Runtime analysis of a heavy-tailed $$(1+(\lambda,\lambda ))$$ Genetic Algorithm on Jump Functions

It was recently observed that the $$(1+(\lambda,\lambda ))$$ genetic algorithm can comparably easily escape the local optimum of the jump functions benchmark. Consequently, this algorithm can optimize the jump function with jump size k in an expected runtime of only $$n^{(k + 1)/2}k^{-k/2}e^{O(k)}$$ fitness evaluations (Antipov, Doerr, Karavaev (GECCO 2020)). This performance, however, was obtained with non-standard parameter setting depending on the jump size k. To overcome this difficulty, we propose to choose two parameters of the $$(1+(\lambda,\lambda ))$$ genetic algorithm randomly from a power-law distribution. Via a mathematical runtime analysis, we show that this algorithm with natural instance-independent choices of the power-law parameters on all jump functions with jump size at most n/4 has a performance close to what the best instance-specific parameters in the previous work obtained. This price for instance-independence can be made as small as an $$O(n\log (n))$$ factor. Given the difficulty of the jump problem and the runtime losses from using mildly suboptimal fixed parameters (also discussed in this work), this appears to be a fair price.