Blending Dynamic Programming with Monte Carlo Simulation for Bounding the Running Time of Evolutionary Algorithms
This work addresses the need for more general theoretical bounds in evolutionary computation, though it is incremental as it builds on prior dynamic programming methods.
The authors tackled the problem of providing absolute lower bounds for the running times of evolutionary algorithms by extending a dynamic programming approach to handle problems where transition probabilities are approximated via Monte Carlo simulation, applying it to a concatenated jump function to analyze parameter control schemes.
With the goal to provide absolute lower bounds for the best possible running times that can be achieved by $(1+λ)$-type search heuristics on common benchmark problems, we recently suggested a dynamic programming approach that computes optimal expected running times and the regret values inferred when deviating from the optimal parameter choice. Our previous work is restricted to problems for which transition probabilities between different states can be expressed by relatively simple mathematical expressions. With the goal to cover broader sets of problems, we suggest in this work an extension of the dynamic programming approach to settings in which the transition probabilities cannot necessarily be computed exactly, but in which they can be approximated numerically, up to arbitrary precision, by Monte Carlo sampling. We apply our hybrid Monte Carlo dynamic programming approach to a concatenated jump function and demonstrate how the obtained bounds can be used to gain a deeper understanding into parameter control schemes.