input language; (3) finally move into an input-language-specific region of the embedding space. We cast these results into a conceptual model where the three phases operate in "input space", "concept space", and "output space", respectively. Crucially, our evidence suggests that the abstract "concept space" lies closer to English than to other languages, which may have important consequences regarding the biases held by multilingual language models.

Living things, computers, societies, and even books are part of a grand evolutionary struggle to survive. That struggle shapes nature, nations, religions, art, science, and you. What you think, feel, and do is determined by it. Darwinian evolution does not apply solely to the genes that are stored in DNA. Using the insights of Alan Turing and Richard Dawkins, we will see that it also applies to the memes we store in our brains and the information we store in our computers. The next time you run for president, fight a war, or just deal with the ordinary problems humans are heir to, perhaps this book will be of use. If you want to understand why and when you will die, or if you want to achieve greatness this book may help. If you are concerned about where the computer revolution is headed, this book may provide some answers.