Asymptotic Convergence Rate Research Articles

Asymptotic theory of dependent Bayesian multiple testing procedures under possible model misspecification

We study asymptotic properties of Bayesian multiple testing procedures and provide sufficient conditions for strong consistency under general dependence structure. We also consider a novel Bayesian multiple testing procedure and associated error measures that coherently accounts for the dependence structure present in the model. We advocate posterior versions of FDR and FNR as appropriate error rates and show that their asymptotic convergence rates are directly associated with the Kullback–Leibler divergence from the true model. The theories hold regardless of the class of postulated models being misspecified. We illustrate our results in a variable selection problem with autoregressive response variables and compare our procedure with some existing methods through simulation studies. Superior performance of the new procedure compared to the others indicates that proper exploitation of the dependence structure by multiple testing methods is indeed important. Moreover, we obtain encouraging results in a maize dataset, where we select influential marker variables.

Local Linear Forests

Random forests are a powerful method for nonparametric regression, but are limited in their ability to fit smooth signals. Taking the perspective of random forests as an adaptive kernel method, we pair the forest kernel with a local linear regression adjustment to better capture smoothness. The resulting procedure, local linear forests, enables us to improve on asymptotic rates of convergence for random forests with smooth signals, and provides substantial gains in accuracy on both real and simulated data. We prove a central limit theorem valid under regularity conditions on the forest and smoothness constraints, and propose a computationally efficient construction for confidence intervals. Moving to a causal inference application, we discuss the merits of local regression adjustments for heterogeneous treatment effect estimation, and give an example on a dataset exploring the effect word choice has on attitudes to the social safety net. Last, we include simulation results on real and generated data. A software implementation is available in the R package grf. Supplementary materials for this article are available online.

Zames–Falb multipliers for convergence rate: motivating example and convex searches

The bound of convergence rates for discrete-time Lurye (sometimes written Lur'e) systems has recently attracted much attention. The contributions of this technical note are twofold. Firstly, we show an example where the asymptotic convergence rate is slower than the linear case. This example can be seen as a counterexample of the Kalman conjecture extended to the analysis of convergence rates and motivates a rigorous analysis. Secondly, we develop a result to bound the convergence rate as a convex search over the suitable subclass of noncausal FIR Zames–Falb multipliers. The convex search is demonstrated with several numerical examples.

A dynamical mean-field theory for learning in restricted Boltzmann machines

We define a message-passing algorithm for computing magnetizations in restricted Boltzmann machines, which are Ising models on bipartite graphs introduced as neural network models for probability distributions over spin configurations. To model nontrivial statistical dependencies between the spins’ couplings, we assume that the rectangular coupling matrix is drawn from an arbitrary bi-rotation invariant random matrix ensemble. Using the dynamical functional method of statistical mechanics we exactly analyze the dynamics of the algorithm in the large system limit. We prove the global convergence of the algorithm under a stability criterion and compute asymptotic convergence rates showing excellent agreement with numerical simulations.

Covariance Matrix Estimation With Non Uniform and Data Dependent Missing Observations

In this paper we study covariance estimation with missing data. We consider missing data mechanisms that can be independent of the data, or have a time varying dependency. Additionally, observed variables may have arbitrary (non uniform) and dependent observation probabilities. For each mechanism, we construct an unbiased estimator and obtain bounds for the expected value of their estimation error in operator norm. Our bounds are equivalent, up to constant and logarithmic factors, to state of the art bounds for complete and uniform missing observations. Furthermore, for the more general non uniform and dependent cases, the proposed bounds are new or improve upon previous results. Our error estimates depend on quantities we call scaled effective rank, which generalize the effective rank to account for missing observations. All the estimators studied in this work have the same asymptotic convergence rate (up to logarithmic factors).

Asymptotic Convergence Rate of Alternating Minimization for Rank One Matrix Completion

We study alternating minimization for matrix completion in the simplest possible setting: completing a rank-one matrix from a revealed subset of the entries. We bound the asymptotic convergence rate by the variational characterization of eigenvalues of a reversible consensus problem. This leads to a polynomial upper bound on the asymptotic rate in terms of number of nodes as well as the largest degree of the graph of revealed entries.

The Convergence of the Legendre–Galerkin Spectral Method for Constructing Atmospheric Acoustic Normal Modes

The asymptotic rate of convergence of the Legendre–Galerkin spectral approximation to an atmospheric acoustic eigenvalue problem is established, as the dimension of the approximating subspace approaches infinity. Convergence is in the [Formula: see text] Sobolev norm and is based on the existing theory [F. Chatelin, Spectral Approximations of Linear Operators (SIAM, 2011)]. The assumption is made that the eigenvalues are simple. Numerical results that help interpret the theory are presented. Eigenvalues corresponding to acoustic modes with smaller [Formula: see text] norms are especially accurately approximated, even with lower dimensioned basis sets of Legendre polynomials. The deficiencies in the potential applications of the theoretical results are noted in connection with the numerical examples.

On New Modification of Bernstein Operators: Theory and Applications

Approximation theory has a significant place in the studies in mathematics, and the researches on it are increasing with each passing day. Accordingly, a number of studies have been presented about Bernstein approximation which is one of the most known linear positive operators. In this study, a new modification of Bernstein operators which fix constant and preserve Korovkin’s other test functions in limit case has been introduced. Then, the approximation properties of the newly defined operators such as asymptotic formulas, weighted approximation and rate of convergence have been presented. Moreover, numerical simulations are included. Finally, discussion and conclusions are presented.

Numerical Performance of Optimized Frolov Lattices in Tensor Product Reproducing Kernel Sobolev Spaces

In this paper, we deal with several aspects of the universal Frolov cubature method, which is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable asymptotic behavior, there are significant differences concerning the precise numerical behavior of the worst-case error. To this end, we propose new generating polynomials that promise a significant reduction in the integration error compared to the classical polynomials. Moreover, we develop a new algorithm to enumerate the Frolov points from non-orthogonal lattices for numerical cubature in the d-dimensional unit cube [0,1]^d. Finally, we study Sobolev spaces with anisotropic mixed smoothness and compact support in [0,1]^d and derive explicit formulas for their reproducing kernels. This allows for the simulation of exact worst-case errors which numerically validate our theoretical results.

Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line

<p style='text-indent:20px;'>In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the one-dimensional compressible isentropic micropolar fluid model in a half line <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}_{+}: = (0, \infty). $\end{document}</tex-math></inline-formula> We mainly investigate the unique existence, the asymptotic stability and convergence rates of stationary solutions to the outflow problem for this model. We obtain the convergence rates of global solutions towards corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proof is based on the weighted energy method by taking into account the effect of the microrotational velocity on the viscous compressible fluid.

Risk concentration under second order regular variation

Measures of risk concentration and their asymptotic behavior for portfolios with heavy-tailed risk factors is of interest in risk management. Second order regular variation is a structural assumption often imposed on such risk factors to study their convergence rates. In this paper, we provide the asymptotic rate of convergence of the measure of risk concentration for a portfolio of heavy-tailed risk factors, when the portfolio admits the so-called second order regular variation property. Moreover, we explore the relationship between multivariate second order regular variation for a vector (e.g., risk factors) and the second order regular variation property for the sum of its components (e.g., the portfolio of risk factors). Results are illustrated with a variety of examples.

Verification of Unstructured Grid Adaptation Components

Unstructured grid techniques have the potential of minimizing discretization errors for production analysis workflows where the control of errors is critical to obtaining reliable simulation results. Recent progress has matured a number of independent implementations of flow solvers, error estimation methods, and anisotropic grid adaptation mechanics. Here, the interoperability of several separately developed unstructured grid adaptation tools is verified using analytic functions and the Code Comparison Principle. Three analytic functions with different smoothness properties are adapted to show the impact of smoothness on implementation differences. A scalar advection-diffusion problem with an analytic solution that models a boundary layer is adapted to test individual grid adaptation components. While optimal asymptotic error convergence rates are achieved with many grid adaptation tool combinations for the scalar problems, the scalar problems also illustrate known differences in grid adaptation component implementations and a previously unknown interaction between components. Laminar flow over a delta wing is verified with multiple, independent grid adaptation procedures to show consistent convergence to fine-grid forces and pitching moment. These verification efforts form the nucleus of a benchmark to verify the integration of unstructured grad adaptation components and support production analysis workflows.

Asymptotic distribution and convergence rates of stochastic algorithms for entropic optimal transportation between probability measures

This paper is devoted to the stochastic approximation of entropically regularized Wasserstein distances between two probability measures, also known as Sinkhorn divergences. The semi-dual formulation of such regularized optimal transportation problems can be rewritten as a nonstrongly concave optimisation problem. It allows to implement a Robbins–Monro stochastic algorithm to estimate the Sinkhorn divergence using a sequence of data sampled from one of the two distributions. Our main contribution is to establish the almost sure convergence and the asymptotic normality of a new recursive estimator of the Sinkhorn divergence between two probability measures in the discrete and semi-discrete settings. We also study the rate of convergence of the expected excess risk of this estimator in the absence of strong concavity of the objective function. Numerical experiments on synthetic and real datasets are also provided to illustrate the usefulness of our approach for data analysis.

A partially inexact ADMM with o(1/n) asymptotic convergence rate, 𝒪(1/n) complexity, and immediate relative error tolerance

In this work, we propose a new partially inexact Alternating Direction Method of Multipliers (ADMM) with relative error tolerance. This method departs from previous semi-inexact variants of the ADMM by allowing the second subproblem to be solved inexactly, instead of the first one. Relative error tolerance criteria for inexact ADMM methods require, to be verified, the solution of the two proximal subproblems in each iteration; therefore, by allowing for inexactness in the approximate solutions of the second subproblem, the error criterion is immediately testable during the computation of that approximate solution and no backtracking is required. In this sense, the proposed method uses an immediate error criterion. We also establish iteration complexity and asymptotic convergence rate for a modified ergodic mean, with respect to the residuals in Karush–Kuhn–Tucker conditions. As far as we know, this is the first asymptotic convergence rate, with respect to this error measure, for ADMM applied to a general problem.

The steepest descent of gradient-based iterative method for solving rectangular linear systems with an application to Poisson\u2019s equation

We introduce an effective iterative method for solving rectangular linear systems, based on gradients along with the steepest descent optimization. We show that the proposed method is applicable with any initial vectors as long as the coefficient matrix is of full column rank. Convergence analysis produces error estimates and the asymptotic convergence rate of the algorithm, which is governed by the term sqrt {1-kappa^{-2}}, where κ is the condition number of the coefficient matrix. Moreover, we apply the proposed method to a sparse linear system arising from a discretization of the one-dimensional Poisson equation. Numerical simulations illustrate the capability and effectiveness of the proposed method in comparison to the well-known and recent methods.

Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem

Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm follows a (noisy) descent direction along a continuous stream of data. The parameter updates occur in continuous time and satisfy a stochastic differential equation. This paper analyzes the asymptotic convergence rate of the SGDCT algorithm by proving a central limit theorem for strongly convex objective functions and, under slightly stronger conditions, for nonconvex objective functions as well. An [Formula: see text] convergence rate is also proven for the algorithm in the strongly convex case. The mathematical analysis lies at the intersection of stochastic analysis and statistical learning.

Synchronized Dynamics of Hyperchaotic Flows via an Improved Finite-Time Adaptive Controller Design

We proposed a performance-improved finite-time adaptive synchronizing controllers and parameter update laws for coupling the dynamics of identical 4D hyperchaotic flows. The four-dimensional hyperchaotic flows consists of 12 terms and 11 system parameters and possessed very rich dynamics and larger parameter space. The performance of the proposed finite-time adaptive synchronizing controller was enhanced by the introduction of scalar quantities known as global controller strength coefficients and parameter update strength coefficients respectively, into the algebraically-derived control and parameter update structures, in order to constrained overshoots of the trajectories of the coupled systems and accelerate their rate of uniform convergence in finite time. Numerical simulation results obtained confirmed that the uniform asymptotic convergence rate of the coupling trajectories was faster, while the parameter update laws give a stable identification of the unknown system parameters in a global synchronizing time. A comparative analysis of the convergence time of the proposed adaptive controllers with recently published works indicated that the proposed controller has faster rates of uniform convergence of system trajectories.

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

SummaryWe describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.

Wavelet pointwise estimations under multiplicative censoring

In this paper, we investigate the wavelet-based estimators of a kind of censored mixture density and discuss their pointwise asymptotic convergence rates over Hölder spaces. We first consider the linear wavelet estimator and give its upper bound. However, the linear one is nonadaptive and not applicable since it is related to the unknown space parameter. Finally, we use the hard threshold method to explore adaptive nonlinear wavelet estimator and obtain the same convergence order as the linear one up to a logarithmic penalty.

On asymptotic behaviors and convergence rates related to weak limiting distributions of geometric random sums

On asymptotic behaviors and convergence rates related to weak limiting distributions of geometric random sums