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We give here a proof of the convergence of the Stochastic Gradient Descent (SGD) in a self-contained manner.

The integrated nested Laplace approximations (INLA) method has become a widely utilized tool for researchers and practitioners seeking to perform approximate Bayesian inference across various fields of application. To address the growing demand for incorporating more complex models and enhancing the method's capabilities, this paper introduces a novel framework that leverages dense matrices for performing approximate Bayesian inference based on INLA across multiple computing nodes using HPC. When dealing with non-sparse precision or covariance matrices, this new approach scales better compared to the current INLA method, capitalizing on the computational power offered by multiprocessors in shared and distributed memory architectures available in contemporary computing resources and specialized dense matrix algebra. To validate the efficacy of this approach, we conduct a simulation study then apply it to analyze cancer mortality data in Spain, employing a three-way spatio-temporal interaction model.

Approximate inference methods like the Laplace method, Laplace approximations and variational methods, amongst others, are popular methods when exact inference is not feasible due to the complexity of the model or the abundance of data. In this paper we propose a hybrid approximate method called Low-Rank Variational Bayes correction (VBC), that uses the Laplace method and subsequently a Variational Bayes correction in a lower dimension, to the joint posterior mean. The cost is essentially that of the Laplace method which ensures scalability of the method, in both model complexity and data size. Models with fixed and unknown hyperparameters are considered, for simulated and real examples, for small and large datasets.

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our method has the same symbol $|\xi|^\alpha$ as the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This {\it unique feature} distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature, which are usually limited to periodic boundary conditions (see Remark \ref{remark0}). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector products. The computational complexity is ${\mathcal O}(2N\log(2N))$, and the memory storage is ${\mathcal O}(N)$ with $N$ the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems.

Large Language Models (LLMs) have reshaped natural language processing with their impressive capabilities. Their ever-increasing size, however, raised concerns about their effective deployment and the need for LLM compressions. This study introduces the Divergent Token metrics (DTMs), a novel approach for assessing compressed LLMs, addressing the limitations of traditional perplexity or accuracy measures that fail to accurately reflect text generation quality. DTMs focus on token divergence, that allow deeper insights into the subtleties of model compression, i.p. when evaluating component's impacts individually. Utilizing the First Divergent Token metric (FDTM) in model sparsification reveals that a quarter of all attention components can be pruned beyond 90% on the Llama-2 model family, still keeping SOTA performance. For quantization FDTM suggests that over 80% of parameters can naively be transformed to int8 without special outlier management. These evaluations indicate the necessity of choosing appropriate compressions for parameters individually-and that FDTM can identify those-while standard metrics result in deteriorated outcomes.

We propose an individual claims reserving model based on the conditional Aalen--Johansen estimator, as developed in Bladt and Furrer (2023b). In our approach, we formulate a multi-state problem, where the underlying variable is the individual claim size, rather than time. The states in this model represent development periods, and we estimate the cumulative density function of individual claim costs using the conditional Aalen--Johansen method as transition probabilities to an absorbing state. Our methodology reinterprets the concept of multi-state models and offers a strategy for modeling the complete curve of individual claim costs. To illustrate our approach, we apply our model to both simulated and real datasets. Having access to the entire dataset enables us to support the use of our approach by comparing the predicted total final cost with the actual amount, as well as evaluating it in terms of the continuously ranked probability score, as discussed in Gneiting and A. E. Raftery (2007)

This paper investigates goal-oriented communication for remote estimation of multiple Markov sources in resource-constrained networks. An agent selects the update order of the sources and transmits the packet to a remote destination over an unreliable delay channel. The destination is tasked with source reconstruction for the purpose of actuation. We utilize the metric cost of actuation error (CAE) to capture the significance (semantics) of error at the point of actuation. We aim to find an optimal sampling policy that minimizes the time-averaged CAE subject to average resource constraints. We formulate this problem as an average-cost constrained Markov Decision Process (CMDP) and transform it into an unconstrained MDP by utilizing Lyapunov drift techniques. Then, we propose a low-complexity drift-plus-penalty(DPP) policy for systems with known source/channel statistics and a Lyapunov optimization-based deep reinforcement learning (LO-DRL) policy for unknown environments. Our policies achieve near-optimal performance in CAE minimization and significantly reduce the number of uninformative transmissions.

Electrodermal activity (EDA) is considered a standard marker of sympathetic activity. However, traditional EDA measurement requires electrodes in steady contact with the skin. Can sympathetic arousal be measured using only an optical sensor, such as an RGB camera? This paper presents a novel approach to infer sympathetic arousal by measuring the peripheral blood flow on the face or hand optically. We contribute a self-recorded dataset of 21 participants, comprising synchronized videos of participants' faces and palms and gold-standard EDA and photoplethysmography (PPG) signals. Our results show that we can measure peripheral sympathetic responses that closely correlate with the ground truth EDA. We obtain median correlations of 0.57 to 0.63 between our inferred signals and the ground truth EDA using only videos of the participants' palms or foreheads or PPG signals from the foreheads or fingers. We also show that sympathetic arousal is best inferred from the forehead, finger, or palm.

The satisfiability problem is NP-complete but there are subclasses where all the instances are satisfiable. For this, restrictions on the shape of the formula are made. Darman and D\"ocker show that the subclass MONOTONE $3$-SAT-($k$,1) with $k \geq 5$ proves to be NP-complete and pose the open question whether instances of MONOTONE $3$-SAT-(3,1) are satisfiable. This paper shows that all instances of MONOTONE $3$-SAT-(3,1) are satisfiable using the new concept of a color-structures.

Although measuring held-out accuracy has been the primary approach to evaluate generalization, it often overestimates the performance of NLP models, while alternative approaches for evaluating models either focus on individual tasks or on specific behaviors. Inspired by principles of behavioral testing in software engineering, we introduce CheckList, a task-agnostic methodology for testing NLP models. CheckList includes a matrix of general linguistic capabilities and test types that facilitate comprehensive test ideation, as well as a software tool to generate a large and diverse number of test cases quickly. We illustrate the utility of CheckList with tests for three tasks, identifying critical failures in both commercial and state-of-art models. In a user study, a team responsible for a commercial sentiment analysis model found new and actionable bugs in an extensively tested model. In another user study, NLP practitioners with CheckList created twice as many tests, and found almost three times as many bugs as users without it.