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This paper focus on investigating the distributed Riemannian stochastic optimization problem on the Stiefel manifold for multi-agent systems, where all the agents work collaboratively to optimize a function modeled by the average of their expectation-valued local costs. Each agent only processes its own local cost function and communicate with neighboring agents to achieve optimal results while ensuring consensus. Since the local Riemannian gradient in stochastic regimes cannot be directly calculated, we will estimate the gradient by the average of a variable number of sampled gradient, which however brings about noise to the system. We then propose a distributed Riemannian stochastic optimization algorithm on the Stiefel manifold by combining the variable sample size gradient approximation method with the gradient tracking dynamic. It is worth noticing that the suitably chosen increasing sample size plays an important role in improving the algorithm efficiency, as it reduces the noise variance. In an expectation-valued sense, the iterates of all agents are proved to converge to a stationary point (or neighborhood) with fixed step sizes. We further establish the convergence rate of the iterates for the cases when the sample size is exponentially increasing, polynomial increasing, or a constant, respectively. Finally, numerical experiments are implemented to demonstrate the theoretical results.

We consider an $n$ agents distributed optimization problem with imperfect information characterized in a parametric sense, where the unknown parameter can be solved by a distinct distributed parameter learning problem. Though each agent only has access to its local parameter learning and computational problem, they mean to collaboratively minimize the average of their local cost functions. To address the special optimization problem, we propose a coupled distributed stochastic approximation algorithm, in which every agent updates the current beliefs of its unknown parameter and decision variable by stochastic approximation method; and then averages the beliefs and decision variables of its neighbors over network in consensus protocol. Our interest lies in the convergence analysis of this algorithm. We quantitatively characterize the factors that affect the algorithm performance, and prove that the mean-squared error of the decision variable is bounded by $\mathcal{O}(\frac{1}{nk})+\mathcal{O}\left(\frac{1}{\sqrt{n}(1-\rho_w)}\right)\frac{1}{k^{1.5}}+\mathcal{O}\big(\frac{1}{(1-\rho_w)^2} \big)\frac{1}{k^2}$, where $k$ is the iteration count and $(1-\rho_w)$ is the spectral gap of the network weighted adjacency matrix. It reveals that the network connectivity characterized by $(1-\rho_w)$ only influences the high order of convergence rate, while the domain rate still acts the same as the centralized algorithm. In addition, we analyze that the transient iteration needed for reaching its dominant rate $\mathcal{O}(\frac{1}{nk})$ is $\mathcal{O}(\frac{n}{(1-\rho_w)^2})$. Numerical experiments are carried out to demonstrate the theoretical results by taking different CPUs as agents, which is more applicable to real-world distributed scenarios., Comment: 27 pages, 6 figures

This paper considers a distributed adaptive optimization problem, where all agents only have access to their local cost functions with a common unknown parameter, whereas they mean to collaboratively estimate the true parameter and find the optimal solution over a connected network. A general mathematical framework for such a problem has not been studied yet. We aim to provide valuable insights for addressing parameter uncertainty in distributed optimization problems and simultaneously find the optimal solution. Thus, we propose a novel Prediction while Optimization scheme, which utilizes distributed fractional Bayesian learning through weighted averaging on the log-beliefs to update the beliefs of unknown parameters, and distributed gradient descent for renewing the estimation of the optimal solution. Then under suitable assumptions, we prove that all agents' beliefs and decision variables converge almost surely to the true parameter and the optimal solution under the true parameter, respectively. We further establish a sublinear convergence rate for the belief sequence. Finally, numerical experiments are implemented to corroborate the theoretical analysis., Comment: 16 pages, 6 figures

This work studies the parameter identification problem of a generalized non-cooperative game, where each player's cost function is influenced by an observable signal and some unknown parameters. We consider the scenario where equilibrium of the game at some observable signals can be observed with noises, whereas our goal is to identify the unknown parameters with the observed data. Assuming that the observable signals and the corresponding noise-corrupted equilibriums are acquired sequentially, we construct this parameter identification problem as online optimization and introduce a novel online parameter identification algorithm. To be specific, we construct a regularized loss function that balances conservativeness and correctiveness, where the conservativeness term ensures that the new estimates do not deviate significantly from the current estimates, while the correctiveness term is captured by the Karush-Kuhn-Tucker conditions. We then prove that when the players' cost functions are linear with respect to the unknown parameters and the learning rate of the online parameter identification algorithm satisfies \mu_k \propto 1/\sqrt{k}, along with other assumptions, the regret bound of the proposed algorithm is O(\sqrt{K}). Finally, we conduct numerical simulations on a Nash-Cournot problem to demonstrate that the performance of the online identification algorithm is comparable to that of the offline setting., Comment: 10 pages, 5 figures

We consider a class of nonsmooth aggregative games over networks in stochastic regimes, where each player is characterized by a composite cost function $f_i+r_i$, $f_i$ is a smooth expectation-valued function dependent on its own strategy and an aggregate function of rival strategies, and $r_i$ is a nonsmooth convex function of its strategy with an efficient prox-evaluation. We design a fully distributed iterative proximal stochastic gradient method overlaid by a Tikhonov regularization, where each player may independently choose its steplengths and regularization parameters while meeting some coordination requirements. Under a monotonicity assumption on the pseudo-gradient mapping, we prove the almost sure convergence to the least-norm Nash equilibrium. In addition, when each $r_i$ is an indicator function of a compact convex set, we establish the convergence rate associated with the expected gap function at the time-averaged sequence. We further establish high probability bounds for the gap function via both Markov's inequality as well as a more refined argument that leverages Azuma's inequality. Furthermore, we consider the extension to the private hierarchical regime where each player is a leader with respect to a collection of private followers competing in a strongly monotone game, parametrized by leader decisions. By leveraging a convolution-smoothing framework, we present amongst the first fully distributed schemes for computing a Nash equilibrium of a game complicated by such a hierarchical structure. Based on this framework, we extend the rate statements to accommodate the computation of a hierarchical stochastic Nash equilibrium by using a Fitzpatrick gap function. Finally, we validate the proposed methods on a networked Nash-Cournot equilibrium problem and a hierarchical generalization, observing that regularization has a beneficial impact on empirical behavior.

We consider distributed learning problem in games with an unknown cost-relevant parameter, and aim to find the Nash equilibrium while learning the true parameter. Inspired by the social learning literature, we propose a distributed non-Bayesian rule to learn the parameter (each agent maintains a belief on the parameter and updates the belief according to the received noisy cost), combined with best response dynamics for strategy update. The difficulty of the analysis lies mainly in the fact that the parameter learning process and strategy update process are coupled. We first prove that agents' beliefs converge to a common belief and the strategy profiles converge to a Nash equilibrium under this common belief. On this basis, we further show that the beliefs eventually concentrate on the true parameter.

No-regret learning has been widely used to compute a Nash equilibrium in two-person zero-sum games. However, there is still a lack of regret analysis for network stochastic zero-sum games, where players competing in two subnetworks only have access to some local information, and the cost functions include uncertainty. Such a game model can be found in security games, when a group of inspectors work together to detect a group of evaders. In this paper, we propose a distributed stochastic mirror descent (D-SMD) method, and establish the regret bounds $O(\sqrt{T})$ and $O(\log T)$ in the expected sense for convex-concave and strongly convex-strongly concave costs, respectively. Our bounds match those of the best known first-order online optimization algorithms. We then prove the convergence of the time-averaged iterates of D-SMD to the set of Nash equilibria. Finally, we show that the actual iterates of D-SMD almost surely converge to the Nash equilibrium in the strictly convex-strictly concave setting.

This paper aims to design a distributed coordination algorithm for solving a multi-agent decision problem with a hierarchical structure. The primary goal is to search the Nash equilibrium of a noncooperative game such that each player has no incentive to deviate from the equilibrium under its private objective. Meanwhile, the agents can coordinate to optimize the social cost within the set of Nash equilibria of the underlying game. Such an optimal Nash equilibrium problem can be modeled as a distributed optimization problem with variational inequality constraints. We consider the scenario where the objective functions of both the underlying game and social cost optimization problem have a special aggregation structure. Since each player only has access to its local objectives while cannot know all players' decisions, a distributed algorithm is highly desirable. By utilizing the Tikhonov regularization and dynamical averaging tracking technique, we propose a distributed coordination algorithm by introducing an incentive term in addition to the gradient-based Nash equilibrium seeking, so as to intervene players' decisions to improve the system efficiency. We prove its convergence to the optimal Nash equilibrium of a monotone aggregative game with simulation studies., Comment: 7 pages, 3 figures

This paper considers no-regret learning for repeated continuous-kernel games with lossy bandit feedback. Since it is difficult to give the explicit model of the utility functions in dynamic environments, the players' action can only be learned with bandit feedback. Moreover, because of unreliable communication channels or privacy protection, the bandit feedback may be lost or dropped at random. Therefore, we study the asynchronous online learning strategy of the players to adaptively adjust the next actions for minimizing the long-term regret loss. The paper provides a novel no-regret learning algorithm, called Online Gradient Descent with lossy bandits (OGD-lb). We first give the regret analysis for concave games with differentiable and Lipschitz utilities. Then we show that the action profile converges to a Nash equilibrium with probability 1 when the game is also strictly monotone. We further provide the mean square convergence rate $\mathcal{O}\left(k^{-2\min\{\beta, 1/6\}}\right)$ when the game is $\beta-$ strongly monotone. In addition, we extend the algorithm to the case when the loss probability of the bandit feedback is unknown, and prove its almost sure convergence to Nash equilibrium for strictly monotone games. Finally, we take the resource management in fog computing as an application example, and carry out numerical experiments to empirically demonstrate the algorithm performance., Comment: 14 pages,11 figures

This paper studies a distributed policy gradient in collaborative multi-agent reinforcement learning (MARL), where agents over a communication network aim to find the optimal policy to maximize the average of all agents' local returns. Due to the non-concave performance function of policy gradient, the existing distributed stochastic optimization methods for convex problems cannot be directly used for policy gradient in MARL. This paper proposes a distributed policy gradient with variance reduction and gradient tracking to address the high variances of policy gradient, and utilizes importance weight to solve the {distribution shift} problem in the sampling process. We then provide an upper bound on the mean-squared stationary gap, which depends on the number of iterations, the mini-batch size, the epoch size, the problem parameters, and the network topology. We further establish the sample and communication complexity to obtain an $\epsilon$-approximate stationary point. Numerical experiments are performed to validate the effectiveness of the proposed algorithm.

The paper considers distributed stochastic optimization over randomly switching networks, where agents collaboratively minimize the average of all agents' local expectation-valued convex cost functions. Due to the stochasticity in gradient observations, distributedness of local functions, and randomness of communication topologies, distributed algorithms with a convergence guarantee under fixed step-sizes have not been achieved yet. This work incorporates variance reduction scheme into the distributed stochastic gradient tracking algorithm, where local gradients are estimated by averaging across a variable number of sampled gradients. With an identically and independently distributed (i.i.d.) random network, we show that all agents' iterates converge almost surely to the same optimal solution under fixed step-sizes. When the global cost function is strongly convex and the sample size increases at a geometric rate, we prove that the iterates geometrically converge to the unique optimal solution, and establish the iteration, oracle, and communication complexity. The algorithm performance including rate and complexity analysis are further investigated with constant step-sizes and a polynomially increasing sample size. Finally, the empirical algorithm performance are illustrated with numerical examples.

In this paper, we consider a distributed learning problem in a subnetwork zero-sum game, where agents are competing in different subnetworks. These agents are connected through time-varying graphs where each agent has its own cost function and can receive information from its neighbors. We propose a distributed mirror descent algorithm for computing a Nash equilibrium and establish a sublinear regret bound on the sequence of iterates when the graphs are uniformly strongly connected and the cost functions are convex-concave. Moreover, we prove its convergence with suitably selected diminishing stepsizes for a strictly convex-concave cost function. We also consider a constant step-size variant of the algorithm and establish an asymptotic error bound between the cost function values of running average actions and a Nash equilibrium. In addition, we apply the algorithm to compute a mixed-strategy Nash equilibrium in subnetwork zero-sum finite-strategy games, which have merely convex-concave (to be specific, multilinear) cost functions, and obtain a final-iteration convergence result and an ergodic convergence result, respectively, under different assumptions.

Distributed linear algebraic equation over networks, where nodes hold a part of problem data and cooperatively solve the equation via node-to-node communications, is a basic distributed computation task receiving an increasing research attention. Communications over a network have a stochastic nature, with both temporal and spatial dependence due to link failures, packet dropouts or node recreation, etc. In this paper, we study the convergence and convergence rate of distributed linear equation protocols over a $\ast$-mixing random network, where the temporal and spatial dependencies between the node-to-node communications are allowed. When the network linear equation admits exact solutions, we prove the mean-squared exponential convergence rate of the distributed projection consensus algorithm, while the lower and upper bound estimations of the convergence rate are also given for independent and identically distributed (i.i.d.) random graphs. Motivated by the randomized Kaczmarz algorithm, we also propose a distributed randomized projection consensus algorithm, where each node randomly selects one row of local linear equations for projection per iteration, and establish an exponential convergence rate. When the network linear equation admits no exact solution, we prove that a distributed gradient-descent-like algorithm with diminishing step-sizes can drive all nodes' states to a least-squares solution at a sublinear rate. These results collectively illustrate that distributed computations may overcome communication correlations if the prototype algorithms enjoy certain contractive properties or are designed with suitable parameters., Comment: 17 pages

In this paper, we study a stochastic strongly convex optimization problem and propose three classes of variable sample-size stochastic first-order methods including the standard stochastic gradient descent method, its accelerated variant, and the stochastic heavy ball method. In the iterates of each scheme, the unavailable exact gradients are approximated by averaging across an increasing batch size of sampled gradients. We prove that when the sample-size increases geometrically, the generated estimates converge in mean to the optimal solution at a geometric rate. Based on this result, we provide a unified framework to show that the rescaled estimation errors converge in distribution to a normal distribution, in which the covariance matrix depends on the Hessian matrix, covariance of the gradient noise, and the steplength. If the sample-size increases at a polynomial rate, we show that the estimation errors decay at the corresponding polynomial rate and establish the corresponding central limit theorems (CLTs). Finally, we provide an avenue to construct confidence regions for the optimal solution based on the established CLTs, and test the theoretic findings on a stochastic parameter estimation problem.

This paper considers a distributed stochastic strongly convex optimization, where agents connected over a network aim to cooperatively minimize the average of all agents' local cost functions. Due to the stochasticity of gradient estimation and distributedness of local objective, fast linearly convergent distributed algorithms have not been achieved yet. This work proposes a novel distributed stochastic gradient tracking algorithm with variance reduction, where the local gradients are estimated by an increasing batch-size of sampled gradients. With an undirected connected communication graph and a geometrically increasing batch-size, the iterates are shown to converge in mean to the optimal solution at a geometric rate (achieving linear convergence). The iteration, communication, and oracle complexity for obtaining an $\epsilon$-optimal solution are established as well. Particulary, the communication complexity is $\mathcal{O}(\ln (1/\epsilon))$ while the oracle complexity (number of sampled gradients) is $\mathcal{O}(1/\epsilon^2)$, which is of the same order as that of centralized approaches. Hence, the proposed scheme is communication-efficient without requiring extra sampled gradients. Numerical simulations are given to demonstrate the theoretic results., Comment: 8 pages,3 figure, the paper will be presented on European Control Conference,2020

This paper develops a continuous-time primal-dual accelerated method with an increasing damping coefficient for a class of convex optimization problems with affine equality constraints. This paper analyzes critical values for parameters in the proposed method and prove that the rate of convergence in terms of the duality gap function is $O(\tfrac{1}{t^2})$ by choosing suitable parameters. As far as we know, this is the first continuous-time primal-dual accelerated method that can obtain the optimal rate. Then this work applies the proposed method to two network optimization problems, a distributed optimization problem with consensus constraints and a distributed extended monotropic optimization problem, and obtains two variant distributed algorithms. Finally, numerical simulations are given to demonstrate the efficacy of the proposed method.

This paper considers a stochastic Nash game in which each player minimizes an expectation valued composite objective. We make the following contributions. (I) Under suitable monotonicity assumptions on the concatenated gradient map, we derive optimal rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme when the sample-size increases at a geometric rate. If the sample-size increases at a polynomial rate of degree $v > 0$, the mean-squared errordecays at a corresponding polynomial rate while the iteration and oracle complexities to obtain an $\epsilon$-NE are $\mathcal{O}(1/\epsilon^{1/v})$ and $\mathcal{O}(1/\epsilon^{1+1/v})$, respectively. (II) We then overlay (VS-PGR) with a consensus phase with a view towards developing distributed protocols for aggregative stochastic Nash games. In the resulting scheme, when the sample-size and the consensus steps grow at a geometric and linear rate, computing an $\epsilon$-NE requires similar iteration and oracle complexities to (VS-PGR) with a communication complexity of $\mathcal{O}(\ln^2(1/\epsilon))$; (III) Under a suitable contractive property associated with the proximal best-response (BR) map, we design a variable sample-size proximal BR (VS-PBR) scheme, where each player solves a sample-average BR problem. Akin to (I), we also give the rate statements, oracle and iteration complexity bounds. (IV) Akin to (II), the distributed variant achieves similar iteration and oracle complexities to the centralized (VS-PBR) with a communication complexity of $\mathcal{O}(\ln^2(1/\epsilon))$ when the communication rounds per iteration increase at a linear rate. Finally, we present some preliminary numerics to provide empirical support for the rate and complexity statements.

This paper considers an $N$-player stochastic Nash game in which the $i$th player minimizes a composite objective $f_i(x) + r_i(x_i)$, where $f_i$ is expectation-valued and $r_i$ has an efficient prox-evaluation. In this context, we make the following contributions. (i) Under a strong monotonicity assumption on the concatenated gradient map, we derive ({\bf optimal}) rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme; (ii) We overlay (VS-PGR) with a consensus phase with a view towards developing distributed protocols for aggregative stochastic Nash games. Notably, when the sample-size and the number of consensus steps at each iteration grow at a suitable rate, a linear rate of convergence can be achieved; (iii) Finally, under a suitable contractive property associated with the proximal best-response (BR) map, we design a variable sample-size proximal BR (VS-PBR) scheme, where the proximal BR is computed by solving a sample-average problem. If the batch-size for computing the sample-average is raised at a suitable rate, we show that the resulting iterates converge at a linear rate and derive the oracle complexity.

In this paper, we study network linear equations subject to digital communications with a finite data rate, where each node is associated with one equation from a system of linear equations. Each node holds a dynamic state and interacts with its neighbors through an undirected connected graph, where along each link the pair of nodes share information. Due to the data-rate constraint, each node builds an encoder-decoder pair, with which it produces transmitted message with a zooming-in finite-level uniform quantizer and also generates estimates of its neighbors' states from the received signals. We then propose a distributed quantized algorithm and show that when the network linear equations admit a unique solution, each node's state is driven to that solution exponentially. We further establish the asymptotic rate of convergence, which shows that a larger number of quantization levels leads to a faster convergence rate but is still fundamentally bounded by the inherent network structure and the linear equations. When a unique least-squares solution exists, we show that the algorithm can compute such a solution with a suitably selected time-varying step size inherited from the encoder and zooming-in quantizer dynamics. In both cases, a minimal data rate is shown to be enough for guaranteeing the desired convergence when the step sizes are properly chosen. These results assure the applicability of various network linear equation solvers in the literature when peer-to-peer communication is digital., Comment: 32 pages, 9 figures, A preliminary version of this work will appear in the Proceedings of the IEEE Conference on Decision and Control

We consider the minimization of a sum of an expectation-valued coordinate-wise $L_i$-smooth nonconvex function and a nonsmooth block-separable convex regularizer. We propose an asynchronous variance-reduced algorithm, where in each iteration, a single block is randomly chosen to update its estimates by a proximal variable sample-size stochastic gradient scheme, while the remaining blocks are kept invariant. Notably, each block employs a steplength that is in accordance with its block-specific Lipschitz constant while block-specific batch-sizes are random variables updated at a rate that grows either at a geometric or polynomial rate with the (random) number of times that block is selected. We show that every limit point for almost every sample path is a stationary point and establish the ergodic non-asymptotic rate $\mathcal{O}(1/K) $. Iteration and oracle complexity to obtain an $\epsilon$-stationary point are shown to be $\mathcal{O}(1/\epsilon)$ and $\mathcal{O}(1/\epsilon^2)$, respectively. Furthermore, under a $ \mu $-proximal Polyak-{\L}ojasiewicz (PL) condition with the batch size increasing at a geometric rate, we prove that the suboptimality diminishes at a {\em geometric} rate, the {\em optimal} deterministic rate while iteration and oracle complexity to obtain an $\epsilon$-optimal solution are proven to be $\mathcal{O}( (L_{\rm max}/\mu) \ln(1/\epsilon))$ and $\mathcal{O}\left((L_{\rm ave}/\mu) (1/\epsilon)^{1+c} \right)$ with $c\geq 0$, respectively. In pursuit of less aggressive sampling rates, when the batch sizes increase at a polynomial rate of degree $v \geq 1$, suboptimality decays at a corresponding polynomial rate while the iteration and oracle complexity to obtain an $\epsilon-$optimal solution are provably $\mathcal{O} ( v(1/\epsilon)^{1/v})$ and $\mathcal{O} \left(e^v v^{2v+1}(1/\epsilon)^{1+1/v}\right)$, respectively., Comment: arXiv admin note: substantial text overlap with arXiv:1711.03963

The distributed computation of equilibria and optima has seen growing interest in a broad collection of networked problems. We consider the computation of equilibria of convex stochastic Nash games characterized by a possibly nonconvex potential function. Our focus is on two classes of stochastic Nash games: (P1): A potential stochastic Nash game, in which each player solves a parameterized stochastic convex program; and (P2): A misspecified generalization, where the player-specific stochastic program is complicated by a parametric misspecification. In both settings, exact proximal BR solutions are generally unavailable in finite time since they necessitate solving parameterized stochastic programs. Consequently, we design two asynchronous inexact proximal BR schemes to solve the problems, where in each iteration a single player is randomly chosen to compute an inexact proximal BR solution with rivals' possibly outdated information. Yet, in the misspecified regime (P2), each player possesses an extra estimate of the misspecified parameter and updates its estimate by a projected stochastic gradient (SG) algorithm. By Since any stationary point of the potential function is a Nash equilibrium of the associated game, we believe this paper is amongst the first ones for stochastic nonconvex (but block convex) optimization problems equipped with almost-sure convergence guarantees. These statements can be extended to allow for accommodating weighted potential games and generalized potential games. Finally, we present preliminary numerics based on applying the proposed schemes to congestion control and Nash-Cournot games.

This work considers a stochastic Nash game in which each player solves a parameterized stochastic optimization problem. In deterministic regimes, best-response schemes have been shown to be convergent under a suitable spectral property associated with the proximal best-response map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization at each iteration. Instead, we propose an inexact generalization in which an inexact solution is computed via an increasing number of projected stochastic gradient steps. Based on this framework, we present three inexact best-response schemes: (i) First, we propose a synchronous scheme where all players simultaneously update their strategies; (ii) Subsequently, we extend this to a randomized setting where a subset of players is randomly chosen to their update strategies while the others keep their strategies invariant; (iii) Finally, we propose an asynchronous scheme, where each player determines its own update frequency and may use outdated rival-specific data in updating its strategy. Under a suitable contractive property of the proximal best-response map, we derive a.s. convergence of the iterates for (i) and (ii) and mean-convergence for (i) -- (iii). In addition, we show that for (i) -- (iii), the iterates converge to the unique equilibrium in mean at a prescribed linear rate. Finally, we establish the overall iteration complexity in terms of projected stochastic gradient steps for computing an $\epsilon-$Nash equilibrium and in all settings, the iteration complexity is ${\cal O}(1/\epsilon^{2(1+c) + \delta})$ where $c = 0$ in the context of (i) and represents the positive cost of randomization (in (ii)) and asynchronicity and delay (in (iii)). The schemes are further extended to linear and quadratic recourse-based stochastic Nash games.

This paper studies a distributed stochastic optimization problem over random networks with imperfect communications subject to a global constraint, which is the intersection of local constraint sets assigned to agents. The global cost function is the sum of local cost functions, each of which is the expectation of a random cost function. By incorporating the augmented Lagrange technique with the projection method, a stochastic approximation based distributed primal-dual algorithm is proposed to solve the problem. Each agent updates its estimate by using the local observations and the information derived from neighbors. For the constrained problem, the estimates are first shown to be bounded almost surely (a.s.), and then are proved to converge to the optimal solution set a.s. Furthermore, the asymptotic normality and efficiency of the algorithm are addressed for the unconstrained case. The results demonstrate the influence of random networks, communication noises, and gradient errors on the performance of the algorithm. Finally, numerical simulations demonstrate the theoretic results.

In this paper, a stochastic approximation (SA) based distributed algorithm is proposed to solve the resource allocation (RA) with uncertainties. In this problem, a group of agents cooperatively optimize a separable optimization problem with a linear network resource constraint and allocation feasibility constraints, where the global objective function is the sum of agents' local objective functions. Each agent can only get noisy observations of its local function's gradient and its local resource, which cannot be shared by other agents or transmitted to a center. Moreover, there are communication uncertainties such as time-varying topologies (described by random graphs) and additive channel noises. To solve the RA, we propose an SA-based distributed algorithm, and prove that agents can collaboratively achieve the optimal allocation with probability one by virtue of ordinary differential equation (ODE) method for SA. Finally, simulations related to the demand response management in power systems verify the effectiveness of the proposed algorithm., Comment: 9 pages,3 figures, submitted to Systems & Control Letters

The paper studies a distributed constrained optimization problem, where multiple agents connected in a network collectively minimize the sum of individual objective functions subject to a global constraint being an intersection of the local constraint sets assigned to the agents. Based on the augmented Lagrange method, a distributed primal-dual algorithm with a projection operation included is proposed to solve the problem. It is shown that with appropriately chosen constant step size, the local estimates derived at all agents asymptotically reach a consensus at an optimal solution. In addition, the value of the cost function at the time-averaged estimate converges with rate $O(\frac{1}{k})$ to the optimal value for the unconstrained problem. By these properties the proposed primal-dual algorithm is distinguished from the existing algorithms for distributed constrained optimization. The theoretical analysis is justified by numerical simulations., Comment: 8 pages, 3 figures

In the paper, a novel distributed stochastic approximation algorithm (DSAA) is proposed to seek roots of the sum of local functions, each of which is associated with an agent from the multiple agents connected in a network. At any time, each agent updates its estimate for the root utilizing the observation of its local function and the information derived from the neighboring agents. The key difference of the proposed algorithm from the existing ones consists in the expanding truncations (so it is called as DSAAWET), by which the boundedness of the estimates can be guaranteed without imposing the growth rate constraints on the local functions. The convergence of the estimates generated by DSAAWET to a consensus set belonging to the root set of the sum function is shown under weaker conditions on the local functions and on the observation noise in comparison with the existing results. We illustrate our approach by two applications, one from signal processing and the other one from distributed optimization. Numerical simulation results are also included., Comment: 18 pages, 2 pictures