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This work addresses the problem of high-dimensional classification by exploring the generalized Bayesian logistic regression method under a sparsity-inducing prior distribution. The method involves utilizing a fractional power of the likelihood resulting the fractional posterior. Our study yields concentration results for the fractional posterior, not only on the joint distribution of the predictor and response variable but also for the regression coefficients. Significantly, we derive novel findings concerning misclassification excess risk bounds using sparse generalized Bayesian logistic regression. These results parallel recent findings for penalized methods in the frequentist literature. Furthermore, we extend our results to the scenario of model misspecification, which is of critical importance.

We propose a novel time stepping method for linear poroelasticity by extending a recent iterative decoupling approach to the second-order case. This results in a two-step scheme with an inner iteration and a relaxation step. We prove second-order convergence for a prescribed number of inner iteration steps, only depending on the coupling strength of the elastic and the flow equation. The efficiency of the scheme is illustrated by a number of numerical experiments, including a simulation of three-dimensional brain tissue.

This paper studies the fundamental limits of availability and throughput for independent and heterogeneous demands of a limited resource. Availability is the probability that the demands are below the capacity of the resource. Throughput is the expected fraction of the resource that is utilized by the demands. We offer a concentration inequality generator that gives lower bounds on feasible availability and throughput pairs with a given capacity and independent but not necessarily identical distributions of up-to-unit demands. We show that availability and throughput cannot both be poor. These bounds are analogous to tail inequalities on sums of independent random variables, but hold throughout the support of the demand distribution. This analysis gives analytically tractable bounds supporting the unit-demand characterization of Chawla, Devanur, and Lykouris (2023) and generalizes to up-to-unit demands. Our bounds also provide an approach towards improved multi-unit prophet inequalities (Hajiaghayi, Kleinberg, and Sandholm, 2007). They have applications to transaction fee mechanism design (for blockchains) where high availability limits the probability of profitable user-miner coalitions (Chung and Shi, 2023).

Preference modelling lies at the intersection of economics, decision theory, machine learning and statistics. By understanding individuals' preferences and how they make choices, we can build products that closely match their expectations, paving the way for more efficient and personalised applications across a wide range of domains. The objective of this tutorial is to present a cohesive and comprehensive framework for preference learning with Gaussian Processes (GPs), demonstrating how to seamlessly incorporate rationality principles (from economics and decision theory) into the learning process. By suitably tailoring the likelihood function, this framework enables the construction of preference learning models that encompass random utility models, limits of discernment, and scenarios with multiple conflicting utilities for both object- and label-preference. This tutorial builds upon established research while simultaneously introducing some novel GP-based models to address specific gaps in the existing literature.

This paper focuses on the numerical scheme for multiple-delay stochastic differential equations with partially H\"older continuous drifts and locally H\"older continuous diffusion coefficients. To handle with the superlinear terms in coefficients, the truncated Euler-Maruyama scheme is employed. Under the given conditions, the convergence rates at time $T$ in both $\mathcal{L}^{1}$ and $\mathcal{L}^{2}$ senses are shown by virtue of the Yamada-Watanabe approximation technique. Moreover, the convergence rates over a finite time interval $[0,T]$ are also obtained. Additionally, it should be noted that the convergence rates will not be affected by the number of delay variables. Finally, we perform the numerical experiments on the stochastic volatility model to verify the reliability of the theoretical results.

This paper presents a time-causal analogue of the Gabor filter, as well as a both time-causal and time-recursive analogue of the Gabor transform, where the proposed time-causal representations obey both temporal scale covariance and a cascade property with a simplifying kernel over temporal scales. The motivation behind these constructions is to enable theoretically well-founded time-frequency analysis over multiple temporal scales for real-time situations, or for physical or biological modelling situations, when the future cannot be accessed, and the non-causal access to future in Gabor filtering is therefore not viable for a time-frequency analysis of the system. We develop the theory for these representations, obtained by replacing the Gaussian kernel in Gabor filtering with a time-causal kernel, referred to as the time-causal limit kernel, which guarantees simplification properties from finer to coarser levels of scales in a time-causal situation, similar as the Gaussian kernel can be shown to guarantee over a non-causal temporal domain. In these ways, the proposed time-frequency representations guarantee well-founded treatment over multiple scales, in situations when the characteristic scales in the signals, or physical or biological phenomena, to be analyzed may vary substantially, and additionally all steps in the time-frequency analysis have to be fully time-causal.

This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.

This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region and (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data. We study the properties of these three main discretization methods both theoretically and experimentally, and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and derivatives as well as the integrated Gaussian kernels and derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing.

This paper proposes a methodology for constructing such corpora of child directed speech (CDS) paired with sentential logical forms, and uses this method to create two such corpora, in English and Hebrew. The approach enforces a cross-linguistically consistent representation, building on recent advances in dependency representation and semantic parsing. Specifically, the approach involves two steps. First, we annotate the corpora using the Universal Dependencies (UD) scheme for syntactic annotation, which has been developed to apply consistently to a wide variety of domains and typologically diverse languages. Next, we further annotate these data by applying an automatic method for transducing sentential logical forms (LFs) from UD structures. The UD and LF representations have complementary strengths: UD structures are language-neutral and support consistent and reliable annotation by multiple annotators, whereas LFs are neutral as to their syntactic derivation and transparently encode semantic relations. Using this approach, we provide syntactic and semantic annotation for two corpora from CHILDES: Brown's Adam corpus (English; we annotate ~80% of its child-directed utterances), all child-directed utterances from Berman's Hagar corpus (Hebrew). We verify the quality of the UD annotation using an inter-annotator agreement study, and manually evaluate the transduced meaning representations. We then demonstrate the utility of the compiled corpora through (1) a longitudinal corpus study of the prevalence of different syntactic and semantic phenomena in the CDS, and (2) applying an existing computational model of language acquisition to the two corpora and briefly comparing the results across languages.

As language models (LMs) become more capable, it is increasingly important to align them with human preferences. However, the dominant paradigm for training Preference Models (PMs) for that purpose suffers from fundamental limitations, such as lack of transparency and scalability, along with susceptibility to overfitting the preference dataset. We propose Compositional Preference Models (CPMs), a novel PM framework that decomposes one global preference assessment into several interpretable features, obtains scalar scores for these features from a prompted LM, and aggregates these scores using a logistic regression classifier. Through these simple steps, CPMs allow to control which properties of the preference data are used to train the preference model and to build it based on features that are believed to underlie the human preference judgment. Our experiments show that CPMs not only improve generalization and are more robust to overoptimization than standard PMs, but also that best-of-n samples obtained using CPMs tend to be preferred over samples obtained using conventional PMs. Overall, our approach demonstrates the benefits of endowing PMs with priors about which features determine human preferences while relying on LM capabilities to extract those features in a scalable and robust way.