A Switched Dynamical System Framework for Analysis of Massively Parallel Asynchronous Numerical Algorithms

In the near future, massively parallel computing systems will be necessary to solve computation intensive applications. The key bottleneck in massively parallel implementation of numerical algorithms is the synchronization of data across processing elements (PEs) after each iteration, which results in significant idle time. Thus, there is a trend towards relaxing the synchronization and adopting an asynchronous model of computation to reduce idle time. However, it is not clear what is the effect of this relaxation on the stability and accuracy of the numerical algorithm. In this paper we present a new framework to analyze such algorithms. We treat the computation in each PE as a dynamical system and model the asynchrony as stochastic switching. The overall system is then analyzed as a switched dynamical system. However, modeling of massively parallel numerical algorithms as switched dynamical systems results in a very large number of modes, which makes current analysis tools available for such systems computationally intractable. We develop new techniques that circumvent this scalability issue. The framework is presented on a one-dimensional heat equation as a case study and the proposed analysis framework is verified by solving the partial differential equation (PDE) in a $\mathtt{nVIDIA\: Tesla^{\scriptsize{TM}}}$ GPU machine, with asynchronous communication between cores.Comment: ACC 201

Convergence Analysis of Asynchronous Consensus in Discrete-time Multi-agent Systems with Fixed Topology

In this paper, we study a convergence condition for asynchronous consensus problems in multi-agent systems. The convergence in this context implies the asynchronous consensus value converges to the synchronous one and is unique. Although it is reported in the literature that the consensus value under asynchronous communications may not coincide with the synchronous consensus value, it has not received much attention. In some applications, the discrepancy between them may result in serious consequences. For such applications it is critical to determine under what conditions the asynchronous consensus value is the same as the synchronous consensus value. We illustrate these issues with a few examples and then provide a condition, which guarantees that the asynchronous consensus value converges to the synchronous one. The validity of the proposed result is verified with simulations

On the Convergence Analysis of Asynchronous Distributed Quadratic Programming via Dual Decomposition

In this paper, we analyze the convergence as well as the rate of convergence of asynchronous distributed quadratic programming (QP) with dual decomposition technique. In general, distributed optimization requires synchronization of data at each iteration step due to the interdependency of data. This synchronization latency may incur a large amount of waiting time caused by an idle process during computation. We aim to attack this synchronization penalty in distributed QP problems by implementing asynchronous update of dual variable. The price to pay for adopting asynchronous computing algorithms is unpredictability of the solution, resulting in a tradeoff between speedup and accuracy. Thus, the convergence to an optimal solution is not guaranteed owing to the stochastic behavior of asynchrony. In this paper, we employ the switched system framework as an analysis tool to investigate the convergence of asynchronous distributed QP. This switched system will facilitate analysis on asynchronous distributed QP with dual decomposition, providing necessary and sufficient conditions for the mean square convergence. Also, we provide an analytic expression for the rate of convergence through the switched system, which enables performance analysis of asynchronous algorithms as compared with synchronous case. To verify the validity of the proposed methods, numerical examples are presented with an implementation of asynchronous parallel QP using OpenMP

A Stabilizing Control Algorithm for Asynchronous Parallel Quadratic Programming via Dual Decomposition

This paper proposes a control algorithm for stable implementation of asynchronous parallel quadratic programming (PQP) through dual decomposition technique. In general, distributed and parallel optimization requires synchronization of data at each iteration step due to the interdependency of data. The synchronization latency may incur a large amount of waiting time caused by an idle process during computation. We aim to mitigate this synchronization penalty in PQP problems by implementing asynchronous updates of the dual variable. The price to pay for adopting asynchronous computing algorithms is the unpredictability of the solution, resulting in a tradeoff between speedup and accuracy. In the worst case, the state of interest may become unstable owing to the stochastic behavior of asynchrony. We investigate the stability condition of asynchronous PQP problems by employing the switched system framework. A formal algorithm is provided to ensure the asymptotic stability of dual variables. Further, it is shown that the implementation of the proposed algorithm guarantees the uniqueness of optimal solutions, irrespective of asynchronous behavior. To verify the validity of the proposed methods, simulation results are presented.Comment: arXiv admin note: substantial text overlap with arXiv:1506.0548