Sharp bounds by probability-generating functions and variable drift

Jansen, T., Zarges, C. (2013). Performance analysis of randomised search heuristics operating with a fixed budget. Theoretical Computer Science, 545, 39-58When for a difficult real-world optimisation problem no good problem-specific algorithm is available often randomised search heuristics are used. They are hoped to deliver good solutions in acceptable time. The theoretical analysis usually concentrates on the average time needed to find an optimal or approximately optimal solution. This matches neither the application in practice nor the empirical analysis since usually optimal solutions are not known and even if found cannot be recognised. More often the algorithms are stopped after some time. This motivates a theoretical analysis to concentrate on the quality of the best solution obtained after a pre-specified number of function evaluations called budget. Using this perspective two simple randomised search heuristics, random local search and the (1+1) evolutionary algorithm, are analysed on some well-known example problems. Upper and lower bounds on the expected quality of a solution for a fixed budget of function evaluations are proven. The analysis shows novel and challenging problems in the study of randomised search heuristics. It demonstrates the potential of this shift in perspective from expected run time to expected solution quality.authorsversionPeer reviewe

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“…We point out that obtaining such precise results is also a current trend in the analysis of expected optimisation times [12,13].…”

Section: Models and Notationmentioning

confidence: 83%

Performance analysis of randomised search heuristics operating with a fixed budget

Jansen

Zarges

2014

Theoretical Computer Science

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“…Unfortunately, in our case the drift displays an uncommon behavior, being strongest for medium range pheromone values and weaker at the extremes. This makes all of the additive drift method of He and Yao, best for uniform drift, the multiplicative drift method by Doerr, Johannsen and Winzen [6], best for drift proportional to the distance from the optimum, and the variable drift method of Johannsen [23] (cited, e.g., in [5]), applicable whenever the drift is monotone with the distance from the optimum, give only bad run-time bounds. For this reason, we prove a generalization of the variable drift theorem (in a sense, of all drift theorems so far) that does not need the assumption that the drift is non-decreasing with the distance from the goal.…”

Section: Ant Colony Optimizationmentioning

confidence: 99%

Ants easily solve stochastic shortest path problems

Doerr

Hota

Kötzing

2012

Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation

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The recent theoretical analysis (Horoba, Sudholt (GECCO 2010)) of an ant colony optimizer for the stochastic shortest path problem suggests that ant system experiences significant difficulties when the input data is prone to noise. In this work, we propose a slightly different ant optimizer to deal with noise.We prove that, under mild conditions, it finds the paths with shortest expected length efficiently, despite the fact that we do not have convergence in the classic sense. To prove our results, we introduce a stronger drift theorem that can also deal with the situation that the progress is faster when one is closer to the goal.

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“…To prove lower bounds on the hitting time by variable drift, we need additional assumptions like the one in the following lemma, a special case of which was first proposed in [DFW11].…”

Section: Drift Theoremsmentioning

confidence: 99%

The Interplay of Population Size and Mutation Probability in the ( $$1+\lambda $$ 1 + λ ) EA on OneMax

Gieβen

Witt

2016

Algorithmica

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The (1+λ) EA with mutation probability c/n, where c > 0 is an arbitrary constant, is studied for the classical OneMax function. Its expected optimization time is analyzed exactly (up to lower order terms) as a function of c and λ. It turns out that 1/n is the only optimal mutation probability if λ = o(ln n ln ln n/ln ln ln n), which is the cut-off point for linear speed-up. However, if λ is above this cut-off point then the standard mutation probability 1/n is no longer the only optimal choice. Instead, the expected number of generations is (up to lower order terms) independent of c, irrespectively of it being less than 1 or greater.The theoretical results are obtained by a careful study of order statistics of the binomial distribution and variable drift theorems for upper and lower bounds. Experimental supplements shed light on the optimal mutation probability for small problem sizes.

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Sharp bounds by probability-generating functions and variable drift

Cited by 54 publications

References 16 publications

Performance analysis of randomised search heuristics operating with a fixed budget

Performance analysis of randomised search heuristics operating with a fixed budget

Ants easily solve stochastic shortest path problems

The Interplay of Population Size and Mutation Probability in the ( $$1+\lambda $$ 1 + λ ) EA on OneMax

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