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The monotone variational inequality is a central problem in mathematical programming that unifies and generalizes many important settings such as smooth convex optimization, two-player zero-sum games, convex-concave saddle point problems, etc. The extragradient method by Korpelevich [1976] is one of the most popular methods for solving monotone variational inequalities. Despite its long history and intensive attention from the optimization and machine learning community, the following major problem remains open. What is the last-iterate convergence rate of the extragradient method for monotone and Lipschitz variational inequalities with constraints? We resolve this open problem by showing a tight $O\left(\frac{1}{\sqrt{T}}\right)$ last-iterate convergence rate for arbitrary convex feasible sets, which matches the lower bound by Golowich et al. [2020]. Our rate is measured in terms of the standard gap function. The technical core of our result is the monotonicity of a new performance measure -- the tangent residual, which can be viewed as an adaptation of the norm of the operator that takes the local constraints into account. To establish the monotonicity, we develop a new approach that combines the power of the sum-of-squares programming with the low dimensionality of the update rule of the extragradient method. We believe our approach has many additional applications in the analysis of iterative methods.

Linear mixed models (LMMs) are instrumental for regression analysis with structured dependence, such as grouped, clustered, or multilevel data. However, selection among the covariates--while accounting for this structured dependence--remains a challenge. We introduce a Bayesian decision analysis for subset selection with LMMs. Using a Mahalanobis loss function that incorporates the structured dependence, we derive optimal linear coefficients for (i) any given subset of variables and (ii) all subsets of variables that satisfy a cardinality constraint. Crucially, these estimates inherit shrinkage or regularization and uncertainty quantification from the underlying Bayesian model, and apply for any well-specified Bayesian LMM. More broadly, our decision analysis strategy deemphasizes the role of a single "best" subset, which is often unstable and limited in its information content, and instead favors a collection of near-optimal subsets. This collection is summarized by key member subsets and variable-specific importance metrics. Customized subset search and out-of-sample approximation algorithms are provided for more scalable computing. These tools are applied to simulated data and a longitudinal physical activity dataset, and demonstrate excellent prediction, estimation, and selection ability.

SVD (singular value decomposition) is one of the basic tools of machine learning, allowing to optimize basis for a given matrix. However, sometimes we have a set of matrices $\{A_k\}_k$ instead, and would like to optimize a single common basis for them: find orthogonal matrices $U$, $V$, such that $\{U^T A_k V\}$ set of matrices is somehow simpler. For example DCT-II is orthonormal basis of functions commonly used in image/video compression - as discussed here, this kind of basis can be quickly automatically optimized for a given dataset. While also discussed gradient descent optimization might be computationally costly, there is proposed CSVD (common SVD): fast general approach based on SVD. Specifically, we choose $U$ as built of eigenvectors of $\sum_i (w_k)^q (A_k A_k^T)^p$ and $V$ of $\sum_k (w_k)^q (A_k^T A_k)^p$, where $w_k$ are their weights, $p,q>0$ are some chosen powers e.g. 1/2, optionally with normalization e.g. $A \to A - rc^T$ where $r_i=\sum_j A_{ij}, c_j =\sum_i A_{ij}$.

Let $X^{(n)}$ be an observation sampled from a distribution $P_{\theta}^{(n)}$ with an unknown parameter $\theta,$ $\theta$ being a vector in a Banach space $E$ (most often, a high-dimensional space of dimension $d$). We study the problem of estimation of $f(\theta)$ for a functional $f:E\mapsto {\mathbb R}$ of some smoothness $s>0$ based on an observation $X^{(n)}\sim P_{\theta}^{(n)}.$ Assuming that there exists an estimator $\hat \theta_n=\hat \theta_n(X^{(n)})$ of parameter $\theta$ such that $\sqrt{n}(\hat \theta_n-\theta)$ is sufficiently close in distribution to a mean zero Gaussian random vector in $E,$ we construct a functional $g:E\mapsto {\mathbb R}$ such that $g(\hat \theta_n)$ is an asymptotically normal estimator of $f(\theta)$ with $\sqrt{n}$ rate provided that $s>\frac{1}{1-\alpha}$ and $d\leq n^{\alpha}$ for some $\alpha\in (0,1).$ We also derive general upper bounds on Orlicz norm error rates for estimator $g(\hat \theta)$ depending on smoothness $s,$ dimension $d,$ sample size $n$ and the accuracy of normal approximation of $\sqrt{n}(\hat \theta_n-\theta).$ In particular, this approach yields asymptotically efficient estimators in some high-dimensional exponential models.

We introduce Universal Solution Manifold Network (USM-Net), a novel surrogate model, based on Artificial Neural Networks (ANNs), which applies to differential problems whose solution depends on physical and geometrical parameters. Our method employs a mesh-less architecture, thus overcoming the limitations associated with image segmentation and mesh generation required by traditional discretization methods. Indeed, we encode geometrical variability through scalar landmarks, such as coordinates of points of interest. In biomedical applications, these landmarks can be inexpensively processed from clinical images. Our approach is non-intrusive and modular, as we select a data-driven loss function. The latter can also be modified by considering additional constraints, thus leveraging available physical knowledge. Our approach can also accommodate a universal coordinate system, which supports the USM-Net in learning the correspondence between points belonging to different geometries, boosting prediction accuracy on unobserved geometries. Finally, we present two numerical test cases in computational fluid dynamics involving variable Reynolds numbers as well as computational domains of variable shape. The results show that our method allows for inexpensive but accurate approximations of velocity and pressure, avoiding computationally expensive image segmentation, mesh generation, or re-training for every new instance of physical parameters and shape of the domain.

The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $\delta$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $\delta$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

We present a novel static analysis technique to derive higher moments for program variables for a large class of probabilistic loops with potentially uncountable state spaces. Our approach is fully automatic, meaning it does not rely on externally provided invariants or templates. We employ algebraic techniques based on linear recurrences and introduce program transformations to simplify probabilistic programs while preserving their statistical properties. We develop power reduction techniques to further simplify the polynomial arithmetic of probabilistic programs and define the theory of moment-computable probabilistic loops for which higher moments can precisely be computed. Our work has applications towards recovering probability distributions of random variables and computing tail probabilities. The empirical evaluation of our results demonstrates the applicability of our work on many challenging examples.

In the pooled data problem we are given a set of $n$ agents, each of which holds a hidden state bit, either $0$ or $1$. A querying procedure returns for a query set the sum of the states of the queried agents. The goal is to reconstruct the states using as few queries as possible. In this paper we consider two noise models for the pooled data problem. In the noisy channel model, the result for each agent flips with a certain probability. In the noisy query model, each query result is subject to random Gaussian noise. Our results are twofold. First, we present and analyze for both error models a simple and efficient distributed algorithm that reconstructs the initial states in a greedy fashion. Our novel analysis pins down the range of error probabilities and distributions for which our algorithm reconstructs the exact initial states with high probability. Secondly, we present simulation results of our algorithm and compare its performance with approximate message passing (AMP) algorithms that are conjectured to be optimal in a number of related problems.