On the convergence rates of asynchronous iterations

Abstract

Abstract-This paper presents a unifying convergence result for asynchronous iterations involving pseudo-contractions in the block-maximum norm. Contrary to previous results which only established asymptotic convergence or studied simplified models of asynchronism, our result allows to bound the convergence rates for both partially and totally asynchronous implementations. Several examples are worked out to demonstrate that our theorem recovers and improves on existing results, and that it allows to characterize the solution times for several classes of asynchronous iterations that have not been addressed before. I. INTRODUCTIONAsynchronous algorithms appear naturally in parallel and distributed systems and are heavily exploited applications ranging from large-scale linear algebra and optimization to distributed coordination of small embedded devices. Allowing nodes to operate in an asynchronous manner simplifies the implementation of distributed algorithms and eliminates the overhead associated with synchronization. However, care has to be taken, since asynchrony runs the risk of rendering an otherwise stable iteration unstable.The dynamics of asynchronous iterations are much richer than their synchronous counterparts, and quantifying the impact of asynchrony on the convergence properties of iterative algorithms remains challenging. Some of the first results on the convergence of asynchronous iterations were derived by Chazan and Miranker [1], who studied chaotic relaxations for solving linear systems of equations. Several authors have proposed extensions of this pioneering work to nonlinear iterations involving maximum norm contractions (e.g., [2], [3]) and for monotone iterations (e.g., [4], [5]). Powerful convergence results for broad classes of asynchronous algorithms, including maximum norm contractions and monotone mappings, under different assumptions on communication delays and update rates were presented by Bertsekas [6] and Bertsekas and Tsitsiklis [7]. Most of the results in the literature only guarantee asymptotic convergence. This paper complements the existing work by developing convergence theorems that characterize the rate of convergence of asynchronous iterations and quantify how these rates depend on the update intervals and information delays in the system.We focus on iterations involving block-maximum norm pseudo-contractions under the general asynchronous model introduced in [6], [7], which allows for heterogeneous and time-varying update rates and communication delays. Such iterations arise in a variety of algorithms, such as certain classes of linear fixed-point iterations and gradient

Asynchronous Iterations in Optimization: New Sequence Results and Sharper Algorithmic Guarantees

“…For each family, we derive convergence results that allow to characterize how the degree of asynchrony affects the convergence rate guarantees. These results extend and streamline our earlier work [5,23,25], and recover the optimal convergence results for synchronous iterations in the absence of asynchrony.…”

“…where Q = L/µ [9]. To solve (25), we apply ARock to S = I d − T prox . Then, the update rule of Algorithm 3 at the kth iteration becomes…”

“…Remark 3.14. Liu and Wright [43] proposed an asynchronous coordinate descent algorithm for solving composite optimization problems of the form (25). They proved that the linear speedup is achievable if τ ≤ O m 1/4 .…”

“…They proved that the linear speedup is achievable if τ ≤ O m 1/4 . For the special case of Problem (25) where R(x) ≡ 0, Liu et al [44] showed that the asynchronous coordinate descent method can enjoy the linear speedup if τ ≤ O m 1/2 . In comparison with [43] and [44], our requirement for the linear speedup property is τ ≤ o (m) and, hence, allows a larger value for τ .…”

On the connectivity of unions of random graphs

“…For example, distributed agreement problems over random graphs are studied in [37] and connected graph. A partial sampling of works using this assumption (or a related variant) includes [3], [7], [14], [20], [22], [27]- [31], [33], [34], [36], [39], [40], [44]. In addition, some works derive convergence rates or other results that explicitly use the length of such intervals, including [3], [7], [14], [28], [29], [31], [36], [39], [40].…”