Iterative Stochastic Method Research Articles

Stochastic iterative methods for online rank aggregation from pairwise comparisons

Stochastic iterative methods for online rank aggregation from pairwise comparisons

Stochastic Chebyshev-Picard Iteration Method for Nonlinear Differential Equations with Random Inputs

Stochastic Chebyshev-Picard Iteration Method for Nonlinear Differential Equations with Random Inputs

Markov chain Monte Carlo for seismic facies classification

Seismic facies classification aims to predict a facies model, or a set of facies models, from measured seismic data. We focus on stochastic classification methods to estimate the probability distribution of facies conditioned on seismic data. Bayesian classification methods based on analytical solutions are generally applied to seismically inverted elastic attributes or petrophysical properties rather than measured seismic data, and they do not include spatial correlation models in the prior distribution. Instead, iterative stochastic methods, such as Monte Carlo acceptance-rejection sampling and approximate Bayesian computation, can be directly applied to seismic data and can be implemented with complex prior models; however, their convergence is generally slow because it requires the simulation of a large number of facies models from the prior distribution. We develop an efficient implementation of a Markov chain Monte Carlo (MCMC) approach that adopts complex prior models, such as multiple-point statistics simulations based on a training image, to generate geologically realistic facies realizations. The novelty of the approach is that the proposal distribution of the proposed MCMC method is based on a gradual deformation algorithm, in which the proposal distribution is expressed as a linear combination of the indicator variables of the previously accepted model and a random prior simulation, according to a uniform random weight. The synthetic examples indicate accurate facies predictions with a success rate of approximately 80% for well-log facies classification based on elastic attributes and approximately 60% for seismic facies classification based on seismic data. The inversion is compared to an MCMC method with prior models sampled from a first-order Markov chain and Bayesian facies classification.

Stochastic Geometric Iterative Method for Loop Subdivision Surface Fitting

In this paper, we propose a stochastic geometric iterative method (S-GIM) to approximate the high-resolution 3D models by finite loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite loop subdivision surface approximate the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of S-GIM and verify it by demonstrating error curves in the experiment. In addition, compared with existing geometric iterative methods, S-GIM has a shorter running time under the same number of iteration steps.

On block accelerations of quantile randomized Kaczmarz for corrupted systems of linear equations

With the growth of large data as well as large-scale learning tasks, the need for efficient and robust linear system solvers is greater than ever. The randomized Kaczmarz method (RK) and similar stochastic iterative methods have received considerable recent attention due to their efficient implementation and memory footprint. These methods can tolerate streaming data, accessing only part of the data at a time, and can also approximate the least squares solution even if the system is affected by noise. However, when data is instead affected by large (possibly adversarial) corruptions, these methods fail to converge, as corrupted data points draw iterates far from the true solution. A recently proposed solution to this is the quantileRK method, which avoids harmful corrupted data by exploring the space carefully as the method iterates. The exploration component requires the computation of quantiles of large samples from the system and is computationally much heavier than the subsequent iteration update. In this paper, we propose an approach that better uses the information obtained during exploration by incorporating an averaged version of the block Kaczmarz method. This significantly speeds up convergence, while still allowing for a constant fraction of the equations to be arbitrarily corrupted. We provide theoretical convergence guarantees as well as experimental supporting evidence. We also demonstrate that the classical projection-based block Kaczmarz method cannot be robust to sparse adversarial corruptions, but rather the blocking has to be carried out by averaging one-dimensional projections.

Selectable Set Randomized Kaczmarz

AbstractThe Randomized Kaczmarz method (RK) is a stochastic iterative method for solving linear systems that has recently grown in popularity due to its speed and low memory requirement. Selectable Set Randomized Kaczmarz is a variant of RK that leverages existing information about the Kaczmarz iterate to identify an adaptive “selectable set” and thus yields an improved convergence guarantee. In this article, we propose a general perspective for selectable set approaches and prove a convergence result for that framework. In addition, we define two specific selectable set sampling strategies that have competitive convergence guarantees to those of other variants of RK. One selectable set sampling strategy leverages information about the previous iterate, while the other leverages the orthogonality structure of the problem via the Gramian matrix. We complement our theoretical results with numerical experiments that compare our proposed rules with those existing in the literature.

Sub-linear convergence of a stochastic proximal iteration method in Hilbert space

We consider a stochastic version of the proximal point algorithm for convex optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in this form. Indeed, most related results are confined to the finite-dimensional setting, where error bounds could depend on the dimension of the space. On the other hand, the few existing results in the infinite-dimensional setting only prove very weak types of convergence, owing to weak assumptions on the problem. In particular, there are no results that show strong convergence with a rate. In this article, we bridge these two worlds by assuming more regularity of the optimization problem, which allows us to prove convergence with an (optimal) sub-linear rate also in an infinite-dimensional setting. In particular, we assume that the objective function is the expected value of a family of convex differentiable functions. While we require that the full objective function is strongly convex, we do not assume that its constituent parts are so. Further, we require that the gradient satisfies a weak local Lipschitz continuity property, where the Lipschitz constant may grow polynomially given certain guarantees on the variance and higher moments near the minimum. We illustrate these results by discretizing a concrete infinite-dimensional classification problem with varying degrees of accuracy.

Dynamic statistical responses of gear drive based on improved stochastic iteration method

Mechanical equipment is always influenced by environmental factors or random excitations, such as random collision, rock fall, sea wave and wind force. It is significant to analyze the statistical information of the system performance for the operation security and kinematic accuracy. Considering time-varying stiffness and nonlinearity, a stochastic iteration method is proposed for dynamic responses in the gear drive subjected to deterministic and stochastic excitations. Also, the convergence analysis is investigated. Considering the parametric correlation, the improved iterative schemes are analyzed for the mathematical expectation and standard deviation of any multi-stage gear drive with the backlash and mesh stiffness, respectively. Also, covariance and correlation coefficient are investigated to represent the dynamic relationship between each gear pair. The crude Monte Carlo simulation is performed to provide benchmark results, and the data of simulation have good coincidence with the iteration results. It is shown that auxiliary hole of gear drive can reduce the fluctuation of stochastic response in operation. The reasonable modification gear can decrease the dispersion of system response at the transient and stable stage.

Finitely convergent deterministic and stochastic iterative methods for solving convex feasibility problems

We propose finitely convergent methods for solving convex feasibility problems defined over a possibly infinite pool of constraints. Following other works in this area, we assume that the interior of the solution set is nonempty and that certain overrelaxation parameters form a divergent series. We combine our methods with a very general class of deterministic control sequences where, roughly speaking, we require that sooner or later we encounter a violated constraint if one exists. This requirement is satisfied, in particular, by the cyclic, repetitive and remotest set controls. Moreover, it is almost surely satisfied for random controls.

Generation of seismic excitations compatible with target spectrum: application to Eurocode 8

PurposeThis paper aims to artificially generate seismic accelerograms compatible with the response spectrum imposed as a function of the given environmental parameters such as magnitude, epicentral distance and type of soil. This study is necessary for the non-linear dynamic analysis of structures in regions where real seismic records are not available.Design/methodology/approachFirst, a stochastic iterative method is used to estimate the spectral densities of acceleration power from the respective target response spectra. Thereafter, based on the superposition of seismic waves, a subsequent iterative procedure, which implicitly takes into account the non-stationary character of temporal intensity content of strong ground motions, is developed to synthesize, from these power spectral density, the corresponding acceleration time histories. The phase contents of the ground acceleration samples, thus obtained, are generated using a probability density function of phase derivatives with characteristic parameters estimated from seismological considerations. When based on seismic codes spectrum compatible criteria, this procedure can be used to generate strong ground motions for structural design.FindingsThe results found show that the forms of acceleration of the target and the simulated signals have similar characteristics in terms of strong motion durations, the peak ground acceleration values, corresponding time of occurrence and also, the corresponding cumulative energy functions follow practically the same pattern of variations.Originality/valueThe aim of this study is to generate seismic accelerograms compatible with regulatory spectra by the composition of the three acceleration duration segments based on environmental parameters (magnitude, epicentral distance and type of soil) and which subsequently serves to control the time envelope of the generated signals, and therefore the random generation of phase derivatives, which has not been previously treated.

Krahn's sales territory for PPG precipitated silica extended to cover Benelux

A well designed territory enhances customer coverage, increases sales, fosters fair performance and rewards systems and lower travel cost. This paper considers a real life case study to design a sales territory for a business sales plan. The business plan consists in assigning the optimal quantity of sellers to a territory including the scheduling and routing plans for each seller. The problem is formulated as a combination of assignment, scheduling and routing optimization problems. The solution approach considers a meta-heuristic using stochastic iterative projection method for large systems. Several real life instances of different sizes were tested with stochastic data to represent raise/fall in the customers demand as well as the appearance/loss of customers.

Stochastic gradient descent for linear systems with sequential matrix entry accumulation

Conventional stochastic iterative methods are often employed for solving linear systems of equations involving large matrix sizes using low memory footprint. However, their performances are often limited by the unavailability of all the matrix entries, which is often termed as the problem of missing data. Although Ma and Needell [1] have recently proposed a method, termed as mSGD, assuming a model for data missing that results in improved convergence, their result is also affected by constant large variance of the stochastic gradient. In this paper we propose a SGD type method termed as cumulative information SGD (CISGD) for solving a linear system with missing data with an additional provision to accumulate a very small number of matrix entries sequentially per iteration, termed as the sequential matrix entry accumulation (SEMEA) mechanism. CISGD uses the data collected by SEMEA mechanism along with the prior model for data missing mechanism of [1] to gradually reduce variance of the stochastic gradient. The convergence of the proposed CISGD is theoretically analyzed and some interesting implications of the result are investigated under a specific SEMEA mechanism. Finally, numerical experiments are performed along with simulations that corroborate the theoretical findings regarding the efficacy of the proposed CISGD method.

Stochastic Successive Convex Approximation for General Stochastic Optimization Problems

One key challenge for solving a general stochastic optimization problem with expectations in the objective and constraint functions using ordinary stochastic iterative methods lies in the infeasibility issue caused by the randomness over iterates. This letter aims to address this main challenge. First, we obtain an equivalent stochastic optimization problem which is to minimize the weighted sum of the original objective and the penalty for violating the original constraints. Then, we propose a stochastic successive convex approximation (SSCA) method to obtain a stationary point of the original stochastic optimization problem. Using similar techniques, we propose a parallel SSCA method to obtain a stationary point of a special case of the general stochastic optimization problem which has decoupled constraint functions. We also provide application examples of the proposed methods in power control for interference networks. The proposed SSCA and parallel SSCA methods achieve empirically higher convergence rates and lower computational complexities than existing ones, benefiting from the elegant way of balancing the objective minimization and constraint satisfaction over random iterates.

Bold diagrammatic Monte Carlo in the lens of stochastic iterative methods

Abstract This work aims at understanding of bold diagrammatic Monte Carlo (BDMC) methods for stochastic summation of Feynman diagrams from the angle of stochastic iterative methods. The convergence enhancement trick of the BDMC is investigated from the analysis of condition number and convergence of the stochastic iterative methods. Numerical experiments are carried out for model systems to compare the BDMC with related stochastic iterative approaches.

Territorial design optimization for business sales plan

A well designed territory enhances customer coverage, increases sales, fosters fair performance and rewards systems and lower travel cost. This paper considers a real life case study to design the sales territory for a business sales plan. The business plan consist of assigning the optimal quantity of sellers to a territory including the scheduling and routing plans for each seller. The problem is formulated as a combination of assignment, scheduling and routing optimization models. The solution approach considers a meta-heuristic using stochastic iterative projection method for large systems. Several real life instances of different sizes were tested with stochastic data to represent raise/fall in the customers demand as well as the appearance/loss of customers.

Iterative Projection approach for solving the Territorial Business Sales optimization problem

A well designed territory enhances customer coverage, increases sales, fosters fair performance and rewards systems and lower travel cost. This paper considers a real life case study to design a sales territory for a business sales plan. The business plan consists in assigning the optimal quantity of sellers to a territory including the scheduling and routing plans for each seller. The problem is formulated as a combination of assignment, scheduling and routing optimization problems. The solution approach considers a meta-heuristic using stochastic iterative projection method for large systems. Several real life instances of different sizes were tested with stochastic data to represent raise/fall in the customers demand as well as the appearance/loss of customers.

Application of Genetic Algorithms for Driverless Subway Train Energy Optimization

After an introduction on the basic aspects of electric railway transports, focusing mainly on driverless subways and their related automation systems (ATC, ATP, and ATO), a technique for energy optimization of the train movement through their control using genetic algorithms will be presented. Genetic algorithms are a heuristic search and iterative stochastic method used in computing to find exact or approximate solutions to optimization problems. This optimization process has been calculated and tested on a real subway line in Milan through the implementation of a dedicated Matlab code. The so-defined algorithm provides the optimization of the trains movement through a coast control table created by the use of a genetic algorithm that minimizes the energy consumption and the train scheduled time. The obtained results suggest that the method is promising in minimizing the energy consumption of the electric trains.

Distributed Estimation and Control of Algebraic Connectivity Over Random Graphs

In this paper, we propose a distributed algorithm for the estimation and control of the connectivity of ad-hoc networks in the presence of a random topology. First, given a generic random graph, we introduce a novel stochastic power iteration method that allows each node to estimate and track the algebraic connectivity of the underlying expected graph. Using results from stochastic approximation theory, we prove that the proposed method converges almost surely (a.s.) to the desired value of connectivity even in the presence of imperfect communication scenarios. The estimation strategy is then used as a basic tool to adapt the power transmitted by each node of a wireless network, in order to maximize the network connectivity in the presence of realistic medium access control (MAC) protocols or simply to drive the connectivity toward a desired target value. Numerical results corroborate our theoretical findings, thus illustrating the main features of the algorithm and its robustness to fluctuations of the network graph due to the presence of random link failures.

Stabilization of Stochastic Iterative Methods for Singular and Nearly Singular Linear Systems

We consider linear systems of equations, Ax = b, with an emphasis on the case where A is singular. Under certain conditions, necessary as well as sufficient, linear deterministic iterative methods generate sequences {xk} that converge to a solution as long as there exists at least one solution. This convergence property can be impaired when these methods are implemented with stochastic simulation, as is often done in important classes of large-scale problems. We introduce additional conditions and novel algorithmic stabilization schemes under which {xk} converges to a solution when A is singular and may also be used with substantial benefit when A is nearly singular.

Regularized Iterative Stochastic Approximation Methods for Stochastic Variational Inequality Problems

We consider a Cartesian stochastic variational inequality problem with a monotone map. Monotone stochastic variational inequalities arise naturally, for instance, as the equilibrium conditions of monotone stochastic Nash games over continuous strategy sets or multiuser stochastic optimization problems. We introduce two classes of stochastic approximation methods, each of which requires exactly one projection step at every iteration, and provide convergence analysis for each of them. Of these, the first is a stochastic iterative Tikhonov regularization method which necessitates the update of the regularization parameter after every iteration. The second method is a stochastic iterative proximal-point method, where the centering term is updated after every iteration. The Cartesian structure lends itself to constructing distributed multi-agent extensions and conditions are provided for recovering global convergence in limited coordination variants where agents are allowed to choose their steplength sequences, regularization and centering parameters independently, while meeting a suitable coordination requirement. We apply the proposed class of techniques and their limited coordination versions to a stochastic networked rate allocation problem.