Blending Dynamic Programming with Monte Carlo Simulation for Bounding the Running Time of Evolutionary Algorithms

TL;DR

This paper introduces a hybrid approach combining dynamic programming and Monte Carlo sampling to estimate lower bounds on the running times of evolutionary algorithms, enabling analysis of more complex problems.

Contribution

It extends previous dynamic programming methods to approximate transition probabilities via Monte Carlo sampling, broadening applicability to complex benchmark problems.

Findings

Successfully applied to a concatenated jump function

Provided bounds that enhance understanding of parameter control schemes

Demonstrated the method's effectiveness in complex problem settings

Abstract

With the goal to provide absolute lower bounds for the best possible running times that can be achieved by $(1+\lambda)$-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.