Anytime Analysis on BinVal: Adaptive Parameters Help

TL;DR

This paper studies the anytime performance of evolutionary algorithms on BinVal, showing that adaptive parameter strategies can significantly improve run times across all target precisions.

Contribution

It introduces an analysis of fixed-target run times for various algorithms on BinVal, demonstrating benefits of adaptive parameters over fixed ones.

Findings

Standard (1+1) EA has fixed-target run time in Θ(n log k) for all k in o(n).

Using an EDA, the expected evaluations are Θ(k log n).

Self-adjusting mutation rates yield run times in O(k^{1+ε}), independent of n.

Abstract

While most theoretical run time analyses of discrete randomized search heuristics provide bounds on the expected number of evaluations to find the global optimum, we consider the anytime performance of evolutionary and estimation-of-distribution algorithms. For this purpose, we analyze the fixed-target run time of various algorithms using BinVal as fitness function and bound the run time to optimize the most significant $k \in o(n)$ bits of a bit string with length $n$. We analyze the run times such that they hold not only for a fixed $k$, but simultaneously for all $k \in o(n)$. For the standard (1+1) EA with fixed mutation rate $1/n$, we show that the fixed-target run time for all $k \in o(n)$ is in $\Theta(n \log k)$. Using an EDA instead, we get an expected number of evaluations of $\Theta(k \log n)$ for the sig-cGA. Replacing in the standard (1+1) EA the fixed mutation rate with a self-adjusting rate, we show that the fixed-target run time for $k \in o(n)$ and a constant $\varepsilon >0$ arbitrarily close to zero is in $\mathcal{O}\left(k^{1+\varepsilon}\right)$ for this algorithm. In particular, this run time is independent of $n$, holds simultaneously for all $k \in o(n)$, and is close to the run time of $\Theta(k \log k)$ for the (1+1) EA with the best fixed mutation rate if $k$ is known.