Stalling in Space: Attractor Analysis for any Algorithm

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.

A Deep Dive into Effects of Structural Bias on CMA-ES Performance along Affine Trajectories

To guide the design of better iterative optimisation heuristics, it is imperative to understand how inherent structural biases within algorithm components affect the performance on a wide variety of search landscapes. This study explores the impact of structural bias in the modular Covariance Matrix Adaptation Evolution Strategy (modCMA), focusing on the roles of various modulars within the algorithm. Through an extensive investigation involving 435,456 configurations of modCMA, we identified key modules that significantly influence structural bias of various classes. Our analysis utilized the Deep-BIAS toolbox for structural bias detection and classification, complemented by SHAP analysis for quantifying module contributions. The performance of these configurations was tested on a sequence of affine-recombined functions, maintaining fixed optimum locations while gradually varying the landscape features. Our results demonstrate an interplay between module-induced structural bias and algorithm performance across different landscape characteristics.

Understanding fitness landscapes in morpho-evolution via local optima networks

Morpho-evolution (ME) refers to the simultaneous optimisation of a robot's design and controller to maximise performance given a task and environment. Many genetic encodings have been proposed which are capable of representing design and control. Previous research has provided empirical comparisons between encodings in terms of their performance with respect to an objective function and the diversity of designs that are evaluated, however there has been no attempt to explain the observed findings. We address this by applying Local Optima Network (LON) analysis to investigate the structure of the fitness landscapes induced by three different encodings when evolving a robot for a locomotion task, shedding new light on the ease by which different fitness landscapes can be traversed by a search process. This is the first time LON analysis has been applied in the field of ME despite its popularity in combinatorial optimisation domains; the findings will facilitate design of new algorithms or operators that are customised to ME landscapes in the future.

Information flow and Laplacian dynamics on local optima networks

We propose a new way of looking at local optima networks (LONs). LONs represent fitness landscapes; the nodes are local optima, and the edges are search transitions between them. Many metrics computed on LONs have been proposed and shown to be linked to metaheuristic search difficulty. These have typically considered LONs as describing static structures. In contrast to this, Laplacian dynamics (LD) is an approach to consider the information flow across a network as a dynamical process. We adapt and apply LD to the context of LONs. As a testbed, we consider instances from the quadratic assignment problem (QAP) library. Metrics related to LD are proposed and these are compared with existing LON metrics. The results show that certain LD metrics are strong predictors of metaheuristic performance for iterated local search and tabu search.