An important benefit of multi-objective search is that it maintains a diverse population of candidates, which helps in deceptive problems in particular. Not all diversity is useful, however: candidates that optimize only one objective while ignoring others are rarely helpful. This paper proposes a solution: The original objectives are replaced by their linear combinations, thus focusing the search on the most useful tradeoffs between objectives. To compensate for the loss of diversity, this transformation is accompanied by a selection mechanism that favors novelty. In the highly deceptive problem of discovering minimal sorting networks, this approach finds better solutions, and finds them faster and more consistently than standard methods. It is therefore a promising approach to solving deceptive problems through multi-objective optimization.
Multi-objective optimization is most commonly useful in discovering a Pareto front from which solutions that represent useful tradeoffs between objectives can be selected (Coello Coello, 2007;Deb et al. 2002;Deb and Jain, 2014;Deb et al. 2016;Jain and Deb, 2014). Evolutionary methods are a natural fit for such problems because the Pareto front naturally emerges in the population maintained in these methods. Interestingly, multi-objectivity can also improve evolutionary optimization because it encourages populations with more diversity. Even when the focus of optimization is find good solutions along a primary performance metric, it is useful to create secondary dimensions that reward solutions that are different, e.g. in terms of structure, size, cost, consistency etc. Multi-objective optimization then discovers stepping stones that can be combined to achieve high fitness along the primary dimension (Meyerson and Miikkulainen, 2017). The stepping stones are useful in particular in problems where the fitness landscape is deceptive, i.e. where the optima are surrounded by inferior solutions (Lehman and Miikkulainen, 2014).
However, not all such diversity is useful. In particular, candidates that optimize one objective only and ignore the others are less likely to lead to useful tradeoffs, and are less likely to escape deception. The main idea evaluated in this paper is to replace the objectives with their linear combinations, thus focusing the search in more useful areas of the search space. In effect, the Pareto axes become angled, and search focuses more on tradeoffs instead of single objectives, allowing it to search around deceptive areas.
Naturally, some diversity is lost with such a focus. The second idea in this paper is that diversity can be encouraged more directly in the remaining space by utilizing a novelty metric for selection. Among the best candidates, those that are most different from the others are selected for reproduction; among the worst candidates, those that are the least different from the others will be discarded. Such a bias for diversity creates synergetic focus on tradeoffs. Together they result in a powerful method for optimization in domains where a primary performance objective can be supplemented with secondary objectives for diversity.
These ideas are tested in this paper in the highly deceptive domain of sorting networks (Knuth, 1998), i.e. networks of comparators that map any set of numbers represented in their input lines to a sorted order in their output lines. These networks have to be correct, i.e. sort all possible cases of input. The goal is to discover networks that are as small as possible, i.e. have as few comparators organized in as few sequential layers as possible. While correctness is the primary objective, it is actually not that difficult to achieve, because it is not deceptive. Minimality on the other hand, is highly deceptive and makes the sorting network design an interesting benchmark problem.
The composite novelty method is implemented in this domain and evaluated in four steps. As a baseline, a single objective combining correctness and minimality is first run. It lacks diversity and is effective only with the simplest networks. Second, the standard multiobjective approach is then implemented based with NSGA-II (Deb et al. 2002), with inaccuracy, number of layers, and number of comparators as the dimensions to be minimized. The approach has increased diversity, and finds solutions faster and to harder problems, but it also finds many solutions that are not useful. Third, these objectives are replaced with composites: one objective consists primarily of inaccuracy, with some layer and comparator fitness included; the other two consist a proportional combination of primarily layer and comparator fitness, with some correctness included. The solutions are found even faster and more consistently, but they are not yet optimal quality, presumably due to lost diversity in search. Fourth, novelty-based selection is included in the method, improving the search and resulting in solutions with better quality. This method finds optimal or near-optimal solutions to sorting networks with 8 to16 lines, and could likely find more with more extensive computational resources.
The composite novelty method is thus a promising approach to a range of problems where secondary objective is available to diversify search.
Evolutionary methods for optimizing single-objective and multiobjective problems are discussed, as well as the idea of using novelty to encourage diversity. The problem of minimal sorting networks is introduced and prior work in it reviewed.
When the optimization problem has a smooth and non-deceptive search space, evolutionary optimization of a single objective is usually convenient and effective. However, we are increasingly faced with problems with more than one objective and with a rugged and deceptive search space. The first approach often is to combi
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