Escaping Local Optima using Crossover with Emergent or Reinforced Diversity

Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous run time analysis to gain insight into population dynamics and GA performance for the (μ+1) GA and the Jumpk test function. We show that the interplay of crossover followed by mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to improvements of the expected optimisation time of order Ω(n/ logn) compared to mutation-only algorithms like the (1+1) EA. Moreover, increasing the mutation rate by an arbitrarily small constant factor can facilitate the generation of diversity, leading to speedups of order Ω(n). We also compare seven commonly used diversity mechanisms and evaluate their impact on run time bounds for the (μ+1) GA. All previous results in this context only hold for unrealistically low crossover probability pc = O(k/n), while we give analyses for the setting of constant pc < 1 in all but one case. For the typical case of constant k > 2 and constant pc, we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of μ: • O ( nk−1 ) for duplicate elimination/minimisation, • O ( n logn ) for maximising the convex hull, • O(n logn) for deterministic crowding (assuming pc = k/n), • O(n logn) for maximising the Hamming distance, • O(n logn) for fitness sharing, • O(n logn) for the single-receiver island model. This proves a sizeable advantage of all variants of the (μ+1) GA compared to the (1+1) EA, which requires time Θ(n). Experiments complement our theoretical findings and further highlight the benefits of crossover and diversity on Jumpk. 1 ar X iv :1 60 8. 03 12 3v 1 [ cs .N E ] 1 0 A ug 2 01 6