Monotonic functions in EC

Abstract

To understand how evolutionary algorithms optimize the simple class of monotonic functions, Jansen (FOGA 2007) introduced the partially-ordered evolutionary algorithm (PO-EA) model and analyzed its runtime. The PO-EA is a pessimistic model of the true optimization process, hence performance guarantees for it immediately take over to the true optimization process.Based on the observation that Jansen's model leads to a process more pessimistic than what any monotonic function would, we extend his model by parameterizing the degree of pessimism. For all degrees of pessimism, and all mutation rates c/n, we give a precise runtime analysis of this process. For all degrees of pessimism lower than that of Jansen, we observe a Θ(n log n) runtime for the standard mutation probability of 1/n. However, we also observe a strange double-jump behavior in terms of the mutation probability. For all non-zero degrees of pessimism, there is a threshold c ∈ R such that (i) for mutation rates c /n with c < c we have a Θ(n log n) runtime, (ii) for the mutation rate c/n we have a runtime of Θ(n 3/2 ), and (iii) for mutation rates c /n with c > c we have an exponential runtime.To overcome the complicated interplay of mutation and selection in the PO-EA, by define artificial algorithms which provably (via a coupling argument) have the same asymptotic runtime, but allow a much easier computation of the drift towards the optimum.

Two-Dimensional Drift Analysis: Optimizing Two Functions Simultaneously Can Be Hard

“…Our positive result overlaps with the result of [29]. 2 He gives a sufficient condition for fast convergence that works in arbitrary dimensions: convergence is always fast if the matrix A has an eigenvector which contains only real, positive entries, and which has a corresponding positive real eigenvalue. In two dimensions with b, c < 0 and a, d > 0, we show in Lemma 2 that this is the case if and only if both eigenvalues of A are positive.…”

“…In particular, the failure mode is not related to the trivial problems that occur for extremely large mutation rates, χ log n, where the algorithm fails to produce neighbors in Hamming distance one [32]. Subsequently, the results on how the mutation rate and related parameters affect optimization of monotone functions have been refined [2,20,24] and extended to a large collection of other EAs [18,26]. To highlight just one result, the (μ + 1)-EA with standard mutation rate 1/n fails on some monotone functions if the population size μ is too large (but still constant in n) [26].…”

“…These environments are very simple, so they allow us to study failure modes in greater detail. Crucially, failure in such environments is due to the algorithms, not the problems: they all fall within a general class of problems introduced by Jansen [13] and called partially-ordered EA (PO-EA) by Colin, Doerr and Ferey [2], which can be solved efficiently by Random Local Search (RLS) and the (1 + 1)-EA. More precisely, the (1 + 1)-EA with mutation rate χ /n is known to have optimization time O (n log n) for χ < 1, and O (n 3/2 ) for χ = 1.…”

Unified Approach to (1 + 1) Evolutionary Algorithm on Discrete Linear Functions

“…Colin et al [5] have shown that PO-EA can be divided into two parts. Ma et al [6] have named it PO-mutation and ZeroMax models.…”

“…The Partial Order Evolutionary Algorithm (PO-EA) model introduced by Jansen [3] contributes to the analysis of performance in linear functions and is expected to simulate the optimization of monotonic functions [4,5]. PO-EA is a pessimistic model of the true optimization process, which can be used to derive an upper bound on the expected hitting time of monotonic functions.…”