performance of a distributed stochastic approximation algorithm in this paper a distributed stochastic approximation algorithm is studied applications of such algorithms include decentralized estimation optimization control or computing the algorithm consists in two steps a local step where each node in a network updates a local estimate using a stochastic approximation algorithm with decreasing step size and a gossip step where a node computes a local weighted average between its estimates and those of its neighbors convergence of the estimates toward a consensus is established under weak assumptions the approach relies on two main ingredients the existence of a lyapunov function for the mean field in the agreement subspace and a contraction property of the random matrices of weights in the subspace orthogonal to the agreement subspace a second order analysis of the algorithm is also performed under the form of a central limit theorem the polyak averaged version of the algorithm is also considered

incremental stochastic subgradient algorithms for convex optimization 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

stability of stochastic approximation under verifiable conditions in this paper we address the problem of the stability and convergence of the stochastic approximation procedure theta n 1 theta n gamma n 1 h theta n xi n 1 the stability of such sequences theta n is known to heavily rely on the behavior of the mean field h at the boundary of the parameter set and the magnitude of the stepsizes used the conditions typically required to ensure convergence and in particular the boundedness or stability of theta n are either too difficult to check in practice or not satisfied at all this is the case even for very simple models the most popular technique for circumventing the stability problem consists of constraining theta n to a compact subset mathcal k in the parameter space this is obviously not a satisfactory solution as the choice of mathcal k is a delicate one in this paper we first prove a deterministic stability result which relies on simple conditions on the sequences xi n and gamma n we then propose and analyze an algorithm based on projections on adaptive truncation sets which ensures that the aforementioned conditions required for stability are satisfied we focus in particular on the case where xi n is a so called markov state dependent noise we establish both the stability and convergence with probability 1 w p 1 of the algorithm under a set of simple and verifiable assumptions we illustrate our results with an example related to adaptive markov chain monte carlo algorithms

parallel gaussian interference channels are not always separable it is known that the capacity of parallel multicarrier gaussian point to point multiple access and broadcast channels can be achieved by separate encoding for each subchannel carrier subject to a power allocation across carriers in this paper we show that such a separation does not apply to parallel gaussian interference channels in general a counterexample is provided in the form of a 3 user interference channel where separate encoding can only achieve a sum capacity of 2 log 1 3 snr while the actual capacity achieved only by joint encoding across carriers is 3 log 1 2 snr as a byproduct of our analysis we propose a class of multiple access outer bounds on the capacity of the 3 user interference channel

distributed parameter estimation in sensor networks nonlinear observation models and imperfect communication the paper studies distributed static parameter vector estimation in sensor networks with nonlinear observation models and noisy intersensor communication it introduces separably estimable observation models that generalize the observability condition in linear centralized estimation to nonlinear distributed estimation it studies two distributed estimation algorithms in separably estimable models the nu with its linear counterpart lu and the nlu their update rule combines a consensus step where each sensor updates the state by weight averaging it with its neighbors states and an innovation step where each sensor processes its local current observation this makes the three algorithms of the consensus innovations type very different from traditional consensus this paper proves consistency all sensors reach consensus almost surely and converge to the true parameter value efficiency and asymptotic unbiasedness for lu and nu it proves asymptotic normality and provides convergence rate guarantees the three algorithms are characterized by appropriately chosen decaying weight sequences algorithms lu and nu are analyzed in the framework of stochastic approximation theory algorithm nlu exhibits mixed time scale behavior and biased perturbations and its analysis requires a different approach that is developed in this paper

rate balancing in multiuser mimo ofdm systems recently the capacity region of the gaussian broadcast channel has been characterized for a given transmit power constraint those points on the boundary of the capacity region can be regarded as the set of optimal operational points the present work addresses the problem of selecting the point within this set that satisfies given constraints on the ratios between rates achieved by the different users in the network this problem is usually known as rate balancing to this end the optimum iterative approach for general mimo channels is revisited and adapted to an ofdm transmission scheme specifically an algorithm is proposed that exploits the structure of the ofdm channel and whose convergence speed is essentially insensitive to the number of subcarriers this is in contrast to a straightforward extension of the general mimo algorithm to an ofdm scheme still relatively high complexity and the need of a time sharing policy to reach certain rates are at least two obstacles for a practical implementation of the optimum solution based on a novel decomposition technique for broadcast channels a suboptimum non iterative algorithm is introduced that does not require time sharing and very closely approaches the optimum solution

optimum linear joint transmit receive processing for mimo channels with qos constraints this paper considers vector communications through multiple input multiple output mimo channels with a set of quality of service qos requirements for the simultaneously established substreams linear transmit receive processing also termed linear precoder at the transmitter and linear equalizer at the receiver is designed to satisfy the qos constraints with minimum transmitted power the exact conditions under which the problem becomes unfeasible are given although the original problem is a complicated nonconvex problem with matrix valued variables with the aid of majorization theory we reformulate it as a simple convex optimization problem with scalar variables we then propose a practical and efficient multilevel water filling algorithm to optimally solve the problem for the general case of different qos requirements the optimal transmit receive processing is shown to diagonalize the channel matrix only after a very specific prerotation of the data symbols for situations in which the resulting transmit power is too large we give the precise way to relax the qos constraints in order to reduce the required power based on a perturbation analysis we also propose a robust design under channel estimation errors that has an important interest for practical systems numerical results from simulations are given to support the mathematical development of the problem

similarity between the mandelbrot set and julia sets the mandelbrot setm is self similar about any misiurewicz pointc in the sense that if we examine a neighborhood ofc inm with a very powerful microscope and then increase the magnification by a carefully chosen factor the picture will be unchanged except for a rotation the corresponding julia setj c is also self similar in the same sense with the same magnification factor moreover the two setsm andj c are similar in the sense that if we use a very powerful microscope to look atm andj c both focused atc the structures we see look like very much the same

stochastic gradient push for strongly convex functions on time varying directed graphs we investigate the convergence rate of the recently proposed subgradient push method for distributed optimization over time varying directed graphs the subgradient push method can be implemented in a distributed way without requiring knowledge of either the number of agents or the graph sequence each node is only required to know its out degree at each time our main result is a convergence rate of o ln t t for strongly convex functions with lipschitz gradients even if only stochastic gradient samples are available this is asymptotically faster than the o ln t sqrt t rate previously known for general convex functions

asymptotic properties of distributed and communication stochastic approximation algorithms the asymptotic properties of extensions of the type of distributed or decentralized stochastic approximation proposed in 1 are developed such algorithms have numerous potential applications in decentralized estimation detection and adaptive control or in decentralized monte carlo simulation for system optimization where they can exploit the possibilities of parallel processing the structure involves several isolated processors recursive algorithms that communicate to each other asynchronously and at random intervals the asymptotic small gain properties are derived the communication intervals need not be strictly bounded and they and the system noise can depend on the communicating system state state space constraints are also handled in many applications the dynamical terms are merely indicator functions or have other types of discontinuities the typical such case is also treated as is the case where there is noise in the communication the linear stochastic differential equation

on the inseparability of parallel mimo broadcast channels with linear transceivers parallel multiple input multiple output mimo broadcast channels have been shown to be separable from an information theoretic point of view i e capacity can be achieved with a strategy that performs separate encoding and decoding on each subchannel subject to a power allocation across subchannels in this paper we show that separability does not necessarily hold if the transmit strategy is restricted to linear transceivers the proof will be done by identifying a rate tuple that is achievable in a certain set of channels with joint treatment of the subchannels but lies outside the achievable rate region of separate linear precoding the implications of this result are that any algorithm to optimize transmit strategies in parallel mimo broadcast channels that is based on both linear transceivers and separate encoding on each subchannel does in fact introduce two suboptimalities aandnd not only one

constrained consensus and optimization in multi agent networks we present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time varying connectivity our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem where the global objective function is a combination of local agent objective functions our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets to highlight the effects of constraints we first consider a constrained consensus problem and present a distributed projected consensus algorithm in which agents combine their local averaging operation with projection on their individual constraint sets this algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents we establish convergence and convergence rate results for the projected consensus algorithm we next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets we present a distributed projected subgradient algorithm which involves each agent performing a local averaging operation taking a subgradient step to minimize its own objective function and projecting on its constraint set we show that with an appropriately selected stepsize rule the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal and when the weights are time varying but all agents have the same constraint set

stability and performance limits of adaptive primal dual networks this paper studies distributed primal dual strategies for adaptation and learning over networks from streaming data two first order methods are considered based on the arrow hurwicz ah and augmented lagrangian al techniques several revealing results are discovered in relation to the performance and stability of these strategies when employed over adaptive networks the conclusions establish that the advantages that these methods exhibit for deterministic optimization problems do not necessarily carry over to stochastic optimization problems it is found that they have narrower stability ranges and worse steady state mean square error performance than primal methods of the consensus and diffusion type it is also found that the ah technique can become unstable under a partial observation model while the other techniques are able to recover the unknown under this scenario a method to enhance the performance of al strategies is proposed by tying the selection of the step size to their regularization parameter it is shown that this method allows the al algorithm to approach the performance of consensus and diffusion strategies but that it remains less stable than these other strategies

distributed linear parameter estimation asymptotically efficient adaptive strategies this paper considers the problem of distributed adaptive linear parameter estimation in multiagent inference networks local sensing model information is only partially available at the agents and interagent communication is assumed to be unpredictable the paper develops a generic mixed time scale stochastic procedure consisting of simultaneous distributed learning and estimation in which the agents adaptively assess their relative observation quality over time and fuse the innovations accordingly under rather weak assumptions on the statistical model and the interagent communication it is shown that by properly tuning the consensus potential with respect to the innovation potential the asymptotic information rate loss incurred in the learning process may be made negligible as such it is shown that the agent estimates are asymptotically efficient in that their asymptotic covariance coincides with that of a centralized estimator the inverse of the centralized fisher information rate for gaussian

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

convergence of adaptive and interacting markov chain monte carlo algorithms adaptive and interacting markov chain monte carlo algorithms mcmc have been recently introduced in the literature these novel simulation algorithms are designed to increase the simulation efficiency to sample complex distributions motivated by some recently introduced algorithms such as the adaptive metropolis algorithm and the interacting tempering algorithm we develop a general methodological and theoretical framework to establish both the convergence of the marginal distribution and a strong law of large numbers this framework weakens the conditions introduced in the pioneering paper by roberts and rosenthal j appl probab 44 2007 458 475 it also covers the case when the target distribution is sampled by using markov transition kernels with a stationary distribution that differs from

rate optimization for multiuser mimo systems with linear processing this paper focuses on linear transceiver design for rate optimization in multiuser gaussian multple input multiple output mimo systems we focus on two design criteria 1 maximizing the weighted sum rate subject to a total power constraint 2 maximizing the minimum user rate subject to a total power constraint for these problems new power allocation strategies are derived which can be formulated as geometric programs gps involving mean square errors mses based on these solutions we propose iterative algorithms where each iteration contains the optimization of the uplink power uplink receive filters and downlink receive filters monotonic convergence of the algorithms is proved simulations show that the algorithms outperform existing linear schemes additionally we extend the results to other variations of the problems such as the problem of sum rate constrained or user rate constrained power minimization and the problem of sum rate maximization subject to user rate constraints and a total power constraint

distributed consensus algorithms in sensor networks quantized data and random link failures the paper studies the problem of distributed average consensus in sensor networks with quantized data and random link failures to achieve consensus dither small noise is added to the sensor states before quantization when the quantizer range is unbounded countable number of quantizer levels stochastic approximation shows that consensus is asymptotically achieved with probability one and in mean square to a finite random variable we show that the mean squared error mse can be made arbitrarily small by tuning the link weight sequence at a cost of the convergence rate of the algorithm to study dithered consensus with random links when the range of the quantizer is bounded we establish uniform boundedness of the sample paths of the unbounded quantizer this requires characterization of the statistical properties of the supremum taken over the sample paths of the state of the quantizer this is accomplished by splitting the state vector of the quantizer in two components one along the consensus subspace and the other along the subspace orthogonal to the consensus subspace the proofs use maximal inequalities for submartingale and supermartingale sequences from these we derive probability bounds on the excursions of the two subsequences from which probability bounds on the excursions of the quantizer state vector follow the paper shows how to use these probability bounds to design the quantizer parameters and to explore tradeoffs among the number of quantizer levels the size of the quantization steps the desired probability of saturation and the desired level of accuracy away from consensus finally the paper illustrates the quantizer design with a numerical study

gradient based power minimization in mimo broadcast channels with linear precoding we study the problem of minimizing the sum transmit power in mutliple input multiple output mimo downlink channels with linear transceivers if per user quality of service qos constraints expressed in terms of rates have to be fulfilled to find a suboptimal solution of the arising non convex optimization problem we introduce new auxiliary variables representing the di vision of the per user rate constraints into per stream rate targets and we optimize these variables by means of gradient projection steps this new method is combined with alternating updates of the transmit and receive filters furthermore the proposed algorithm ensures that the mean square error mse matrices of all users are diagonal and in the course of the execution of the algorithm it is possible that inactive streams get activated if this leads to a decreased sum transmit power in numerical simulations the new algorithm turns out to be superior to the various existing methods since these methods either lead to a higher sum transmit power than the proposed scheme or have a higher computational complexity or make restrictive assumptions on the system parameters

carrier cooperation can reduce the transmit power in parallel mimo broadcast channels with zero forcing even though parallel multiple input multiple output mimo broadcast channels are known to be separable from an information theoretic point of view performing separate encoding and decoding on each of the parallel channels has been shown to be potentially suboptimal in broadcast channels with linear transceivers in this paper we show that suboptimality of such a carrier noncooperative transmission also occurs in broadcast channels with zero forcing and quality of service constraints if time sharing is not allowed the proof is given by constructing a minimal example and identifying a rate tuple that is achievable using carrier cooperative zero forcing with a certain sum power but requires a higher sum power with carrier noncooperative zero forcing this observation is of practical relevance since zero forcing without time sharing is a popular assumption in the design of low complexity optimization algorithms

success and failure of adaptation diffusion algorithms with decaying step size in multiagent networks this paper investigates the problem of distributed stochastic approximation in multiagent systems the algorithm under study consists of two steps a local stochastic approximation step and a diffusion step which drives the network to a consensus the diffusion step uses row stochastic matrices to weight the network exchanges as opposed to previous works exchange matrices are not supposed to be doubly stochastic and may also depend on the past estimate we prove that nondoubly stochastic matrices generally influence the limit points of the algorithm nevertheless the limit points are not affected by the choice of the matrices provided that the latter are doubly stochastic in expectation this conclusion legitimates the use of broadcast like diffusion protocols which are easier to implement next by means of a central limit theorem we prove that doubly stochastic protocols perform asymptotically as well as centralized algorithms and we quantify the degradation caused by the use of nondoubly stochastic matrices throughout this paper a special emphasis is put on the special case of distributed nonconvex optimization as an illustration of our results

the scenery flow for hyperbolic julia sets we define the scenery flow space at a point z in the julia set j of a hyperbolic rational map t c c with degree at least 2 and more generally for t a conformal mixing repellor we prove that for hyperbolic rational maps except for a few exceptional cases listed below the scenery flow is ergodic we also prove ergodicity for almost all conformal mixing repellors here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real analytic curves and furthermore that for the collection of such maps based on a fixed open set u the ergodic cases form a dense open subset of that collection scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the hausdorff measure hd where d is the dimension of j and that the flow has a unique measure of maximal entropy for all conformal mixing repellors the flow is loosely bernoulli and has topological entropy at most d moreover the flow at almost every point is the same up to a rotation and so as a corollary one has an analogue of the lebesgue density theorem for the fractal set giving a different proof of a theorem of falconer

decentralized parameter estimation by consensus based stochastic approximation in this paper an algorithm for decentralized multi agent estimation of parameters in linear discrete time regression models is proposed in the form of a combination of local stochastic approximation algorithms and a global consensus strategy an analysis of the asymptotic properties of the proposed algorithm is presented taking into account both the multi agent network structure and the probabilities of getting local measurements and implementing exchange of inter agent messages in the case of non vanishing gains in the stochastic approximation algorithms an asymptotic estimation error covariance matrix bound is defined as the solution of a lyapunov like matrix equation in the case of asymptotically vanishing gains the mean square convergence is proved and the rate of convergence estimated in the discussion the problem of additive communication noise is treated in a methodologically consistent way it is also demonstrated how the consensus scheme in the algorithm can contribute to the overall reduction of measurement noise influence some simulation results illustrate the obtained theoretical results

diffusion strategies outperform consensus strategies for distributed estimation over adaptive networks adaptive networks consist of a collection of nodes with adaptation and learning abilities the nodes interact with each other on a local level and diffuse information across the network to solve estimation and inference tasks in a distributed manner in this work we compare the mean square performance of two main strategies for distributed estimation over networks consensus strategies and diffusion strategies the analysis in the paper confirms that under constant step sizes diffusion strategies allow information to diffuse more thoroughly through the network and this property has a favorable effect on the evolution of the network diffusion networks are shown to converge faster and reach lower mean square deviation than consensus networks and their mean square stability is insensitive to the choice of the combination weights in contrast and surprisingly it is shown that consensus networks can become unstable even if all the individual nodes are stable and able to solve the estimation task on their own when this occurs cooperation over the network leads to a catastrophic failure of the estimation task this phenomenon does not occur for diffusion networks we show that stability of the individual nodes always ensures stability of the diffusion network irrespective of the combination topology simulation results support the theoretical findings

minimal transmit power in parallel vector broadcast channels with linear precoding we consider the communication over a set of parallel multiple input single output miso broadcast channels with separate linear precoding on each of them in this setup we study the problem of fulfilling per user quality of service constraints expressed in terms of rates using minimal transmit power by means of a dual decomposition approach and a branch and bound algorithm solving the arising nonconvex subproblems we find the so far unknown globally optimal solution for the case of linear precoding with time sharing although prohibitively complex for implementation in a practical system the new power minimization method is highly interesting from a theoretical point of view as it can be used to quantify the performance gap between the outcome of heuristic algorithms and the theoretical limitations of the system with linear transceivers we also extend the approach such that zero forcing constraints can be handled

consensus in ad hoc wsns with noisy links part i distributed estimation of deterministic signals we deal with distributed estimation of deterministic vector parameters using ad hoc wireless sensor networks wsns we cast the decentralized estimation problem as the solution of multiple constrained convex optimization subproblems using the method of multipliers in conjunction with a block coordinate descent approach we demonstrate how the resultant algorithm can be decomposed into a set of simpler tasks suitable for distributed implementation different from existing alternatives our approach does not require the centralized estimator to be expressible in a separable closed form in terms of averages thus allowing for decentralized computation even of nonlinear estimators including maximum likelihood estimators mle in nonlinear and non gaussian data models we prove that these algorithms have guaranteed convergence to the desired estimator when the sensor links are assumed ideal furthermore our decentralized algorithms exhibit resilience in the presence of receiver and or quantization noise in particular we introduce a decentralized scheme for least squares and best linear unbiased estimation blue and establish its convergence in the presence of communication noise our algorithms also exhibit potential for higher convergence rate with respect to existing schemes corroborating simulations demonstrate the merits of the novel distributed estimation algorithms