Frequency Fitness Assignment: Making Optimization Algorithms Invariant under Bijective Transformations of the Objective Function Value

Abstract

Under Frequency Fitness Assignment (FFA), the fitness corresponding to an objective value is its encounter frequency in fitness assignment steps and is subject to minimization. FFA renders optimization processes invariant under bijective transformations of the objective function value. On TwoMax, Jump, and Trap functions of dimension s, the classical (1+1)-EA with standard mutation at rate 1/s can have expected runtimes exponential in s. In our experiments, a (1+1)-FEA, the same algorithm but using FFA, exhibits mean runtimes that seem to scale as $s^2\ln{s}$. Since Jump and Trap are bijective transformations of OneMax, it behaves identical on all three. On OneMax, LeadingOnes, and Plateau problems, it seems to be slower than the (1+1)-EA by a factor linear in s. The (1+1)-FEA performs much better than the (1+1)-EA on W-Model and MaxSat instances. We further verify the bijection invariance by applying the Md5 checksum computation as transformation to some of the above problems and yield the same behaviors. Finally, we show that FFA can improve the performance of a memetic algorithm for job shop scheduling.