Trading accuracy for speed in parallel simulated annealing with simultaneous moves

Abstract

A common approach to parallelizing simulated annealing to generate several perturbations to the current s o l u t i o n s i m ultaneously, requiring synchronization to guarantee correct evaluation of the cost function. The cost of this synchronization may be reduced by allowing inaccuracies in the cost calculations. We provide a framework for understanding the theoretical implications of this approach based on a model of processor interaction under reduced synchronization that demonstrates how errors in cost calculations occur and how to estimate them. We show h o w bounds on error in the cost calculations in a s i m ulated annealing algorithm can be translated into worst-case bounds on perturbations in the parameters which describe the behavior of the algorithm.Keywords: Parallel simulated annealing Metropolis algorithm Boltzmann distribution.Simulated annealing (SA) is an iterative method for nding approximate solutions to intractable combinatorial optimization problems. For parallel implementations of SA based on simultaneous moves, synchronization is a bottleneck because it reduces the amount of available parallelism and increases the overhead. Synchronization costs can be reduced by allowing processors to use out-ofdate state information to evaluate the quality o f i n termediate solutions at the expense of introducing errors into the evaluation function. This approach assumes that SA will still generate a good solution if the size of the errors is controlled.Since 1986, parallel implementations of SA using simultaneous moves have been reported in the literature, using many di erent heuristics to control the size of the error. However, to our knowledge, very little has been published that analyzes the nature and impact of these errors theoretically. In this paper, we model the error that occurs on multiprocessors and analyze this error using techniques from statistical physics. First, a formal model of parallel simulated annealing (PSA) based on simultaneous moves with reduced synchronization is introduced. We describe how e r r o r i n t h e c hange in energy occurs in this model and discuss implementation issues that a ect its size. The e ect of such error on the behavior of SA can be understood through its impact on the frequency with which the algorithm visits states in the solution space. We prove a theorem that

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“…K b is the Boltzmann constant. For a single system with a certain temperature, Boltzmann distribution gives the probability that the system is in the specified state (Vasan and Raju, 2009;Misevicius, 2003;Durand and White, 2000;Ingber, 1993). Temperature-like parameter (u) is used as controlling parameter like temperature in the annealing process.…”

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