Stalling in Space: Attractor Analysis for any Algorithm

TL;DR

Stalling in Space introduces attractor networks (ANs) as a temporal generalization of landscape networks to capture stalling regions in optimization trajectories. ANs define attractors as locations where an algorithm stalls for at least $\beta$ evaluations and are constructed from logs of improvements, enabling analysis for any algorithm. The study builds ANs for CMA-ES, differential evolution, and random search on the 24 BBOB noiseless functions and compares them to Local Optima Networks (LONs) and Search Trajectory Networks (STNs), revealing attractor structures overlooked by traditional models. Key findings show that ANs produce sparser, temporally informative representations that persist across high $\beta$ values and expose intermediate attractors not visible in LON or STN analyses, offering a new lens on algorithm dynamics. The work highlights potential extensions to multimodal and discrete spaces and provides code and data for reproducibility.

Abstract

Network-based representations of fitness landscapes have grown in popularity in the past decade; this is probably because of growing interest in explainability for optimisation algorithms. Local optima networks (LONs) have been especially dominant in the literature and capture an approximation of local optima and their connectivity in the landscape. However, thus far, LONs have been constructed according to a strict definition of what a local optimum is: the result of local search. Many evolutionary approaches do not include this, however. Popular algorithms such as CMA-ES have therefore never been subject to LON analysis. Search trajectory networks (STNs) offer a possible alternative: nodes can be any search space location. However, STNs are not typically modelled in such a way that models temporal stalls: that is, a region in the search space where an algorithm fails to find a better solution over a defined period of time. In this work, we approach this by systematically analysing a special case of STN which we name attractor networks. These offer a coarse-grained view of algorithm behaviour with a singular focus on stall locations. We construct attractor networks for CMA-ES, differential evolution, and random search for 24 noiseless black-box optimisation benchmark problems. The properties of attractor networks are systematically explored. They are also visualised and compared to traditional LONs and STN models. We find that attractor networks facilitate insights into algorithm behaviour which other models cannot, and we advocate for the consideration of attractor analysis even for algorithms which do not include local search.

Figures (7)

• Figure 1: Attractor networks for CMA on 2D Gallagher function f21 across different values of $\beta$.
• Figure 2: Attractor networks [$\epsilon$=$10^{-5}$] for CMA on the 10-dimensional Sphere function $f1$ across different values of $\beta$. Node size is proportional to the number of runs which reached that location
• Figure 3: Attractor networks for 10D Schaffer function f17, $\beta$ = 320, $\epsilon$=$10^{-5}$ for different algorithms
• Figure 4: 10D ANs [constructed with 30 runs] number of nodes with increasing $\beta$ and various $\epsilon$.
• Figure 5: The median proportion [over 24 functions] of node matches [i.e. fraction of mutual solutions] between CMA (2D) attractor networks with different network construction settings $\beta$ and $\epsilon$. In the plot labels, the integer part is the setting for $\beta$ and the float is for $\epsilon$