Exact Rate Of Convergence Research Articles

Exact convergence rate of the central limit theorem and polynomial convergence rate for branching processes in a random environment

Let (Zn) be a supercritical branching process in an independent and identically distributed (i.i.d.) random environment. The paper studies the properties of the estimator Mn=n−1∑k=0n−1(Zk+1/Zk) introduced by Dion and Esty in 1979. We introduce a related martingale and discuss its convergence and exponential convergence rate. On this basis the exact convergence rate of the central limit theorem for normalized Mn is given.

Exact Convergence Rate of the Local Limit Theorem for a Branching Random Walk in ℤd with a Random Environment in Time

Exact Convergence Rate of the Local Limit Theorem for a Branching Random Walk in ℤd with a Random Environment in Time

A graph decomposition-based approach for the graph-fused lasso

We propose a new algorithm for solving the graph-fused lasso (GFL), a regularized model that operates under the assumption that the signal tends to be locally constant over a predefined graph structure. The proposed method applies a novel decomposition of the objective function for the alternating direction method of multipliers (ADMM) algorithm. While ADMM has been widely used in fused lasso problems, existing works such as the network lasso decompose the objective function into the loss function component and the total variation penalty component. In contrast, based on the graph matching technique in graph theory, we propose a new method of decomposition that separates the objective function into two components, where one component is the loss function plus part of the total variation penalty, and the other component is the remaining total variation penalty. We develop an exact convergence rate of the proposed algorithm by developing a general theory on the local convergence of ADMM. Compared with the network lasso algorithm, our algorithm has a faster exact linear convergence rate (although in the same order as for the network lasso). It also enjoys a smaller computational cost per iteration, thus converges overall faster in most numerical examples.

Linear convergence of event‐triggered distributed optimization with metric subregularity condition

AbstractThis paper designs a continuous‐time algorithm with event‐triggered communication (ETC) for solving a class of distributed convex optimization problems with a metric subregularity condition. First, we develop an event‐triggered continuous‐time optimization algorithm to overcome the bandwidth limitation of multi‐agent systems. Besides, with the aid of Lyapunov theory, we prove that the distributed event‐triggered algorithm converges to the optimum set with an exact linear convergence rate, without the strongly convex condition. Moreover, we provide the discrete version of the continuous‐time algorithm and show its exact linear convergence rate. Finally, we give a comparison example to validate the effectiveness of the designed algorithm in communication resource saving.

Nonequilibrium–diffusion limit of the compressible Euler radiation model in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$

This article studies the nonequilibrium–diffusion limit of the compressible Euler model arising in radiation hydrodynamics in with the general initial data. Combining the moment method with Hilbert expansion, we show that the radiative intensity can be approximated by the sum of interior solution and initial layer. We also show that the solution satisfied by the density, temperature, and velocity can be approximated by the interior solutions. Our results can be considered as an extension from in arXiv.2312.15208 by Ju, Li, and Zhang to . In contrast to arXiv.2312.15208, we get the exact convergence rates by studying the error system derived from the primitive system, the zeroth‐order to the second‐order about the radiative intensity, and the zeroth‐order about the hydrodynamics.

Exact convergence rate of solutions for a semilinear heat equation with a critical and a supercritical exponent revisited

Exact convergence rate of solutions for a semilinear heat equation with a critical and a supercritical exponent revisited

On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants

Abstract In this article, we analyze a rate of attraction of poles of an approximated function to poles of incomplete multipoint Padé approximants and use it to derive a sharp bound on the geometric rate of convergence of multipoint Hermite-Padé approximants to a vector of approximated functions in the Montessus de Ballore theorem when a table of interpolation points is Newtonian.

The exact worst-case convergence rate of the alternating direction method of multipliers

AbstractRecently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We also study the linear and R-linear convergence of ADMM in terms of dual objective. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak–Łojasiewicz (PŁ) inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor. Moreover, we study the R-linear convergence of ADMM under two scenarios.

Exact convergence rate in central limit theorem for a supercritical branching process with immigration in a random environment

Let {Z n } be a supercritical branching process with immigration in an independent, identically distributed(i.i.d.) environment ξ. The Berry-Esseen bound for log⁡Z n has been established by Wang et al. (2021). To refine that, under the less restrictive moment conditions, we calculate the exact convergence rate in the central limit theorem for log⁡Z n under the annealed law.

Event-triggered primal–dual design with linear convergence for distributed nonstrongly convex optimization

This paper designs continuous-time algorithms with linear convergence for solving distributed convex optimization problems without a strongly convex condition. The proposed primal–dual algorithms operate under weight-balanced digraphs and the dual variables are not exchanged with neighbors, which makes the algorithm more efficient. To save communication resources, we propose a class of event-triggered communication (ETC) schemes, which includes static and dynamic counterparts with the latter being more effective in communication resource saving. Furthermore, using Lyapunov theory, we prove that the distributed event-triggered algorithms converge to the optimum set with exact linear convergence rates. Finally, we present a comparison example that validates the effectiveness of the proposed algorithms in reducing communication burdens.

On the exact rate of convergence of digits in Engel expansions

Let ϕ:N→R+ be a function satisfying ϕ(n)→∞ as n→∞. Denote by 〈d1(x),d2(x),⋯,dn(x),⋯〉 the Engel expansion of an irrational number x∈(0,1). We study the Baire classification and Hausdorff dimension ofDϕ(α,β):={x∈(0,1)﹨Q:liminfn→∞log⁡dn(x)−nϕ(n)=α,limsupn→∞log⁡dn(x)−nϕ(n)=β} for α,β∈[−∞,∞] with α≤β. Under the condition that ϕ(n+1)−ϕ(n)→0 as n→∞, we show that the set Dϕ(α,β) is residual if and only if α=−∞ and β=∞. Under the conditions that ϕ is increasing and ϕ(n+1)−ϕ(n)→0 as n→∞, we prove that Dϕ(α,β) has full Hausdorff dimension for any −∞≤α≤β≤∞, which improves the result of Liu and Wu (2003). We also do some similar analyses for the gap of consecutive digits.

Extremal polynomials on the unit circle with preassigned zeros and Two-point Partial Padé approximation

It is well known that for a general class of measures the sequence of paraorthogonal polynomials {Bn}n∈N satisfies limn→∞|Bn(z)|1n=max(|z|,1),uniformly on compact subsets of ℂˆ∖T. This is an essential property to obtain the exact rate of convergence for two-point Padé approximants to the Herglotz–Riesz transform of the measure when these paraorthogonal polynomials are used as denominators. In this paper we obtain a similar result when we appropriately preselect some of the zeros of these paraorthogonal polynomials. As an application, we obtain the corresponding exact rate of convergence of the approximants to the Herglotz–Riesz transform with rational perturbation when these polynomials are used as denominators.

On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H>12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H>\\frac{1}{2}$$\\end{document} with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n1-2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{1-2H}$$\\end{document}, which is twice as good as the best known results in the case of discontinuous integrands and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply a change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

Exact convergence analysis for metropolis–hastings independence samplers in Wasserstein distances

AbstractUnder mild assumptions, we show that the exact convergence rate in total variation is also exact in weaker Wasserstein distances for the Metropolis–Hastings independence sampler. We develop a new upper and lower bound on the worst-case Wasserstein distance when initialized from points. For an arbitrary point initialization, we show that the convergence rate is the same and matches the convergence rate in total variation. We derive exact convergence expressions for more general Wasserstein distances when initialization is at a specific point. Using optimization, we construct a novel centered independent proposal to develop exact convergence rates in Bayesian quantile regression and many generalized linear model settings. We show that the exact convergence rate can be upper bounded in Bayesian binary response regression (e.g. logistic and probit) when the sample size and dimension grow together.

Convergence rate of multiple-try Metropolis independent sampler

The multiple-try Metropolis method is an interesting extension of the classical Metropolis–Hastings algorithm. However, theoretical understanding about its usefulness and convergence behavior is still lacking. We here derive the exact convergence rate for the multiple-try Metropolis Independent sampler (MTM-IS) via an explicit eigen analysis. As a by-product, we prove that an naive application of the MTM-IS is less efficient than using the simpler approach of “thinned” independent Metropolis–Hastings method at the same computational cost. We further explore more variants and find it possible to design more efficient algorithms by applying MTM to part of the target distribution or creating correlated multiple trials.

Convergence Rate Analysis and Improved Iterations for Numerical Radius Computation

The current two main methods for computing the numerical are the level-set approach of Mengi and Overton and the cutting-plane method of Uhlig, but until now, no formal convergence rate results have been established for either. In practice, which is faster is also unclear, with Uhlig's method sometimes being much faster than Mengi and Overton's approach, and on other problems, much slower. In this paper, we clarify this issue and also propose three improved methods. We show that Mengi and Overton's method converges quadratically, as has been suspected, while we completely characterize the total cost of Uhlig's method for so-called disk matrices. Then, for arbitrary fields of values, we derive the exact Q-linear local convergence rate of Uhlig's cutting procedure. Together, this establishes that Uhlig's method is extremely expensive when the field of values is a disk centered at the origin, but that the local rate of convergence of his cutting procedure actually varies from linear to superlinear depending on the shape and location of the field of values, which we show can be encapsulated by a single parameter via introducing the notion of normalized curvature. These results fully explain why Uhlig's method can both be exceptionally fast and exceptionally slow. With this insight, we propose an improved level-set method and an improved cutting-plane method, both of which can be significantly faster than their earlier counterparts, while also establishing analogous convergence rate results for both. Moreover, in order to remain efficient for any field of values configuration, we introduce a third algorithm that leverages the concept of normalized curvature and combines both of our improved iterations.

Exact convergence rates of alternating projections for nontransversal intersections

Exact convergence rates of alternating projections for nontransversal intersections

Polynomial continued fractions for exp(π)

ABSTRACT We present two (inequivalent) polynomial continued fraction representations of the number with all their elements in ; no such representation was seemingly known before. More generally, a similar result for is obtained for every such that . The proof uses a classical polynomial continued fraction representation of , for and , of which we present a new proof that enables us to obtain the exact rate of convergence of the convergents of the continued fraction for . We also deduce some consequences of arithmetic interest concerning the elements of certain polynomial continued fraction representations of the (transcendental) Gel'fond-Schneider numbers , where and , where is the field of algebraic numbers, embedded into .

A central limit theorem for conservative fragmentation chains

AbstractWe are interested in a fragmentation process. We observe fragments frozen when their sizes are less than $\varepsilon$ ( $\varepsilon>0$ ). It is known (Bertoin and Martínez, 2005) that the empirical measure of these fragments converges in law, under some renormalization. Hoffmann and Krell (2011) showed a bound for the rate of convergence. Here, we show a central limit theorem, under some assumptions. This gives us an exact rate of convergence.

Inexact Primal-Dual Algorithm for DMPC With Coupled Constraints Using Contraction Theory.

This article studies a distributed model-predictive control (DMPC) strategy for a class of discrete-time linear systems subject to globally coupled constraints. To reduce the computational burden, the constraint tightening technique is adopted for enabling the early termination of the distributed optimization algorithm. Using the Lagrangian method, we convert the constrained optimization problem of the proposed DMPC to an unconstrained saddle-point seeking problem. Due to the presence of the global dual variable in the Lagrangian function, we propose a primal-dual algorithm based on the Laplacian consensus to solve such a problem in a distributed manner by introducing the local estimates of the dual variable. We theoretically show the geometric convergence of the primal-dual gradient optimization algorithm by the contraction theory in the context of discrete-time updating dynamics. The exact convergence rate is obtained, leading the stopping number of iterations to be bounded. The recursive feasibility of the proposed DMPC strategy and the stability of the closed-loop system can be established pursuant to the inexact solution. Numerical simulation demonstrates the performance of the proposed strategy.