A Runtime Analysis of Parallel Evolutionary Algorithms in Dynamic Optimization

Abstract

A simple island model with λ islands and migration occurring after every τ iterations is studied on the dynamic fitness function Maze. This model is equivalent to a (1+λ) EA if τ = 1, i. e., migration occurs during every iteration. It is proved that even for an increased offspring population size up to λ = O(n 1− ), the (1 + λ) EA is still not able to track the optimum of Maze. If the migration interval is chosen carefully, the algorithm is able to track the optimum even for logarithmic λ. The relationship of τ, λ, and the ability of the island model to track the optimum is then investigated more closely. Finally, experiments are performed to supplement the asymptotic results, and investigate the impact of the migration topology.

The Impact of a Sparse Migration Topology on the Runtime of Island Models in Dynamic Optimization

“…In the same settings, using a unidirectional ring migration topology of diameter c log n, where c > 0 is a sufficiently large constant, allows the optimum to be tracked through all n phases with high probability. This example illustrates that a less dense migration topology can mitigate the effects of migration occurring during unfavorable iterations of an oscillating fitness function, reducing the need to rely on problem-specific knowledge as in [17]. Moreover, the analysis reveals a crucial dependency of the efficiency of the model on the topology's diameter, for which we have established a sharp-threshold result.…”

“…The following choice of parameters yields a setting similar to the original Maze considered in [11,17]…”

“…Its proof follows the approach used in [17] to analyze the behavior of a single (1 + 1) EA island on Maze. ALT OPT Proof In the absence of migration, the behavior of a non-LOST island can be modeled by a Markov chain over the two states ALT and OPT (corresponding to the currentbest individuals), illustrated in Fig.…”

“…We have based our analysis on a simplified version of the island model studied in [17], which incorporates the major elements of the original setting: an oscillating fitness function, islands performing independent mutation/selection steps, and the effect of Maze phase transitions on the islands' ability to track the optimum based on their current-best individuals at the time of the transition. The simplified model incorporates more randomization, as both the oscillating pattern and migration are randomized, which both simplifies the analysis, and disallows some of the more artificial solutions possible in the original model, such as only performing migrations on iterations that assign a higher fitness value to the desirable solution.…”

Ring Migration Topology Helps Bypassing Local Optima

“…Lässig and Sudholt gave a formal analysis of island models for many different migration topologies in [12] showing how one can gain a speedup from parallelism; Badkobeh, Lehre and Sudholt could even show where the cut-off points are from which on linear speedup is no longer possible and discovered some bounds on different topologies [3]. In 2017, Lissovoi and Witt explored the performance of a parallel approach on dynamic optimization problems [13]. In all these works, the idea is to exploit the computing power that comes along with multiple islands, gaining a speedup from parallelism.…”