No Arabic abstract
Dual decomposition is widely utilized in distributed optimization of multi-agent systems. In practice, the dual decomposition algorithm is desired to admit an asynchronous implementation due to imperfect communication, such as time delay and packet drop. In addition, computational errors also exist when individual agents solve their own subproblems. In this paper, we analyze the convergence of the dual decomposition algorithm in distributed optimization when both the asynchrony in communication and the inexactness in solving subproblems exist. We find that the interaction between asynchrony and inexactness slows down the convergence rate from $mathcal{O} ( 1 / k )$ to $mathcal{O} ( 1 / sqrt{k} )$. Specifically, with a constant step size, the value of objective function converges to a neighborhood of the optimal value, and the solution converges to a neighborhood of the exact optimal solution. Moreover, the violation of the constraints diminishes in $mathcal{O} ( 1 / sqrt{k} )$. Our result generalizes and unifies the existing ones that only consider either asynchrony or inexactness. Finally, numerical simulations validate the theoretical results.
A collection of optimization problems central to power system operation requires distributed solution architectures to avoid the need for aggregation of all information at a central location. In this paper, we study distributed dual subgradient methods to solve three such optimization problems. Namely, these are tie-line scheduling in multi-area power systems, coordination of distributed energy resources in radial distribution networks, and joint dispatch of transmission and distribution assets. With suitable relaxations or approximations of the power flow equations, all three problems can be reduced to a multi-agent constrained convex optimization problem. We utilize a constant step-size dual subgradient method with averaging on these problems. For this algorithm, we provide a convergence guarantee that is shown to be order-optimal. We illustrate its application on the grid optimization problems.
This paper studies the distributed optimization problem where the objective functions might be nondifferentiable and subject to heterogeneous set constraints. Unlike existing subgradient methods, we focus on the case when the exact subgradients of the local objective functions can not be accessed by the agents. To solve this problem, we propose a projected primal-dual dynamics using only the objective functions approximate subgradients. We first prove that the formulated optimization problem can only be solved with an approximate error depending upon the accuracy of the available subgradients. Then, we show the exact solvability of this optimization problem if the accumulated approximation error is not too large. After that, we also give a novel componentwise normalized variant to improve the transient behavior of the convergent sequence. The effectiveness of our algorithms is verified by a numerical example.
Optimization in distributed networks plays a central role in almost all distributed machine learning problems. In principle, the use of distributed task allocation has reduced the computational time, allowing better response rates and higher data reliability. However, for these computational algorithms to run effectively in complex distributed systems, the algorithms ought to compensate for communication asynchrony, network node failures and delays known as stragglers. These issues can change the effective connection topology of the network, which may vary over time, thus hindering the optimization process. In this paper, we propose a new distributed unconstrained optimization algorithm for minimizing a convex function which is adaptable to a parameter server network. In particular, the network worker nodes solve their local optimization problems, allowing the computation of their local coded gradients, which will be sent to different server nodes. Then within this parameter server platform each server node aggregates its communicated local gradients, allowing convergence to the desired optimizer. This algorithm is robust to network s worker node failures, disconnection, or delaying nodes known as stragglers. One way to overcome the straggler problem is to allow coding over the network. We further extend this coding framework to enhance the convergence of the proposed algorithm under such varying network topologies. By using coding and utilizing evaluations of gradients of uniformly bounded delay we further enhance the proposed algorithm performance. Finally, we implement the proposed scheme in MATLAB and provide comparative results demonstrating the effectiveness of the proposed framework
In this work, we revisit a classical incremental implementation of the primal-descent dual-ascent gradient method used for the solution of equality constrained optimization problems. We provide a short proof that establishes the linear (exponential) convergence of the algorithm for smooth strongly-convex cost functions and study its relation to the non-incremental implementation. We also study the effect of the augmented Lagrangian penalty term on the performance of distributed optimization algorithms for the minimization of aggregate cost functions over multi-agent networks.
This paper investigates accelerating the convergence of distributed optimization algorithms on non-convex problems. We propose a distributed primal-dual stochastic gradient descent~(SGD) equipped with powerball method to accelerate. We show that the proposed algorithm achieves the linear speedup convergence rate $mathcal{O}(1/sqrt{nT})$ for general smooth (possibly non-convex) cost functions. We demonstrate the efficiency of the algorithm through numerical experiments by training two-layer fully connected neural networks and convolutional neural networks on the MNIST dataset to compare with state-of-the-art distributed SGD algorithms and centralized SGD algorithms.