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Distributed optimization with open collaboration is a popular field since it provides an opportunity for small groups/companies/universities, and individuals to jointly solve huge-scale problems. However, standard optimization algorithms are fragile in such settings due to the possible presence of so-called Byzantine workers -- participants that can send (intentionally or not) incorrect information instead of the one prescribed by the protocol (e.g., send anti-gradient instead of stochastic gradients). Thus, the problem of designing distributed methods with provable robustness to Byzantine workers has been receiving a lot of attention recently. In particular, several works consider a very promising way to achieve Byzantine tolerance via exploiting variance reduction and robust aggregation. The existing approaches use SAGA- and SARAH-type variance-reduced estimators, while another popular estimator -- SVRG -- is not studied in the context of Byzantine-robustness. In this work, we close this gap in the literature and propose a new method -- Byzantine-Robust Loopless Stochastic Variance Reduced Gradient (BR-LSVRG). We derive non-asymptotic convergence guarantees for the new method in the strongly convex case and compare its performance with existing approaches in numerical experiments.

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A stochastic-gradient-based interior-point algorithm for minimizing a continuously differentiable objective function (that may be nonconvex) subject to bound constraints is presented, analyzed, and demonstrated through experimental results. The algorithm is unique from other interior-point methods for solving smooth (nonconvex) optimization problems since the search directions are computed using stochastic gradient estimates. It is also unique in its use of inner neighborhoods of the feasible region -- defined by a positive and vanishing neighborhood-parameter sequence -- in which the iterates are forced to remain. It is shown that with a careful balance between the barrier, step-size, and neighborhood sequences, the proposed algorithm satisfies convergence guarantees in both deterministic and stochastic settings. The results of numerical experiments show that in both settings the algorithm can outperform a projected-(stochastic)-gradient method.

We present a registration method for model reduction of parametric partial differential equations with dominating advection effects and moving features. Registration refers to the use of a parameter-dependent mapping to make the set of solutions to these equations more amicable for approximation using classical reduced basis methods. The proposed approach utilizes concepts from optimal transport theory, as we utilize Monge embeddings to construct these mappings in a purely data-driven way. The method relies on one interpretable hyper-parameter. We discuss how our approach relates to existing works that combine model order reduction and optimal transport theory. Numerical results are provided to demonstrate the effect of the registration. This includes a model problem where the solution is itself a probability density and one where it is not.

As phasor measurement units (PMUs) become more widely used in transmission power systems, a fast state estimation (SE) algorithm that can take advantage of their high sample rates is needed. To accomplish this, we present a method that uses graph neural networks (GNNs) to learn complex bus voltage estimates from PMU voltage and current measurements. We propose an original implementation of GNNs over the power system's factor graph to simplify the integration of various types and quantities of measurements on power system buses and branches. Furthermore, we augment the factor graph to improve the robustness of GNN predictions. This model is highly efficient and scalable, as its computational complexity is linear with respect to the number of nodes in the power system. Training and test examples were generated by randomly sampling sets of power system measurements and annotated with the exact solutions of linear SE with PMUs. The numerical results demonstrate that the GNN model provides an accurate approximation of the SE solutions. Furthermore, errors caused by PMU malfunctions or communication failures that would normally make the SE problem unobservable have a local effect and do not deteriorate the results in the rest of the power system.

Sparse principal component analysis (SPCA) is widely used for dimensionality reduction and feature extraction in high-dimensional data analysis. Despite many methodological and theoretical developments in the past two decades, the theoretical guarantees of the popular SPCA algorithm proposed by Zou, Hastie & Tibshirani (2006) are still unknown. This paper aims to address this critical gap. We first revisit the SPCA algorithm of Zou et al. (2006) and present our implementation. We also study a computationally more efficient variant of the SPCA algorithm in Zou et al. (2006) that can be considered as the limiting case of SPCA. We provide the guarantees of convergence to a stationary point for both algorithms and prove that, under a sparse spiked covariance model, both algorithms can recover the principal subspace consistently under mild regularity conditions. We show that their estimation error bounds match the best available bounds of existing works or the minimax rates up to some logarithmic factors. Moreover, we demonstrate the competitive numerical performance of both algorithms in numerical studies.

We present a dimension-incremental algorithm for the nonlinear approximation of high-dimensional functions in an arbitrary bounded orthonormal product basis. Our goal is to detect a suitable truncation of the basis expansion of the function, where the corresponding basis support is assumed to be unknown. Our method is based on point evaluations of the considered function and adaptively builds an index set of a suitable basis support such that the approximately largest basis coefficients are still included. For this purpose, the algorithm only needs a suitable search space that contains the desired index set. Throughout the work, there are various minor modifications of the algorithm discussed as well, which may yield additional benefits in several situations. For the first time, we provide a proof of a detection guarantee for such an index set in the function approximation case under certain assumptions on the sub-methods used within our algorithm, which can be used as a foundation for similar statements in various other situations as well. Some numerical examples in different settings underline the effectiveness and accuracy of our method.

In this paper, we propose, analyze and implement efficient time parallel methods for the Cahn-Hilliard (CH) equation. It is of great importance to develop efficient numerical methods for the CH equation, given the range of applicability of the CH equation has. The CH equation generally needs to be simulated for a very long time to get the solution of phase coarsening stage. Therefore it is desirable to accelerate the computation using parallel method in time. We present linear and nonlinear Parareal methods for the CH equation depending on the choice of fine approximation. We illustrate our results by numerical experiments.

This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ($O(1 / k^2)$), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.

High stakes classification refers to classification problems where erroneously predicting the wrong class is very bad, but assigning "unknown" is acceptable. We make the argument that these problems require us to give multiple unknown classes, to get the most information out of our analysis. With imperfect data we refer to covariates with a large number of missing values, large noise variance, and some errors in the data. The combination of high stakes classification and imperfect data is very common in practice, but it is very difficult to work on using current methods. We present a one-class classifier (OCC) to solve this problem, and we call it NBP. The classifier is based on Naive Bayes, simple to implement, and interpretable. We show that NBP gives both good predictive performance, and works for high stakes classification based on imperfect data. The model we present is quite simple; it is just an OCC based on density estimation. However, we have always felt a big gap between the applied classification problems we have worked on and the theory and models we use for classification, and this model closes that gap. Our main contribution is the motivation for why this model is a good approach, and we hope that this paper will inspire further development down this path.

This paper considers the Westervelt equation, one of the most widely used models in nonlinear acoustics, and seeks to recover two spatially-dependent parameters of physical importance from time-trace boundary measurements. Specifically, these are the nonlinearity parameter $\kappa(x)$ often referred to as $B/A$ in the acoustics literature and the wave speed $c_0(x)$. The determination of the spatial change in these quantities can be used as a means of imaging. We consider identifiability from one or two boundary measurements as relevant in these applications. For a reformulation of the problem in terms of the squared slowness $\mathfrak{s}=1/c_0^2$ and the combined coefficient $\eta=\frac{B/A+2}{\varrho_0 c_0^4}$ we devise a frozen Newton method and prove its convergence. The effectiveness (and limitations) of this iterative scheme are demonstrated by numerical examples.

Consider the community detection problem in random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), where each hyperedge appears independently with some given probability depending only on the labels of its vertices. We establish, for the first time in the literature, a sharp threshold for exact recovery under this non-uniform case, subject to minor constraints; in particular, we consider the model with $K$ classes as well as the symmetric binary model ($K=2$). One crucial point here is that by aggregating information from all the uniform layers, we may obtain exact recovery even in cases when this may appear impossible if each layer were considered alone. Two efficient algorithms that successfully achieve exact recovery above the threshold are provided. The theoretical analysis of our algorithms relies on the concentration and regularization of the adjacency matrix for non-uniform random hypergraphs, which could be of independent interest. We also address some open problems regarding parameter knowledge and estimation.

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