In most applications, parallel computing is used to improve code efficiency and expand the scope of problems that can be addressed computationally (for background on parallelization, see references listed at the bottom of this post). For the Borg Many Objective Evolutionary Algorithm (MOEA) however, parallelization can also improve the quality and reliability of many objective search by enabling a multi-population search. The multi-population implementation of Borg is known as Multi-Master Borg, details can be found here. To measure the performance of Multi-Master Borg, we need to go beyond basic parallel efficiency (discussed in my last post, here), which measures the efficiency of computation but not the quality of the many objective search. In this post, I’ll discuss how we measure the performance of Multi-Master Borg using two metrics: hypervolume speedup and reliability.

## Hypervolume speedup

In my last post, I discussed traditional parallel efficiency, which measures the improvement in speed and efficiency that can be achieved through parallelization. For many objective search, speed and efficiency of computation are important, but we are more interested in the speed and efficiency with which the algorithm produces high quality solutions. We often use the hypervolume metric to measure the quality of an approximation set as it captures both convergence and diversity (for a thorough explanation of hypervolume, see this post). Using hypervolume as a measure of search quality, we can then evaluate hypervolume speedup, defined as:

Hypervolume speedup =

where is the time it takes the serial version of the MOEA to achieve a given hypervolume threshold, and is the time it takes the parallel implementation to reach the same threshold. Figure 1 below, adapted from Hadka and Reed, (2014), shows the hypervolume speedup across different parallel implementations of the Borg MOEA for the five objective NSGA II test problem run on 16,384 processors (in this work the parallel epsilon-NSGA II algorithm is used as a baseline rather than a serial implementation). Results from Figure 1 reveal that the Multi-Master implementations of Borg are able to reach each hypervolume threshold faster than the baseline algorithm and the master-worker implementation. For high hypervolume thresholds, the 16 and 32 Master implementations achieve the hypervolume thresholds 10 times faster than the baseline.

## Reliability

MOEAs are inherently stochastic algorithms, they are be initialized with random seeds which may speedup or slow down the efficiency of the search process. To ensure high quality Pareto approximate sets, it’s standard practice to run an MOEA across many random seeds and extract the best solutions across all seeds as the final solution set. Reliability is a measure of the probability that each seed will achieve a high quality set of solutions. Algorithms that have higher reliability allow users to run fewer random seeds which saves computational resources and speeds up the search process. Salazar et al., (2017) examined the performance of 17 configurations of Borg on the Lower Susquehanna River Basin (LSRB) for a fixed 10 hour runtime. Figure 2 shows the performance of each configuration across 50 random seeds. A configuration that is able to achieve the best hypervolume across all seeds would be represented as a blue bar that extends to the top of the plot. The algorithmic configurations are shown in the plot to the right. These results show that though configuration D, which has a high core count and low master count, achieves the best overall hypervolume, it does not do so reliably. Configuration H, which has two masters, is able to achieve nearly the same hypervolume, but has a much higher reliability. Configuration L, which has four masters, achieves a lower maximum hypervolume, but has vary little variance across random seeds.

These results can be further examined by looking at the quality of search across its runtime. In Figure 3, Salazar et al. (2017) compare the performance of the three algorithmic configurations highlighted above (D, H and L). The hypervolume achieved by the maximum and minimum seeds are shown in the shaded areas, and the median hypervolume is shown with each line. Figure 3 clearly demonstrates how improved reliability can enhance search. Though the Multi-Master implementation is able to perform fewer function evaluations in the 10 hour runtime, it has very low variance across seeds. The Master-worker implementation on the other hand achieves better performance with it’s best seed (it gets lucky), but its median performance never achieves the hypervolume of the two or four master configurations.

## Concluding thoughts

The two measures introduced above allow us to quantify the benefits of parallelizing the Multi-Master Borg MOEA. The improvements to search quality not only allow us to reduce the time and resources that we need to expend on many objective search, but may also allow us to discover solutions that would be missed by the serial or Master-Worker implementations of the algorithm. In many objective optimization contexts, this improvement may fundamentally alter our understanding of what is possible in a challenging environmental optimization problems.

## Parallel computing resources

## References

Hadka, D., & Reed, P. (2015). Large-scale parallelization of the Borg multiobjective evolutionary algorithm to enhance the management of complex environmental systems. *Environmental Modelling & Software*, *69*, 353-369.

Salazar, J. Z., Reed, P. M., Quinn, J. D., Giuliani, M., & Castelletti, A. (2017). Balancing exploration, uncertainty and computational demands in many objective reservoir optimization. *Advances in water resources*, *109*, 196-210.