Nested Distributed Gradient Methods with Adaptive Quantized Communication

In this paper, we consider minimizing a sum of local convex objective functions in a distributed setting, where communication can be costly. We propose and analyze a class of nested distributed gradient methods with adaptive quantized communication (NEAR-DGD+Q). We show the effect of performing multiple quantized communication steps on the rate of convergence and on the size of the neighborhood of convergence, and prove R-Linear convergence to the exact solution with increasing number of consensus steps and adaptive quantization. We test the performance of the method, as well as some practical variants, on quadratic functions, and show the effects of multiple quantized communication steps in terms of iterations/gradient evaluations, communication and cost.Comment: 9 pages, 2 figures. arXiv admin note: text overlap with arXiv:1709.0299

Multi-consensus Decentralized Accelerated Gradient Descent

This paper considers the decentralized optimization problem, which has applications in large scale machine learning, sensor networks, and control theory. We propose a novel algorithm that can achieve near optimal communication complexity, matching the known lower bound up to a logarithmic factor of the condition number of the problem. Our theoretical results give affirmative answers to the open problem on whether there exists an algorithm that can achieve a communication complexity (nearly) matching the lower bound depending on the global condition number instead of the local one. Moreover, the proposed algorithm achieves the optimal computation complexity matching the lower bound up to universal constants. Furthermore, to achieve a linear convergence rate, our algorithm \emph{doesn't} require the individual functions to be (strongly) convex. Our method relies on a novel combination of known techniques including Nesterov's accelerated gradient descent, multi-consensus and gradient-tracking. The analysis is new, and may be applied to other related problems. Empirical studies demonstrate the effectiveness of our method for machine learning applications

Asynchronous Distributed Optimization over Lossy Networks via Relaxed ADMM: Stability and Linear Convergence

In this work we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-to-peer network. In particular, we are interested in the case in which communications between nodes are prone to failures and the agents are not synchronized among themselves. We address the problem proposing a modified version of the relaxed ADMM, which corresponds to the Peaceman-Rachford splitting method applied to the dual. By exploiting results from operator theory, we are able to prove the almost sure convergence of the proposed algorithm under general assumptions on the distribution of communication loss and node activation events. By further assuming the cost functions to be strongly convex, we prove the linear convergence of the algorithm in mean to a neighborhood of the optimal solution, and provide an upper bound to the convergence rate. Finally, we present numerical results testing the proposed method in different scenarios.Comment: To appear in IEEE Transactions on Automatic Contro