Emergent behaviour, population-based search and low-pass filtering

In recent work we have formulated a model of emergent coordinated behaviour for a population of interacting entities. The model is a modified spring mass model where masses can perceive the environment and generate external forces. As a result of the interactions the population behaves like a single organism moving under the effect the vector sum of the external forces generated by each entity. When such forces are proportional to the gradient of a resource distribution f(x), the resultant force controlling the single emergent organism is proportional to the gradient of a modified food distribution. This is the result of applying a filtering kernel to f(x). The kernel is typically a low-pass filter. This model can be applied to genetic algorithms (GAs) and other population-based search algorithms. For example, in previous research, we have found kernels (via genetic programming) that allow the single organism model to track the motion of the centre of mass of GAs and particle swarm optimisers accurately for many generations. In this paper we corroborate this model in several ways. Firstly, we provide a mathematical proof that on any problem and for any crossover operator, the effect of crossover is that of reducing the amplitude of the derivatives (slopes) of the population distribution. This implies that a GA perceives an effective fitness landscape which is a smoothed, low-pass filtered version of the original. Then, taking inspiration from this result and our active mass-spring model, we propose a class of fitness functions, OneMix, where there is an area of the landscape with high frequency variations. This area contains the global optimum but a genetic algorithm with high crossover probability should not be able "see" it due to its low-pass behaviour. So, a GA with strong crossover should be deceived and attracted towards a local optimum, while with low crossover probability this should not happen. This is, indeed, what happens as we demonstrate with a variety of empirical runs and with infinite-population model simulations. Finally, following our earlier approach, we also evolved kernels for OneMix, obtaining again a good fit between the behaviour of the "single- organism" hill-climber and the GA.