Geometric Rate Of Convergence Research Articles

Augmentation Samplers for Multinomial Probit Bayesian Additive Regression Trees

The multinomial probit (MNP) (Imai and van Dyk, 2005) framework is based on a multivariate Gaussian latent structure, allowing for natural extensions to multilevel modeling. Unlike multinomial logistic models, MNP does not assume independent alternatives. Kindo et al. (2016) proposed multinomial probit BART (MPBART) to accommodate Bayesian additive regression trees (BART) formulation in MNP. The posterior sampling algorithms for MNP and MPBART are collapsed Gibbs samplers. Because the collapsing augmentation strategy yields a geometric rate of convergence no greater than that of a standard Gibbs sampling step, it is recommended whenever computationally feasible (Liu, 1994a; Imai and van Dyk, 2005). While this strategy necessitates simple sampling steps and a reasonably fast converging Markov chain, the complexity of the stochastic search for posterior trees may undermine its benefit. We address this problem by sampling posterior trees conditional on the constrained parameter space and compare our proposals to that of Kindo et al. (2016), who sample posterior trees based on an augmented parameter space. We also compare to the approach by Sparapani et al. (2021) that specified the multinomial model in terms of conditional probabilities. In terms of MCMC convergence and posterior predictive accuracy, our proposals are comparable to the conditional probability approach and outperform the augmented tree sampling approach. We also show that the theoretical mixing rates of our proposals are guaranteed to be no greater than the augmented tree sampling approach. Appendices and codes for simulations and demonstrations are available online.

Robust Online Tensor Completion for IoT Streaming Data Recovery.

Reliable data measurement is considered to be one of the critical ingredients for variant Internet of Things (IoT) applications. Gaining full knowledge of measurement data is becoming increasingly crucial to ensure a satisfactory user experience. However, data missing and corruption are inevitable in practical applications, which motivates us to study how to accurately recover the missing IoT measurement data in the presence of outliers. The data recovery problem can be formulated as a tensor completion (TC) problem. Existing TC methods are built on the assumption that the rank of the underlying tensor is fixed, which is not suitable for long data sequences in practice. Consequently, based on the characteristics of IoT streaming data, we assume that the data tensor lies in time-varying subspace, and an accurate estimate of the rank is a prerequisite for filling the missing entries and achieving robustness of the variations in both rank and noise. We built up an updatable framework based on dynamic CANDECOMP/PARAFAC (CP) decomposition. In addition, an efficient algorithm, called temporal multi-aspect streaming (T-MUST), is introduced to solve the optimization problem that originates in our developed model. It is worth noting that the proposed algorithm allows time-varying tensor rank and enables the rank changes could be detected and tracked automatically. Theoretical analysis indicates that T-MUST enjoys a geometric convergence rate. Numerical experiments conducted on various synthetic and real-world datasets empirically validate the superiority of the proposed T-MUST in both efficiency and effectiveness.

DC-DistADMM: ADMM Algorithm for Constrained Optimization Over Directed Graphs

This article reports an algorithm for multi-agent distributed optimization problems with a common decision variable, local linear equality and inequality constraints and set constraints with convergence rate guarantees. \textcolor{black}{The algorithm accrues all the benefits of the Alternating Direction Method of Multipliers (ADMM) approach}. It also overcomes the limitations of existing methods on convex optimization problems with linear inequality, equality and set constraints by allowing directed communication topologies. Moreover, the algorithm can be synthesized distributively. The developed algorithm has: (i) a $O(1/k)$ rate of convergence, where $k$ is the iteration counter, when individual functions are convex but not-necessarily differentiable, and (ii) a geometric rate of convergence to any arbitrary small neighborhood of the optimal solution, when the objective functions are smooth and restricted strongly convex at the optimal solution. The efficacy of the algorithm is evaluated by a comparison with state-of-the-art constrained optimization algorithms in solving a constrained distributed $\ell_1$-regularized logistic regression problem, and unconstrained optimization algorithms in solving a $\ell_1$-regularized Huber loss minimization problem. Additionally, a comparison of the algorithm's performance with other algorithms in the literature that utilize multiple communication steps is provided.

Explicit Convergence Rates for the M/G/1 Queue under Perturbation

This paper establishes convergence rates for discrete-time Markov chains on a countable state space that are stochastically ordered starting from a stationary distribution under perturbation. We investigate the explicit criteria to obtain the ordinary ergodicity, geometric ergodicity and polynomial ergodicity for the embedded M/G/1 queue under perturbation. The explicit geometric convergence rates for the original system and the system under perturbation are caculated. Our bounds in the geometric case and polynomial case are closely connected to the first hitting times. Two examples are provided to illustrate our result.

Fast Coordination of Distributed Energy Resources Over Time-Varying Communication Networks

In this article, we consider the problem of optimally coordinating the response of a group of distributed energy resources (DERs) controlled by distributed agents so they collectively meet the electric power demanded by a collection of loads while minimizing the total generation cost and respecting the DER capacity limits. This problem can be cast as a convex optimization problem, where the global objective is to minimize a sum of convex functions corresponding to individual DER generation cost while satisfying 1) linear inequality constraints corresponding to the DER capacity limits and 2) a linear equality constraint corresponding to the total power generated by the DERs being equal to the total power demand. We develop distributed algorithms to solve the DER coordination problem over time-varying communication networks with either bidirectional or unidirectional communication links. The proposed algorithms can be seen as distributed versions of a centralized primal–dual algorithm. One of the algorithms proposed for directed communication graphs has a geometric convergence rate even when communication out-degrees are unknown to agents. We showcase the proposed algorithms using the standard IEEE 39-bus test system and compare their performance against other ones proposed in the literature.

Approximation by Simple Poles – Part I: Density and Geometric Convergence Rate in Hardy Space

Approximation by Simple Poles – Part I: Density and Geometric Convergence Rate in Hardy Space

Convergence of Position-Dependent MALA with Application to Conditional Simulation in GLMMs

We establish conditions under which Metropolis-Hastings (MH) algorithms with a position-dependent proposal covariance matrix will or will not have the geometric rate of convergence. Some of the diffusions based MH algorithms like the Metropolis adjusted Langevin algorithm (MALA) and the pre-conditioned MALA (PCMALA) have a position-independent proposal variance. Whereas, for other modern variants of MALA like the manifold MALA (MMALA) that adapt to the geometry of the target distributions, the proposal covariance matrix changes in every iteration. Thus, we provide conditions for geometric ergodicity of different variations of the Langevin algorithms. These results have important practical implications as these provide crucial justification for the use of asymptotically valid Monte Carlo standard errors for Markov chain based estimates. The general conditions are verified in the context of conditional simulation from the two most popular generalized linear mixed models (GLMMs), namely the binomial GLMM with the logit link and the Poisson GLMM with the log link. Empirical comparison in the framework of some spatial GLMMs shows that the computationally less expensive PCMALA with an appropriately chosen pre-conditioning matrix may outperform the MMALA.

Convergence rate bounds for iterative random functions using one-shot coupling

One-shot coupling is a method of bounding the convergence rate between two copies of a Markov chain in total variation distance, which was first introduced in Roberts and Rosenthal (Process Appl 99:195–208, 2002) and generalized in Madras and Sezer (Bernoulli 16:882–908, 2010). The method is divided into two parts: the contraction phase, when the chains converge in expected distance and the coalescing phase, which occurs at the last iteration, when there is an attempt to couple. One-shot coupling does not require the use of any exogenous variables like a drift function or a minorization constant. In this paper, we summarize the one-shot coupling method into the One-Shot Coupling Theorem. We then apply the theorem to two families of Markov chains: the random functional autoregressive process and the autoregressive conditional heteroscedastic process. We provide multiple examples of how the theorem can be used on various models including ones in high dimensions. These examples illustrate how the theorem’s conditions can be verified in a straightforward way. The one-shot coupling method appears to generate tight geometric convergence rate bounds.

Nonregular and minimax estimation of individualized thresholds in high dimension with binary responses

Given a large number of covariates Z, we consider the estimation of a high-dimensional parameter θ in an individualized linear threshold θTZ for a continuous variable X, which minimizes the disagreement between sign(X−θTZ) and a binary response Y. While the problem can be formulated into the M-estimation framework, minimizing the corresponding empirical risk function is computationally intractable due to discontinuity of the sign function. Moreover, estimating θ even in the fixed-dimensional setting is known as a nonregular problem leading to nonstandard asymptotic theory. To tackle the computational and theoretical challenges in the estimation of the high-dimensional parameter θ, we propose an empirical risk minimization approach based on a regularized smoothed non-convex loss function. The Fisher consistency of the proposed method is guaranteed as the bandwidth of the smoothed loss is shrunk to 0. Statistically, we show that the finite sample error bound for estimating θ in ℓ2 norm is (slogd/n)β/(2β+1), where d is the dimension of θ, s is the sparsity level, n is the sample size and β is the smoothness of the conditional density of X given the response Y and the covariates Z. The convergence rate is nonstandard and slower than that in the classical Lasso problems. Furthermore, we prove that the resulting estimator is minimax rate optimal up to a logarithmic factor. The Lepski’s method is developed to achieve the adaption to the unknown sparsity s or smoothness β. Computationally, an efficient path-following algorithm is proposed to compute the solution path. We show that this algorithm achieves geometric rate of convergence for computing the whole path. Finally, we evaluate the finite sample performance of the proposed estimator in simulation studies and a real data analysis from the ChAMP (Chondral Lesions And Meniscus Procedures) Trial.

Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems

The Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for tough problems, we have to loosen the default Chebfun tolerance in Newton’s solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic, the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how the Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).

On linear and super-linear convergence of Natural Policy Gradient algorithm

Markov Decision Processes are classically solved using Value Iteration and Policy Iteration algorithms. Recent interest in Reinforcement Learning has motivated the study of methods inspired by optimization, such as gradient ascent. Among these, a popular algorithm is the Natural Policy Gradient, which is a mirror descent variant for MDPs. This algorithm forms the basis of several popular RL algorithms such as Natural actor–critic, TRPO, PPO, etc., and so is being studied with growing interest. It has been shown that Natural Policy Gradient with constant step size converges with a sub-linear rate of O(1/k) to the global optimal. In this paper, we present improved finite time convergence bounds, and show that this algorithm has geometric (also known as linear) asymptotic convergence rate. We further improve this convergence result by introducing a variant of Natural Policy Gradient with adaptive step sizes. Finally, we compare different variants of policy gradient methods experimentally.

Dimension free convergence rates for Gibbs samplers for Bayesian linear mixed models

The emergence of big data has led to a growing interest in so-called convergence complexity analysis, which is the study of how the convergence rate of a Monte Carlo Markov chain (for an intractable Bayesian posterior distribution) scales as the underlying data set grows in size. Convergence complexity analysis of practical Monte Carlo Markov chains on continuous state spaces is quite challenging, and there have been very few successful analyses of such chains. One fruitful analysis was recently presented by Qin and Hobert (2022), who studied a Gibbs sampler for a simple Bayesian random effects model. These authors showed that, under regularity conditions, the geometric convergence rate of this Gibbs sampler converges to zero as the data set grows in size. It is shown herein that similar behavior is exhibited by Gibbs samplers for more general Bayesian models that possess both random effects and traditional continuous covariates, the so-called mixed models. The analysis employs the Wasserstein-based techniques introduced by Qin and Hobert (2022).

Bayesian inference for generalized linear model with linear inequality constraints

Bayesian statistical inference for Generalized Linear Models (GLMs) with parameters lying on a constrained space is of general interest (e.g., in monotonic or convex regression), but often constructing valid prior distributions supported on a subspace spanned by a set of linear inequality constraints can be challenging, especially when some of the constraints might be binding leading to a lower dimensional subspace. For the general case with canonical link, it is shown that a generalized truncated multivariate normal supported on a desired subspace can be used. Moreover, it is shown that such prior distribution facilitates the construction of a general purpose product slice sampling method to obtain (approximate) samples from corresponding posterior distribution, making the inferential method computationally efficient for a wide class of GLMs with an arbitrary set of linear inequality constraints. The proposed product slice sampler is shown to be uniformly ergodic, having a geometric convergence rate under a set of mild regularity conditions satisfied by many popular GLMs (e.g., logistic and Poisson regressions with constrained coefficients). One of the primary advantages of the proposed Bayesian estimation method over classical methods is that uncertainty of parameter estimates is easily quantified by using the samples simulated from the path of the Markov Chain of the slice sampler. Numerical illustrations using simulated data sets are presented to illustrate the superiority of the proposed methods compared to some existing methods in terms of sampling bias and variances. In addition, real case studies are presented using data sets for fertilizer-crop production and estimating the SCRAM rate in nuclear power plants.

On the geometric ergodicity for a generalized IFS with probabilities

Main goal of this paper is to formulate possibly simple and easy to verify criteria on existence of the unique attracting probability measure for stochastic process induced by generalized iterated function systems with probabilities (GIFSPs). To do this, we study the long-time behavior of trajectories of Markov-type operators acting on product of spaces of Borel measures on arbitrary Polish space. Precisely, we get the desired geometric rate of convergence of sequences of measures under the action of such operator to the unique distribution in the Hutchinson–Wasserstein distance. We apply the obtained results to study limiting behavior of random trajectories of GIFSPs as well as stochastic difference equations with multiple delays.

A novel convergence analysis of Robin–Robin domain decomposition method for Stokes–Darcy system with Beavers–Joseph interface condition

In this paper, we demonstrate the convergence analysis of Robin–Robin domain decomposition method with finite element discretization for Stokes–Darcy system with Beavers-Joseph interface condition, with particular attention paid to the case which is convergent for small viscosity and hydraulic conductivity in practice. Based on the techniques of the discrete harmonic extension and discrete Stokes extension, the convergence is proved and the almost optimal geometric convergence rate is obtained for the case of γf>γp. Here γf and γp are positive Robin parameters introduced in Cao et al., 2011, which was not able to show the analysis for γf>γp but only numerically illustrated its importance to the convergence for the practical situation with small viscosity and hydraulic conductivity. The analysis result provides a general guideline of choice on the relevant parameters to obtain the convergence and geometric convergence rate. The numerical results verify the theoretical conclusion.

First-order inertial algorithms involving dry friction damping

In a Hilbert space $$ {\mathcal H}$$ , we introduce a new class of first-order algorithms which naturally occur as discrete temporal versions of an inertial differential inclusion jointly involving viscous friction and dry friction. The function $$f:{\mathcal H}\rightarrow {\mathbb {R}}$$ to be minimized is supposed to be differentiable (not necessarily convex), and enters the algorithm via its gradient. The dry friction damping function $$\phi :{\mathcal H}\rightarrow {\mathbb {R}}_+$$ is convex with a sharp minimum at the origin, (typically $$\phi (x) = r \Vert x\Vert $$ with $$r >0$$ ). It enters the algorithm via its proximal mapping, which acts as a soft threshold operator on the velocities. As a result, we obtain a new class of splitting algorithms involving separately the proximal and gradient steps. The sequence of iterates has a finite length, and therefore strongly converges towards an approximate critical point $$x_{\infty }$$ of f (typically $$\Vert \nabla f(x_{\infty })\Vert \le r$$ ). Under a geometric property satisfied by the limit point $$x_{\infty }$$ , we obtain geometric and finite rates of convergence. The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence towards zero. By replacing the function f by its Moreau envelope, we extend the results to the case of nonsmooth convex functions. In this case, the algorithm involves the proximal operators of f and $$\phi $$ separately. Several variants of this algorithm are considered, including the case of the Nesterov accelerated gradient method. We then consider the extension in the case of additive composite optimization, thus leading to new splitting methods. Numerical experiments are given for Lasso-type problems. The performance profiles, as a comparison tool, demonstrate the efficiency of the Nesterov accelerated method with asymptotic vanishing damping combined with dry friction.

Age of Information of Jackson Networks With Finite Buffer Size

We investigate the Age of Information (AoI) of Jackson Networks with finite buffer size. We provide a closed form expression of an upper bound of the average AoI by using Stochastic Hybrid Systems (SHS) along with linear algebra tools. Since the computation of the bound requires inverting a N × N matrix, where N is the number of queues in the network, we provide a simple iterative algorithm and prove its convergence with geometric convergence rate. Numerical results allow us to assess the accuracy of our model in various settings.

Convergence analysis of a collapsed Gibbs sampler for Bayesian vector autoregressions

We study the convergence properties of a collapsed Gibbs sampler for Bayesian vector autoregressions with predictors, or exogenous variables. The Markov chain generated by our algorithm is shown to be geometrically ergodic regardless of whether the number of observations in the underlying vector autoregression is small or large in comparison to the order and dimension of it. In a convergence complexity analysis, we also give conditions for when the geometric ergodicity is asymptotically stable as the number of observations tends to infinity. Specifically, the geometric convergence rate is shown to be bounded away from unity asymptotically, either almost surely or with probability tending to one, depending on what is assumed about the data generating process. This result is one of the first of its kind for practically relevant Markov chain Monte Carlo algorithms. Our convergence results hold under close to arbitrary model misspecification.

Numerical computations of geometric ergodicity for stochastic dynamics

A probabilistic approach to compute the geometric convergence rate of a stochastic process is introduced in this paper. The goal is to quantitatively compute both the upper and lower bounds for rate of the exponential convergence to the stationary distribution of a stochastic dynamical system. By applying the coupling method, we derive an algorithm which does not rely on the discretization of the infinitesimal generator. In this way, our approach works well for many high-dimensional examples. We apply this algorithm to the random perturbations of both iterative maps and differential equations. We show that the rate of geometric ergodicity of a random perturbed system can, to some extent, reveal the degree of chaoticity of the underlying deterministic dynamics. Various SDE models including the ones with degenerate noise or living on the high-dimensional state space are also explored.

Percolation of Estimates for $${{\bar{\partial }}}$$ by the Method of Alternating Projections

The method of alternating projections is used to examine how regularity of operators associated to the $${{\bar{\partial }}}$$ -Neumann problem percolates up the $${{\bar{\partial }}}$$ -complex. The approach revolves around operator identities—rather than estimates—that hold on any Lipschitz domain in $${{\mathbb {C}}}^n$$ , not necessarily bounded or pseudoconvex. We show that a geometric rate of convergence in von Neumann’s alternating projection algorithm, applied to two basic projection operators, is equivalent to $${{\bar{\partial }}}$$ having closed range. This implies that compactness of the $${{\bar{\partial }}}$$ -Neumann operator percolates up the $${{\bar{\partial }}}$$ -complex whenever $${{\bar{\partial }}}$$ has closed range at the corresponding form levels.