How the (1+λ) evolutionary algorithm optimizes linear functions

Selection hyper-heuristics are randomised search methodologies which choose and execute heuristics from a set of low-level heuristics. Recent time complexity analyses for the L O benchmark function have shown that the standard simple random, permutation, random gradient, greedy and reinforcement learning selection mechanisms show no e ects of learning. e idea behind the learning mechanisms is to continue to exploit the currently selected heuristic as long as it is successful. However, the probability that a promising heuristic is successful in the next step is relatively low when perturbing a reasonable solution to a combinatorial optimisation problem. In this paper we generalise the classical selection-perturbation mechanisms so success can be measured over some xed period of length τ , rather than in a single iteration. We present a benchmark function where it is necessary to learn to exploit a particular low-level heuristic, rigorously proving that it makes the di erence between an e cient and an ine cient algorithm. For L O we prove that the generalised random gradient mechanism approaches optimal performance while generalised greedy, although not as fast, still outperforms random local search. An experimental analysis shows that combining the two generalised mechanisms leads to even be er performance.

show abstract

“…, and by applying Wald's equation [9,20], using E (X j ) to denote an upper bound on all E (X j,k ) in stage j:…”

Section: Generalised Mechanismsmentioning

confidence: 99%

On the runtime analysis of generalised selection hyper-heuristics for pseudo-boolean optimisation

Lissovoi

Oliveto

Warwicker

2017

Proceedings of the Genetic and Evolutionary Computation Conference

View full text Add to dashboard Cite

show abstract

“…Lemma 5 (extension of Wald's equation from [6]). Let T be a random variable with bounded expectation and let X1, X2, .…”

Section: Proof Of Theoremmentioning

confidence: 99%

The impact of random initialization on the runtime of randomized search heuristics

Doerr

2014

Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation

Self Cite

View full text Add to dashboard Cite

International audienceIt has often been observed that the expected runtime of an evolutionary algorithm with random initialization does not deviate much from the expected runtime when starting in an initial solution of average fitness. Having this information a priori would greatly simplify the runtime analysis for the algorithm using random initialization. We prove such a result for the optimization of the OneMax test function via the two randomized search heuristics Randomized Local Search (RLS) and the (1+1) Evolutionary Algorithm. For both algorithms, we show that the expected runtime from a random initial solution deviates at most by a constant number of iterations from the expected runtime when starting with a solution having exactly n/2 ones.For RLS we can precisely compute that this constant is -1/2 ± o(1). This leads to an extremely precise bound for the expected runtime. The expected number of fitness evaluations until an optimal search point is found, is n Hn/2 - 1/2 ± o(1), where Hn/2 denotes the (n/2)th harmonic number when n is even, and Hn/2:= (H⌊ n/2 ⌋ + H⌈ n/2 ⌉)/2 when n is odd.The main technique to obtain these results is a coupling of the optimization process starting from different fitness levels. We believe this technique to be interesting also much beyond the specific results mentioned above; e.g., for the study of other optimization problems

show abstract

“…Rowe and Sudholt [17] discussed the running time of (1 + λ) EAs in terms of the offspring population size on unimodal functions. Doerr and Künnemann [18] analysed the time bound of (1 + λ) EAs for optimizing linear pseudo-Boolean functions. Doerr and Künnemann [19] showed that (1 + λ) EAs with even very large offspring populations does not reduce the runtime significantly on the Royal Road function.…”

Section: Introductionmentioning

confidence: 99%

Average Drift Analysis and Population Scalability

Yao

2016

IEEE Trans. Evol. Computat.

View full text Add to dashboard Cite

This paper aims to study how the population size affects the computation time of evolutionary algorithms in a rigorous way. The computation time of evolutionary algorithms can be measured by either the number of generations (hitting time) or the number of fitness evaluations (running time) to find an optimal solution. Population scalability is the ratio of the expected hitting time between a benchmark algorithm and an algorithm using a larger population size. Average drift analysis is introduced to compare the expected hitting time of two algorithms and to estimate lower and upper bounds on the population scalability. Several intuitive beliefs are rigorously analysed. It is proven that (1) using a population sometimes increases rather than decreases the expected hitting time; (2) using a population cannot shorten the expected running time of any elitist evolutionary algorithm on any unimodal function on the time-fitness landscape, however this statement is not true in terms of the distance-based fitness landscape; (3) using a population cannot always reduce the expected running time on deceptive functions, which depends on whether the benchmark algorithm uses elitist selection or random selection.authorsversionPeer reviewe

show abstract

How the (1+λ) evolutionary algorithm optimizes linear functions

Cited by 14 publications

References 35 publications

On the runtime analysis of generalised selection hyper-heuristics for pseudo-boolean optimisation

On the runtime analysis of generalised selection hyper-heuristics for pseudo-boolean optimisation

The impact of random initialization on the runtime of randomized search heuristics

Average Drift Analysis and Population Scalability

Contact Info

Product

Resources

About