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LU and Cholesky matrix factorization algorithms are core subroutines used to solve systems of linear equations (SLEs) encountered while solving an optimization problem. Standard factorization algorithms are highly efficient but remain susceptible to the accumulation roundoff errors, which can lead solvers to return feasibility and optimality certificates that are actually invalid. This paper introduces a novel approach for solving sequences of closely related SLEs encountered in nonlinear programming efficiently and without roundoff errors. Specifically, it introduces rank-one update algorithms for the roundoff-error-free (REF) factorization framework, a toolset built on integer-preserving arithmetic that has led to the development and implementation of fail-proof SLE solution subroutines for linear programming. The formal guarantees of the proposed algorithms are formally established through the derivation of theoretical insights. Their computational advantages are supported with computational experiments, which demonstrate upwards of 75x-improvements over exact factorization run-times on fully dense matrices with over one million entries. A significant advantage of the proposed methodology is that the length of any coefficient calculated via the associated algorithms is bounded polynomially in the size of the inputs without having to resort to greatest common divisor operations, which are required by and thereby hinder an efficient implementation of exact rational arithmetic approaches.

Much recent interest has focused on the design of optimization algorithms from the discretization of an associated optimization flow, i.e., a system of differential equations (ODEs) whose trajectories solve an associated optimization problem. Such a design approach poses an important problem: how to find a principled methodology to design and discretize appropriate ODEs. This paper aims to provide a solution to this problem through the use of contraction theory. We first introduce general mathematical results that explain how contraction theory guarantees the stability of the implicit and explicit Euler integration methods. Then, we propose a novel system of ODEs, namely the Accelerated-Contracting-Nesterov flow, and use contraction theory to establish it is an optimization flow with exponential convergence rate, from which the linear convergence rate of its associated optimization algorithm is immediately established. Remarkably, a simple explicit Euler discretization of this flow corresponds to the Nesterov acceleration method. Finally, we present how our approach leads to performance guarantees in the design of optimization algorithms for time-varying optimization problems.

Properness for supervised losses stipulates that the loss function shapes the learning algorithm towards the true posterior of the data generating distribution. Unfortunately, data in modern machine learning can be corrupted or twisted in many ways. Hence, optimizing a proper loss function on twisted data could perilously lead the learning algorithm towards the twisted posterior, rather than to the desired clean posterior. Many papers cope with specific twists (e.g., label/feature/adversarial noise), but there is a growing need for a unified and actionable understanding atop properness. Our chief theoretical contribution is a generalization of the properness framework with a notion called twist-properness, which delineates loss functions with the ability to "untwist" the twisted posterior into the clean posterior. Notably, we show that a nontrivial extension of a loss function called $\alpha$-loss, which was first introduced in information theory, is twist-proper. We study the twist-proper $\alpha$-loss under a novel boosting algorithm, called PILBoost, and provide formal and experimental results for this algorithm. Our overarching practical conclusion is that the twist-proper $\alpha$-loss outperforms the proper $\log$-loss on several variants of twisted data.

It is common practice to use Laplace approximations to compute marginal likelihoods in Bayesian versions of generalised linear models (GLM). Marginal likelihoods combined with model priors are then used in different search algorithms to compute the posterior marginal probabilities of models and individual covariates. This allows performing Bayesian model selection and model averaging. For large sample sizes, even the Laplace approximation becomes computationally challenging because the optimisation routine involved needs to evaluate the likelihood on the full set of data in multiple iterations. As a consequence, the algorithm is not scalable for large datasets. To address this problem, we suggest using a version of a popular batch stochastic gradient descent (BSGD) algorithm for estimating the marginal likelihood of a GLM by subsampling from the data. We further combine the algorithm with Markov chain Monte Carlo (MCMC) based methods for Bayesian model selection and provide some theoretical results on the convergence of the estimates. Finally, we report results from experiments illustrating the performance of the proposed algorithm.

Bayesian bandit algorithms with approximate inference have been widely used in practice with superior performance. Yet, few studies regarding the fundamental understanding of their performances are available. In this paper, we propose a Bayesian bandit algorithm, which we call Generalized Bayesian Upper Confidence Bound (GBUCB), for bandit problems in the presence of approximate inference. Our theoretical analysis demonstrates that in Bernoulli multi-armed bandit, GBUCB can achieve $O(\sqrt{T}(\log T)^c)$ frequentist regret if the inference error measured by symmetrized Kullback-Leibler divergence is controllable. This analysis relies on a novel sensitivity analysis for quantile shifts with respect to inference errors. To our best knowledge, our work provides the first theoretical regret bound that is better than $o(T)$ in the setting of approximate inference. Our experimental evaluations on multiple approximate inference settings corroborate our theory, showing that our GBUCB is consistently superior to BUCB and Thompson sampling.

The scalar auxiliary variable (SAV) approach \cite{shen2018scalar} and its generalized version GSAV proposed in \cite{huang2020highly} are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems. However, the discrete value of the SAV is not directly linked to the free energy of the dissipative system, and may lead to inaccurate solutions if the time step is not sufficiently small. Inspired by the relaxed SAV method proposed in \cite{jiang2022improving} for gradient flows, we propose in this paper a generalized SAV approach with relaxation (R-GSAV) for general dissipative systems. The R-GSAV approach preserves all the advantages of the GSAV appraoch, in addition, it dissipates a modified energy that is directly linked to the original free energy. We prove that the $k$-th order implicit-explicit (IMEX) schemes based on R-GSAV are unconditionally energy stable, and we carry out a rigorous error analysis for $k=1,2,3,4,5$. We present ample numerical results to demonstrate the improved accuracy and effectiveness of the R-GSAV approach.

The $p$-center problem (pCP) is a fundamental problem in location science, where we are given customer demand points and possible facility locations, and we want to choose $p$ of these locations to open a facility such that the maximum distance of any customer demand point to its closest open facility is minimized. State-of-the-art solution approaches of pCP use its connection to the set cover problem to solve pCP in an iterative fashion by repeatedly solving set cover problems. The classical textbook integer programming (IP) formulation of pCP is usually dismissed due to its size and bad linear programming (LP)-relaxation bounds. We present a novel solution approach that works on a new IP formulation that can be obtained by a projection from the classical formulation. The formulation is solved by means of branch-and-cut, where cuts for demand points are iteratively generated. Moreover, the formulation can be strengthened with combinatorial information to obtain a much tighter LP-relaxation. In particular, we present a novel way to use lower bound information to obtain stronger cuts. We show that the LP-relaxation bound of our strengthened formulation has the same strength as the best known bound in literature, which is based on a semi-relaxation. Finally, we also present a computational study on instances from the literature with up to more than 700000 customers and locations. Our solution algorithm is competitive with highly sophisticated set-cover-based solution algorithms, which depend on various components and parameters.

Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedeness, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with FE methods for local problems. However, nonlocal weak problems involve the computation of a double-integral, which is computationally expensive and presents several challenges. In particular, the inner integral of the variational form associated with the stiffness matrix is defined over the intersections of FE mesh elements with a ball of radius $\delta$, where $\delta$ is the range of nonlocal interaction. Identifying and parameterizing these intersections is a nontrivial computational geometry problem. In this work, we propose a quadrature technique where the inner integration is performed using quadrature points distributed over the full ball, without regard for how it intersects elements, and weights are computed based on the generalized moving least squares method. Thus, as opposed to all previously employed methods, our technique does not require element-by-element integration and fully circumvents the computation of element-ball intersections. This paper considers one- and two-dimensional implementations of piecewise linear continuous FE approximations, focusing on the case where the element size h and the nonlocal radius $\delta$ are proportional, as is typical of practical computations. When boundary conditions are treated carefully and the outer integral of the variational form is computed accurately, the proposed method is asymptotically compatible in the limit of $h \sim \delta \to 0$, featuring at least first-order convergence in L^2 for all dimensions, using both uniform and nonuniform grids.

The aim of this article is further development of the theory of linear difference equations with constant coefficients. We present a new algorithm for calculating the solution to the Cauchy problem for a three-dimensional difference equation with constant coefficients in a parallelepiped at the point using the coefficients of the difference equation and Cauchy data. The implemented algorithm is the next significant achievement in a series of articles justifying the Apanovich and Leinartas' theorems about the solvability and well-posedness of the Cauchy problem. We also use methods of computer algebra since the three-dimensional case usually demands extended calculations.

Lexical simplification (LS) aims to replace complex words in a given sentence with their simpler alternatives of equivalent meaning. Recently unsupervised lexical simplification approaches only rely on the complex word itself regardless of the given sentence to generate candidate substitutions, which will inevitably produce a large number of spurious candidates. We present a simple BERT-based LS approach that makes use of the pre-trained unsupervised deep bidirectional representations BERT. We feed the given sentence masked the complex word into the masking language model of BERT to generate candidate substitutions. By considering the whole sentence, the generated simpler alternatives are easier to hold cohesion and coherence of a sentence. Experimental results show that our approach obtains obvious improvement on standard LS benchmark.