Analysis of the Performance of Algorithm Configurators for Search Heuristics with Global Mutation Operators

💡 Research Summary

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This paper provides a rigorous theoretical analysis of algorithm configurators (also known as parameter tuners) when they are used to tune the mutation rate of the standard bit‑mutation (SBM) operator in the (1+1) Evolutionary Algorithm (EA). The SBM operator flips each bit of an n‑bit string independently with probability χ/n, where χ is a continuous parameter to be optimized. The authors focus on two well‑studied benchmark problem classes: Ridge and LeadingOnes. For Ridge the optimal mutation rate is always χ = 1 (i.e., mutation probability 1/n) regardless of the current search point, whereas for LeadingOnes the optimal static rate that minimizes the expected runtime of the (1+1) EA is approximately χ ≈ 1.59.

The study distinguishes two performance metrics that configurators may use to compare candidate configurations: (i) Optimization Time, which measures the number of iterations required to reach the optimum (with a penalty if the cutoff κ is exceeded), and (ii) Best Fitness, which records the highest fitness attained within a fixed cutoff κ. The authors analyse how the choice of metric influences the required cutoff time κ and the total number of configuration comparisons T needed for a simple configurator, ParamRLS, and its variant ParamRLS‑F (which uses Best Fitness).

Key theoretical contributions

1. Lower bound for Optimization‑Time‑based configurators – For any configurator that relies on Optimization Time, the authors prove that the cutoff κ must be at least as large as the expected optimization time of the optimal configuration. If κ is smaller (specifically κ ≤ (1 − ε)·e·n² for Ridge and κ ≤ 0.772075·n² for LeadingOnes), the configurator will, with overwhelming probability (1 − exp(−Ω(n^α)) for some constant α), return a uniformly random χ from the search space rather than the optimal value. This result holds regardless of the number of runs per comparison r, even when r is polynomial in n.

2. Upper bound for Best‑Fitness‑based configurators – When the Best Fitness metric is used, the situation changes dramatically. The authors show that the simple hill‑climbing configurator ParamRLS‑F can identify the optimal χ = 1 for Ridge with any cutoff κ that grows only linearly with n (κ = Θ(n)). For LeadingOnes, ParamRLS‑F succeeds in finding the optimal static rate χ ≈ 1.6 for any κ ≥ 0.72·n², which is roughly 5 % of the expected runtime of the (1+1) EA with the optimal rate. Moreover, for more than 99 % of cutoff values in the interval