Summary, Extensions, and Conclusions

Abstract

The design of efficient and accurate parallel GAs is a complex problem. One must decide on a configuration among the many choices of topologies, migration rates, deme counts and sizes. Each parameter affects the quality of the search and the efficiency of the algorithm in non-linear ways, which makes the choices difficult. The ultimate goal is to determine the configuration that reaches a solution of the required quality as fast as possible. This book made some contributions towards this goal, and are summarized in this section.

The first part of this book considered GAs with one population, and reports two main results. The first is an accurate model of simple GA performance based on the gambler’s ruin problem. This model integrates previous knowledge of BB supply and correct decision making into an accurate predictor of solution quality. Chapter 2 presented empirical evidence that validates the accuracy of the predictions on various settings of practical interest. The model was successfully tested with functions of varying difficulty, with noisy fitness functions, and with different levels of selection pressure.

The second contribution is a lower bound on the benefits that should be expected from parallel GAs. The analysis of the simple master-slave GA in Chapter 3 showed that there is an optimal number of processors V* = y rf- that minimizes the execution time, and that the maximum parallel speedup is 0.5P*. The critical number of processors that maintain a desired efficiency was calculated, and it was used to determine that processors maintain a near-perfect efficiency of More sophisticated parallel GAs should be more efficient and produce higher speedups or be discarded.

In addition, Chapter 3 described a parallel implementation of a single-population GA that resembles a bounding case of multiple-deme GAs. Although this algorithm is less efficient than the simple master-slave, it has important theoretical implications. Straightforward calculations show that it can use processors to reduce the execution time, which is asymptotically the same number of processors that optimally-configured master-slaves can use.

Most of the book focused on parallel GAs with multiple populations. These algorithms have been very popular, but the effects of their parameters on the solution quality and algorithmic efficiency are not well understood. To begin making progress in the study of multi-deme GAs, Chapter 4 extended the gambler’s ruin model to calculate the expected quality of the best of r isolated or fully-connected demes. This chapter shows that distributing the population to multiple demes has two effects on the quality. The first is that the quality required of each deme may be reduced, because in a successful run only one of the demes has to reach the desired solution. The reduction in the target quality translates into smaller demes, which in turn represent a shorter execution time. However, the reduction is only” marginal.’ The second effect of using multiple demes has a greater impact on the quality and occurs when the populations communicate. After migration, the demes restart with more copies of BBs than when they are initialized randomly, and therefore are expected to reach solutions of higher quality.

One key idea introduced in Chapter 4 is to calculate how many copies of the correct BBs are present in the demes after migration. Then, the gambler’s ruin model is used to predict the outcome of the second epoch. The same idea is at the core of the more refined models in the remainder of the book.

The calculations in Chapter 4 showed that the isolated bounding case does not result in significant performance improvements, and should be avoided in practice. On the other extreme, the size of the fully-connected demes decreases substantially, thereby reducing considerably the time used by computations. However, the reduction in computations is paired with a rapidly increasing cost of communications. This tradeoff was used to find the deme size and deme count that together minimize the execution time.

Chapter 5 used Markov chains to extend the models of solution quality to consider multiple epochs. The first result of the modeling is that the major improvements in quality come after the second epoch, confirming that the results from the previous chapter are relevant and useful. This chapter also showed that the long-term search quality of r fully-connected demes of size n _d is equivalent to a single GA with a population of size rn _d when the migration rate is maximal. As the migration rate decreases, the quality deteriorates, but even moderate migration rates are sufficient to reach the same quality as a panmictic population. In any case, since the cost is independent of the migration rate (because migrations occur after convergence), there is no reason to avoid the highest values.

Chapter 5 also introduced the first model of arbitrary topologies. The model explicitly accounts for all the possible combinations of success and failure of the r demes, and therefore it is very accurate, but it is also impractical for large r. Nevertheless, the model showed that the influence of the topology on the solution quality is significant and should be examined further.

Chapter 6 continues the study of the topology. It presented simple approximate models that relate the deme size, the migration rate, and the degree of the topology with the quality and cost of the search. The models explain why demes with many neighbors are more likely to find the desired solution than sparsely-connected demes. The usual tradeoff between decreasing computations and increasing communications appears, and the analysis showed how to choose the degree of the topology that minimizes the total cost.

An important observation made in Chapter 6 is that the optimal degree, δ*, does not vary considerably as the number of demes is increased. Since the execution time largely depends on the degree, this observation implies that the execution times of optimally-configured topologies do not vary much either. Therefore, any optimally-configured topology would be a good choice to reduce the execution time. Previously, the fully-connected topology was optimized, and the results of Chapter 6 suggest that it may be an adequate choice, even though it cannot connect many demes. In fact, an optimal fully-connected topology uses fewer demes (r* = δ* +1) than any other optimally-configured topology, and therefore it is very efficient (in terms of )

This chapter also established that when the topology is fixed and the algorithm is executed until all the demes converge to the same solution, the optimal number of populations is , which asymptotically is the same as parallel versions of GAs with a single population. This result suggests that parallel GAs—regardless of whether they use a single or multiple populations—can integrate large numbers of processors and reduce significantly the execution time of many practical applications.

Chapter 7 studied the method used to choose migrants and the individuals that they replace in the receiving deme. The chapter showed that some migration policies may cause the algorithm to converge faster. The migration policy that accelerates convergence the most is to choose both the migrants and the replacements according to their fitness, which is a very common policy. The faster convergence may explain some of the controversial claims of superlinear speedups in parallel GAs. In addition, the chapter included calculations of the higher moments of the distribution of fitness. These calculations showed that the degree of the topology and the migration rate affect the population in different ways, even if they result in the same selection intensity.

Chapter 8 has a brief review of fine-grained parallel GAs. These algorithms have an interesting behavior, but have not been studied as much as the other types of parallel GAs. Fine-grained GAs can be used effectively by themselves or as a component of hierarchical algorithms. Chapter 8 described different types of hierarchical parallel GAs, and examined the question of how to choose the fastest configuration when multiple demes are combined with master-slaves. In an ideal situation, there would be enough processors available to use the optimal number of demes (r*) and the optimal number of slaves (S). In this case, the speedup of the best hybrid parallel G A would be the product of the speedup of the optimal master-slave GA by the speedup of the optimal multi-deme. More often, there are not enough processors to implement the best configuration, and those that are available have to be distributed carefully between demes and slaves to obtain the combination that minimizes the execution time. The chapter presented a method that integrates the theory introduced earlier to find the optimal distribution of processors.

When communications are expensive relative to computations, the best algorithm will have few slaves and many demes, because the master-slave GA communicates often. As computations become more costly, the best hierarchical configurations will likely have more slaves per deme.

The last chapter also contained a complete step-by-step example that illustrated how to apply the theory presented in this book to a particular problem. The example discussed how to determine the domain-dependent constants presen. in the equations, and it used the method described earlier to find the best combination of demes and slaves.