Bounding Bloat in Genetic Programming

TL;DR

This paper provides theoretical bounds on bloat in mutation-based genetic programming, analyzing its impact on optimization time for specific test functions and overcoming previous assumptions about bloat magnitude.

Contribution

It introduces new bounds for bloat in GP, removing prior assumptions and analyzing both with and without bloat control for the ORDER and MAJORITY functions.

Findings

Bounded expected optimization time with bloat control: Θ(T_init + n log n).

Unbounded optimization time without bloat control: O(T_init log T_init + n (log n)^3).

Established matching or close bounds for both scenarios.

Abstract

While many optimization problems work with a fixed number of decision variables and thus a fixed-length representation of possible solutions, genetic programming (GP) works on variable-length representations. A naturally occurring problem is that of bloat (unnecessary growth of solutions) slowing down optimization. Theoretical analyses could so far not bound bloat and required explicit assumptions on the magnitude of bloat. In this paper we analyze bloat in mutation-based genetic programming for the two test functions ORDER and MAJORITY. We overcome previous assumptions on the magnitude of bloat and give matching or close-to-matching upper and lower bounds for the expected optimization time. In particular, we show that the (1+1) GP takes (i) $\Theta(T_{init} + n \log n)$ iterations with bloat control on ORDER as well as MAJORITY; and (ii) $O(T_{init} \log T_{init} + n (\log n)^3)$ and $\Omega(T_{init} + n \log n)$ (and $\Omega(T_{init} \log T_{init})$ for $n=1$) iterations without bloat control on MAJORITY.