Minimal walsh structure and ordinal linkage of monotonicity-invariant function classes on bit strings

Abstract

CopyrightItems in 'OpenAIR@RGU', Robert Gordon University Open Access Institutional Repository, are protected by copyright and intellectual property law. If you believe that any material held in 'OpenAIR@RGU' infringes copyright, please contact openair-help@rgu.ac.uk with details. The item will be removed from the repository while the claim is investigated. ABSTRACTProblem structure, or linkage, refers to the interaction between variables in a black-box fitness function. Discovering structure is a feature of a range of algorithms, including estimation of distribution algorithms (EDAs) and perturbation methods (PMs). The complexity of structure has traditionally been used as a broad measure of problem difficulty, as the computational complexity relates directly to the complexity of structure. The EDA literature describes necessary and unnecessary interactions in terms of the relationship between problem structure and the structure of probabilistic graphical models discovered by the EDA. In this paper we introduce a classification of problems based on monotonicity invariance. We observe that the minimal problem structures for these classes often reveal that significant proportions of detected structures are unnecessary. We perform a complete classification of all functions on 3 bits. We consider nonmonotonicity linkage discovery using perturbation methods and derive a concept of directed ordinal linkage associated to optimization schedules. The resulting refined classification factored out by relabeling, shows a hierarchy of nine directed ordinal linkage classes for all 3-bit functions. We show that this classification allows precise analysis of computational complexity and parallelizability and conclude with a number of suggestions for future work.

Generating Easy and Hard Problems using the Proximate Optimality Principle

“…More complex algorithms will operate on the representation in more complex ways and the concept of coherence will need to be generalised. One current line of active research [1,2] is to use probabilistic graphical models on representation space to structure fitness in ways coherent with the action of algorithm operators.…”

Structural coherence of problem and algorithm: An analysis for EDAs on all 2-bit and 3-bit problems

“…In [1] it was shown that a column vector of Walsh coefficients could be computed using the Hadamard matrix. The constant term α ∅ is never necessary to maintain the ordering of function values as it is added to every output.…”

“…We adopt the convention used in [1] and write for the function class of f : C f = [r 1 , r 2 , ..., r 2 n ] where r i is the rank of the ith solution. (The solutions are ordered as binary integers.)…”

“…There are an infinite number of such functions, but if we consider only the rank of solutions (the goal of optimisation being to move towards highly-ranked solutions), there are a finite number of possible rankings. Different to the notation in [1], we define the rank of a solution x, for a function f , as the number of distinct fitness values in the population lower than f (x). In [1], there was some consideration of the separations (there referred to as deltas) between the ranks.…”

“…In previous work [1] we defined rank equivalence classes of monotonicity invariant functions as sets of all functions that rank a set of solutions identically. Restricted to problems with 2 and 3 binary variables, we are able to exhaustively evaluate all rank equivalence classes.…”