Showing papers on "Convergence (routing) published in 2012"

Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling

Abstract: The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent coordination, estimation in sensor networks, and large-scale machine learning. We develop and analyze distributed algorithms based on dual subgradient averaging, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our analysis allows us to clearly separate the convergence of the optimization algorithm itself and the effects of communication dependent on the network structure. We show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network, and confirm this prediction's sharpness both by theoretical lower bounds and simulations for various networks. Our approach includes the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication.

A Unified Framework for High-Dimensional Analysis of $M$-Estimators with Decomposable Regularizers

Abstract: High-dimensional statistical inference deals with models in which the the number of parameters $p$ is comparable to or larger than the sample size $n$. Since it is usually impossible to obtain consistent procedures unless $p/n\rightarrow0$, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices and combinations thereof. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. This paper provides a unified framework for establishing consistency and convergence rates for such regularized $M$-estimators under high-dimensional scaling. We state one main theorem and show how it can be used to re-derive some existing results, and also to obtain a number of new results on consistency and convergence rates, in both $\ell_{2}$-error and related norms. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure corresponding regularized $M$-estimators have fast convergence rates and which are optimal in many well-studied cases.

A Unified Convergence Analysis of Block Successive Minimization Methods for Nonsmooth Optimization

Abstract: The block coordinate descent (BCD) method is widely used for minimizing a continuous function f of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem to be optimized in each iteration needs to be solved exactly to its unique optimal solution. Unfortunately, these requirements are often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of f which are either locally tight upper bounds of f or strictly convex local approximations of f. We focus on characterizing the convergence properties for a fairly wide class of such methods, especially for the cases where the objective functions are either non-differentiable or nonconvex. Our results unify and extend the existing convergence results for many classical algorithms such as the BCD method, the difference of convex functions (DC) method, the expectation maximization (EM) algorithm, as well as the alternating proximal minimization algorithm.

Image restoration: Total variation, wavelet frames, and beyond

Abstract: From the beginning of science, visual observations have been playing important roles. Advances in computer technology have made it possible to apply some of the most sophisticated developments in mathematics and the sciences to the design and implementation of fast algorithms running on a large number of processors to process image data. As a result, image processing and analysis techniques are now applied to virtually all natural sciences and technical disciplines ranging from computer sciences and electronic engineering to biology and medical sciences; and digital images have come into everyone’s life. Image restoration, including image denoising, deblurring, inpainting, computed tomography, etc., is one of the most important areas in image processing and analysis. Its major purpose is to enhance the quality of a given image that is corrupted in various ways during the process of imaging, acquisition and communication, and enables us to see crucial but subtle objects reside in the image. Therefore, image restoration is an important step to take towards the accurate interpretations of the physical world and making the optimal decisions. Mathematics has been playing an important role in image and signal processing from the very beginning; for example, Fourier analysis is one of the main tools in signal and image analysis, processing, and restoration. In fact, mathematics has been one of the driving forces of the modern development of image analysis, processing and restorations. At the same time, the interesting and challenging problems in imaging science also gave birth to new mathematical theories, techniques and methods. The variational methods (e.g. total variation based methods) and wavelets and wavelet frame based methods developed in the last few decades for image and signal processing are two successful recent examples among many. This paper is designed to establish connections between these two major image restoration approaches: variational methods and wavelet frame based methods. Such connections provide new interpretations and understanding of both approaches, and hence, lead to new applications for both approaches. We start with an introduction of both the variational and wavelet frame based methods. The basic linear image restoration model used for variational methods is

Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems

Abstract: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems is the first book in which two new concepts of numerical solutions of multidimensional Coefficient Inverse Problems (CIPs) for a hyperbolic Partial Differential Equation (PDE) are presented: Approximate Global Convergence and the Adaptive Finite Element Method (adaptivity for brevity). Two central questions for CIPs are addressed: How to obtain a good approximation for the exact solution without any knowledge of a small neighborhood of this solution, and how to refine it given the approximation. The book also combines analytical convergence results with recipes for various numerical implementations of developed algorithms. The developed technique is applied to two types of blind experimental data, which are collected both in a laboratory and in the field. The result for the blind backscattering experimental data collected in the field addresses a real-world problem of imaging of shallow explosives.

Push-Sum Distributed Dual Averaging for convex optimization

Abstract: Recently there has been a significant amount of research on developing consensus based algorithms for distributed optimization motivated by applications that vary from large scale machine learning to wireless sensor networks. This work describes and proves convergence of a new algorithm called Push-Sum Distributed Dual Averaging which combines a recent optimization algorithm [1] with a push-sum consensus protocol [2]. As we discuss, the use of push-sum has significant advantages. Restricting to doubly stochastic consensus protocols is not required and convergence to the true average consensus is guaranteed without knowing the stationary distribution of the update matrix in advance. Furthermore, the communication semantics of just summing the incoming information make this algorithm truly asynchronous and allow a clean analysis when varying intercommunication intervals and communication delays are modelled. We include experiments in simulation and on a small cluster to complement the theoretical analysis.

A priori convergence of the greedy algorithm for the parametrized reduced basis method

Abstract: The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the "reduced basis". The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.

Fully Distributed State Estimation for Wide-Area Monitoring Systems

Abstract: This paper presents a fully distributed state estimation algorithm for wide-area monitoring in power systems. Through iterative information exchange with designated neighboring control areas, all the balancing authorities (control areas) can achieve an unbiased estimate of the entire power system's state. In comparison with existing hierarchical or distributed state estimation methods, the novelty of the proposed approach lies in that: 1) the assumption of local observability of all the control areas is no longer needed; 2) the communication topology can be different than the physical topology of the power interconnection; and 3) for DC state estimation, no coordinator is required for each local control area to achieve provable convergence of the entire power system's states to those of the centralized estimation. The performance of both DC and AC state estimation using the proposed algorithm is illustrated in the IEEE 14-bus and 118-bus systems.

Hybrid Deterministic-Stochastic Methods for Data Fitting

Abstract: Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms offer inexpensive iterations by sampling a subset of the terms in the sum; these methods can make great progress initially, but often slow as they approach a solution. In contrast, full-gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate-of-convergence analysis shows that by controlling the sample size in an incremental-gradient algorithm, it is possible to maintain the steady convergence rates of full-gradient methods. We detail a practical quasi-Newton implementation based on this approach. Numerical experiments illustrate its potential benefits.

Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation

Abstract: We present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Frechet derivatives. Efficient offline/online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals. An a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically. We introduce a new algorithm, the PODEI-greedy algorithm, which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way. The approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations. We show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions. In case of symmetries of the problem, the approach realizes an automatic and intuitive space-compression or even space-dimensionality reduction. We perform empirical investigations of the error convergence and run-times. In all cases we obtain a good run-time acceleration.

Average Consensus on General Strongly Connected Digraphs

Abstract: We study the average consensus problem of multi-agent systems for general network topologies with unidirectional information flow. We propose two (linear) distributed algorithms, deterministic and gossip, respectively for the cases where the inter-agent communication is synchronous and asynchronous. Our contribution is that in both cases, the developed algorithms guarantee state averaging on arbitrary strongly connected digraphs; in particular, this graphical condition does not require that the network be balanced or symmetric, thereby extending many previous results in the literature. The key novelty of our approach is to augment an additional variable for each agent, called "surplus", whose function is to locally record individual state updates. For convergence analysis, we employ graph-theoretic and nonnegative matrix tools, with the eigenvalue perturbation theory playing a crucial role.

New Concepts in Adaptive Control Using Multiple Models

Abstract: The concept of using multiple models to cope with transients which arise in adaptive systems with large parametric uncertainties was introduced in the 1990s. Both switching between multiple fixed models, and switching and tuning between fixed and adaptive models was proposed, and the stability of the resulting schemes was established. In all cases, the number of models needed is generally large (cn where n is the dimension of the parameter vector and c an integer), and the models do not “cooperate” in any real sense. In this paper, a new approach is proposed which represents a significant departure from past methods. First, it requires (n+1) models (in contrast to cn) which is significantly smaller, when “n ” is large. Second, while each of the (n+1) models chosen generates an estimate of the plant parameter vector, the new approach provides an estimate which depends on the collective outputs of all the models, and can be viewed as a time-varying convex combination of the estimates. It is then shown that control based on such an estimate results in a stable overall system. Further, arguments are given as to why such a procedure should result in faster convergence of the estimate to the true value of the plant parameter as compared to conventional adaptive controllers, resulting in better performance. Simulation studies are included to practically verify the arguments presented, and demonstrate the improvement in performance.

Routing method in content-centric network

Abstract: Disclosed is a centralized controlling method of the process of delivering a routing packet for content transmission in a content-centric network (CCN). The routing method in a content centric network according to the present invention includes: a content distribution controller in the content centric network receiving a request for specific-content distribution from a user, determining locations of routers storing the content, and finding one of the routers to which the request from the user will be transmitted; and the content distribution controller finding an optimal path in consideration of a traffic distribution status and then transmitting a forwarding information base (FIB) to routers included in the optimal path.

Convergence of the forward-backward sweep method in optimal control

Abstract: The Forward-Backward Sweep Method is a numerical technique for solving optimal control problems. The technique is one of the indirect methods in which the differential equations from the Maximum Principle are numerically solved. After the method is briefly reviewed, two convergence theorems are proved for a basic type of optimal control problem. The first shows that recursively solving the system of differential equations will produce a sequence of iterates converging to the solution of the system. The second theorem shows that a discretized implementation of the continuous system also converges as the iteration and number of subintervals increases. The hypotheses of the theorem are a combination of basic Lipschitz conditions and the length of the interval of integration. An example illustrates the performance of the method.

On convergence of non-linear extended state observer for multi-input multi-output systems with uncertainty

Abstract: In this study, the convergence of non-linear extended state observer (ESO) for a class of multi-input multi-output non-linear systems with uncertainty is studied. The unknown part that comes from either the system itself or the external disturbance is considered as an augmented state. The state variable and augmented state are estimated simultaneously through the ESO. It is shown that with the pertinent choice of non-linear functions for observer, the error between the state and observer can be as small as desired when the high-gain tuning parameter is sufficiently small. The current control for permanent-magnet synchronous motor is applied.

Estimating Sparse Precision Matrix: Optimal Rates of Convergence and Adaptive Estimation

Abstract: Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained $\ell_1$ minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces. The estimator, called ACLIME, is easy to implement and performs well numerically. A major step in establishing the minimax rate of convergence is the derivation of a rate-sharp lower bound. A "two-directional" lower bound technique is applied to obtain the minimax lower bound. The upper and lower bounds together yield the optimal rates ofconvergence for sparse precision matrix estimation and show that the ACLIME estimator is adaptively minimax rate optimal for a collection of parameter spaces and a range of matrix norm losses simultaneously.

State transition algorithm

Abstract: In terms of the concepts of state and state transition, a new heuristic random search algorithm named state transition algorithm is proposed. For continuous function optimization problems, four special transformation operators called rotation, translation, expansion and axesion are designed. Adjusting measures of the transformations are mainly studied to keep the balance of exploration and exploitation. Convergence analysis is also discussed about the algorithm based on random search theory. In the meanwhile, to strengthen the search ability in high dimensional space, communication strategy is introduced into the basic algorithm and intermittent exchange is presented to prevent premature convergence. Finally, experiments are carried out for the algorithms. With 10 common benchmark unconstrained continuous functions used to test the performance, the results show that state transition algorithms are promising algorithms due to their good global search capability and convergence property when compared with some popular algorithms.

Analysis of Max-Consensus Algorithms in Wireless Channels

Abstract: In this paper, we address the problem of estimating the maximal value over a sensor network using wireless links between them. We introduce two heuristic algorithms and analyze their theoretical performance. More precisely, i) we prove that their convergence time is finite with probability one, ii) we derive an upper-bound on their mean convergence time, and iii) we exhibit a bound on their convergence time dispersion.

Universal Estimation of Directed Information

Abstract: Four estimators of the directed information rate between a pair of jointly stationary ergodic finite-alphabet processes are proposed, based on universal probability assignments. The first one is a Shannon--McMillan--Breiman type estimator, similar to those used by Verd\'u (2005) and Cai, Kulkarni, and Verd\'u (2006) for estimation of other information measures. We show the almost sure and $L_1$ convergence properties of the estimator for any underlying universal probability assignment. The other three estimators map universal probability assignments to different functionals, each exhibiting relative merits such as smoothness, nonnegativity, and boundedness. We establish the consistency of these estimators in almost sure and $L_1$ senses, and derive near-optimal rates of convergence in the minimax sense under mild conditions. These estimators carry over directly to estimating other information measures of stationary ergodic finite-alphabet processes, such as entropy rate and mutual information rate, with near-optimal performance and provide alternatives to classical approaches in the existing literature. Guided by these theoretical results, the proposed estimators are implemented using the context-tree weighting algorithm as the universal probability assignment. Experiments on synthetic and real data are presented, demonstrating the potential of the proposed schemes in practice and the utility of directed information estimation in detecting and measuring causal influence and delay.

Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

Abstract: We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods.