Showing papers on "Convergence (routing) published in 2009"

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 → 0, a line of recent work has studied models with various types of structure (e.g., sparse vectors; block-structured matrices; low-rank matrices; Markov assumptions). In such settings, a general approach to estimation is to solve a regularized convex program (known as a regularized M-estimator) which combines a loss function (measuring how well the model fits the data) with some regularization function that encourages the assumed structure. The goal of this paper is to provide 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 several existing results, and also to obtain several new results on consistency and convergence rates. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure the corresponding regularized M-estimators have fast convergence rates.

Convergence Speed in Distributed Consensus and Averaging

Abstract: We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.

On the Convergence of the Concave-Convex Procedure

Abstract: The concave-convex procedure (CCCP) is a majorization-minimization algorithm that solves d.c. (difference of convex functions) programs as a sequence of convex programs. In machine learning, CCCP is extensively used in many learning algorithms like sparse support vector machines (SVMs), transductive SVMs, sparse principal component analysis, etc. Though widely used in many applications, the convergence behavior of CCCP has not gotten a lot of specific attention. Yuille and Rangarajan analyzed its convergence in their original paper, however, we believe the analysis is not complete. Although the convergence of CCCP can be derived from the convergence of the d.c. algorithm (DCA), its proof is more specialized and technical than actually required for the specific case of CCCP. In this paper, we follow a different reasoning and show how Zangwill's global convergence theory of iterative algorithms provides a natural framework to prove the convergence of CCCP, allowing a more elegant and simple proof. This underlines Zangwill's theory as a powerful and general framework to deal with the convergence issues of iterative algorithms, after also being used to prove the convergence of algorithms like expectation-maximization, generalized alternating minimization, etc. In this paper, we provide a rigorous analysis of the convergence of CCCP by addressing these questions: (i) When does CCCP find a local minimum or a stationary point of the d.c. program under consideration? (ii) When does the sequence generated by CCCP converge? We also present an open problem on the issue of local convergence of CCCP.

Convergence speed in distributed consensus and averaging

Abstract: We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter We first consider the case of a fixed communication topology We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm

Incremental Stochastic Subgradient Algorithms for Convex Optimization

Abstract: This paper studies the effect of stochastic errors on two constrained incremental subgradient algorithms. The incremental subgradient algorithms are viewed as decentralized network optimization algorithms as applied to minimize a sum of functions, when each component function is known only to a particular agent of a distributed network. First, the standard cyclic incremental subgradient algorithm is studied. In this, the agents form a ring structure and pass the iterate in a cycle. When there are stochastic errors in the subgradient evaluations, sufficient conditions on the moments of the stochastic errors are obtained that guarantee almost sure convergence when a diminishing step-size is used. In addition, almost sure bounds on the algorithm's performance with a constant step-size are also obtained. Next, the Markov randomized incremental subgradient method is studied. This is a noncyclic version of the incremental algorithm where the sequence of computing agents is modeled as a time nonhomogeneous Markov chain. Such a model is appropriate for mobile networks, as the network topology changes across time in these networks. Convergence results and error bounds for the Markov randomized method in the presence of stochastic errors for diminishing and constant step-sizes are obtained.

Global Convergence of General Derivative-Free Trust-Region Algorithms to First- and Second-Order Critical Points

Abstract: In this paper we prove global convergence for first- and second-order stationary points of a class of derivative-free trust-region methods for unconstrained optimization. These methods are based on the sequential minimization of quadratic (or linear) models built from evaluating the objective function at sample sets. The derivative-free models are required to satisfy Taylor-type bounds, but, apart from that, the analysis is independent of the sampling techniques. A number of new issues are addressed, including global convergence when acceptance of iterates is based on simple decrease of the objective function, trust-region radius maintenance at the criticality step, and global convergence for second-order critical points.

Kalman Filtering with Nonlinear State Constraints

Abstract: An analytic method was developed by D. Simon and T. L. Chia to incorporate linear state equality constraints into the Kalman filter. When the state constraint was nonlinear, linearization was employed to obtain an approximately linear constraint around the current state estimate. This linearized constrained Kalman filter is subject to approximation errors and may suffer from a lack of convergence. We present a method that allows exact use of second-order nonlinear state constraints. It is based on a computational algorithm that iteratively finds the Lagrangian multiplier for the nonlinear constraints. Computer simulation results are presented to illustrate the algorithm.

Gossip consensus algorithms via quantized communication

Abstract: This paper considers the average consensus problem on a network of digital links, and proposes a set of algorithms based on pairwise ''gossip'' communications and updates. We study the convergence properties of such algorithms with the goal of answering two design questions, arising from the literature: whether the agents should encode their communication by a deterministic or a randomized quantizer, and whether they should use, and how, exact information regarding their own states in the update.

Method of successive weighted averages (MSWA) and self-regulated averaging schemes for solving stochastic user equilibrium problem

Abstract: Although stochastic user equilibrium (SUE) problem has been studied extensively in the past decades, the solution convergence of SUE is generally quite slow because of the use of the method of successive averages (MSA), in which the auxiliary flow pattern generated at each iteration contributes equally to the final solution. Realizing that the auxiliary flow pattern is in fact approaching to the solution point when the iteration number is large, in this paper, we introduce the method of successive weighted averages (MSWA) that includes a new step size sequence giving higher weights to the auxiliary flow patterns from the later iterations. We further develop a self-regulated averaging method, in which the step sizes are varying, rather than fixed, depending on the distance between intermediate solution and auxiliary point. The proposed step size sequences in both MSWA and self-regulated averaging method satisfy the Blum theorem, which guarantees the convergence of SUE problem. Computational results demonstrate that convergence speeds of MSWA and self-regulated averaging method are much faster than those of MSA and the speedup factors are in a manner of magnitude for high accuracy solutions. Besides SUE problem, the proposed methods can also be applied to other fixed-point problems where MSA is applicable, which have wide-range applications in the area of transportation networks.

On Combining Shortest-Path and Back-Pressure Routing Over Multihop Wireless Networks

Abstract: Back-pressure based algorithms based on the algorithm by Tassiulas and Ephremides have recently received much attention for jointly routing and scheduling over multi-hop wireless networks. However a significant weakness of this approach has been in routing, because the traditional back-pressure algorithm explores and exploits all feasible paths between each source and destination. While this extensive exploration is essential in order to maintain stability when the network is heavily loaded, under light or moderate loads, packets may be sent over unnecessarily long routes and the algorithm could be very inefficient in terms of end-to-end delay and routing convergence times. This paper proposes new routing/scheduling back-pressure algorithms that not only guarantees network stability (through-put optimality), but also adaptively selects a set of optimal routes based on shortest-path information in order to minimize average path-lengths between each source and destination pair. Our results indicate that under the traditional back-pressure algorithm, the end-to-end packet delay first decreases and then increases as a function of the network load (arrival rate). This surprising low-load behavior is explained due to the fact that the traditional back-pressure algorithm exploits all paths (including very long ones) even when the traffic load is light. On the otherhand, the proposed algorithm adaptively selects a set of routes according to the traffic load so that long paths are used only when necessary, thus resulting in much smaller end-to-end packet delays as compared to the traditional back-pressure algorithm.

Multiple routing configurations for fast IP network recovery

Abstract: As the Internet takes an increasingly central role in our communications infrastructure, the slow convergence of routing protocols after a network failure becomes a growing problem. To assure fast recovery from link and node failures in IP networks, we present a new recovery scheme called Multiple Routing Configurations (MRC). Our proposed scheme guarantees recovery in all single failure scenarios, using a single mechanism to handle both link and node failures, and without knowing the root cause of the failure. MRC is strictly connectionless, and assumes only destination based hop-by-hop forwarding. MRC is based on keeping additional routing information in the routers, and allows packet forwarding to continue on an alternative output link immediately after the detection of a failure. It can be implemented with only minor changes to existing solutions. In this paper we present MRC, and analyze its performance with respect to scalability, backup path lengths, and load distribution after a failure. We also show how an estimate of the traffic demands in the network can be used to improve the distribution of the recovered traffic, and thus reduce the chances of congestion when MRC is used.

Convergence Results for Some Temporal Difference Methods Based on Least Squares

Abstract: We consider finite-state Markov decision processes, and prove convergence and rate of convergence results for certain least squares policy evaluation algorithms of the type known as LSPE(lambda ). These are temporal difference methods for constructing a linear function approximation of the cost function of a stationary policy, within the context of infinite-horizon discounted and average cost dynamic programming. We introduce an average cost method, patterned after the known discounted cost method, and we prove its convergence for a range of constant stepsize choices. We also show that the convergence rate of both the discounted and the average cost methods is optimal within the class of temporal difference methods. Analysis and experiment indicate that our methods are substantially and often dramatically faster than TD(lambda), as well as more reliable.

Polynomial Filtering for Fast Convergence in Distributed Consensus

Abstract: In the past few years, the problem of distributed consensus has received a lot of attention, particularly in the framework of ad hoc sensor networks. Most methods proposed in the literature address the consensus averaging problem by distributed linear iterative algorithms, with asymptotic convergence of the consensus solution. The convergence rate of such distributed algorithms typically depends on the network topology and the weights given to the edges between neighboring sensors, as described by the network matrix. In this paper, we propose to accelerate the convergence rate for given network matrices by the use of polynomial filtering algorithms. The main idea of the proposed methodology is to apply a polynomial filter on the network matrix that will shape its spectrum in order to increase the convergence rate. Such an algorithm is equivalent to periodic updates in each of the sensors by aggregating a few of its previous estimates. We formulate the computation of the coefficients of the optimal polynomial as a semidefinite program that can be efficiently and globally solved for both static and dynamic network topologies. We finally provide simulation results that demonstrate the effectiveness of the proposed solutions in accelerating the convergence of distributed consensus averaging problems.

A Convergent Adaptive Method for Elliptic Eigenvalue Problems

Abstract: We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second-order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in $H^1$. Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation. The error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems, and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem. Because of this nonlinearity, the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine.

Accelerated Distributed Average Consensus via Localized Node State Prediction

Abstract: This paper proposes an approach to accelerate local, linear iterative network algorithms asymptotically achieving distributed average consensus. We focus on the class of algorithms in which each node initializes its ldquostate valuerdquo to the local measurement and then at each iteration of the algorithm, updates this state value by adding a weighted sum of its own and its neighbors' state values. Provided the weight matrix satisfies certain convergence conditions, the state values asymptotically converge to the average of the measurements, but the convergence is generally slow, impeding the practical application of these algorithms. In order to improve the rate of convergence, we propose a novel method where each node employs a linear predictor to predict future node values. The local update then becomes a convex (weighted) sum of the original consensus update and the prediction; convergence is faster because redundant states are bypassed. The method is linear and poses a small computational burden. For a concrete theoretical analysis, we prove the existence of a convergent solution in the general case and then focus on one-step prediction based on the current state, and derive the optimal mixing parameter in the convex sum for this case. Evaluation of the optimal mixing parameter requires knowledge of the eigenvalues of the weight matrix, so we present a bound on the optimal parameter. Calculation of this bound requires only local information. We provide simulation results that demonstrate the validity and effectiveness of the proposed scheme. The results indicate that the incorporation of a multistep predictor can lead to convergence rates that are much faster than those achieved by an optimum weight matrix in the standard consensus framework.

Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise

Abstract: The stochastic heat equation driven by additive noise is discretized in the spatial variables by a standard finite element method. The weak convergence of the approximate solution is investigated and the rate of weak convergence is found to be twice that of strong convergence.

Further results on decentralised coordination in networks of agents with second-order dynamics

Abstract: This study investigates consensus problems for networks of second-order agents, where each agent can only access the relative position information from its neighbours. We first introduce two new protocols with and without time-delay. Then we provide a convergence analysis in three cases: (a) networks with fixed topology; (b) networks with switching topology; (c) networks with switching topology and time-delays. Several conditions are presented to make all agents asymptotically reach consensus while accomplishing some tasks such as moving to a common value and moving together with a constant velocity or with a constant acceleration. Finally, simulation results are provided to demonstrate the effectiveness of our theoretical results.

Analysis of the block Gauss–Seidel solution procedure for a strongly coupled model problem with reference to fluid–structure interaction

Abstract: The block Gauss-Seidel procedure is widely used for the resolution of the strong coupling in the computer simulation of fluid-structure interaction. Based on a simple model problem, this work presents a detailed analysis of the convergence behaviour of the method. In particular, the model problem is used to highlight some aspects that arise in the context of the application of the block Gauss-Seidel method to FSI problems. Thus, the effects of the time integration schemes chosen, of relaxation techniques, of physical constraints and non-linearities on the convergence of the iterations are investigated.