Theoretical and Empirical Analysis of Parameter Control Mechanisms in the (1 + (λ, λ)) Genetic Algorithm

The self-adjusting (1 + (λ, λ)) GA is the best known genetic algorithm for problems with a good fitness-distance correlation as in OneMax. It uses a parameter control mechanism for the parameter λ that governs the mutation strength and the number of offspring. However, on multimodal problems, the parameter control mechanism tends to increase λ uncontrollably. We study this problem for the standard Jumpk benchmark problem class using runtime analysis. The self-adjusting (1 + (λ, λ)) GA behaves like a (1 + n) EA whenever the maximum value for λ is reached. This is ineffective for problems where large jumps are required. Capping λ at smaller values is beneficial for such problems. Finally, resetting λ to 1 allows the parameter to cycle through the parameter space. We show that resets are effective for all Jumpk problems: the self-adjusting (1 + (λ, λ)) GA performs as well as the (1 + 1) EA with the optimal mutation rate and evolutionary algorithms with heavy-tailed mutation, apart from a small polynomial overhead. Along the way, we present new general methods for translating existing runtime bounds from the (1 + 1) EA to the self-adjusting (1 + (λ, λ)) GA. We also show that the algorithm presents a bimodal parameter landscape with respect to λ on Jumpk. For appropriate n and k, the landscape features a local optimum in a wide basin of attraction and a global optimum in a narrow basin of attraction. To our knowledge this is the first proof of a bimodal parameter landscape for the runtime of an evolutionary algorithm on a multimodal problem.