Analysing the Robustness of Evolutionary Algorithms to Noise: Refined Runtime Bounds and an Example Where Noise is Beneficial

Abstract

AbstractWe analyse the performance of well-known evolutionary algorithms, the $$(1+1)$$ ( 1 + 1 )  EA and the $$(1+\lambda )$$ ( 1 + λ )  EA, in the prior noise model, where in each fitness evaluation the search point is altered before the evaluation with probability p. We present refined results for the expected optimisation time of these algorithms on the function Leading-Ones, where bits have to be optimised in sequence. Previous work showed that the $$(1+1)$$ ( 1 + 1 )  EA on Leading-Ones runs in polynomial expected time if $$p = O((\log n)/n^2)$$ p = O ( ( log n ) / n 2 ) and needs superpolynomial expected time if $$p = \omega ((\log n)/n)$$ p = ω ( ( log n ) / n ) , leaving a huge gap for which no results were known. We close this gap by showing that the expected optimisation time is $$\varTheta (n^2) \cdot \exp (\varTheta (\min \{pn^2, n\}))$$ Θ ( n 2 ) · exp ( Θ ( min { p n 2 , n } ) ) for all $$p \le 1/2$$ p ≤ 1 / 2 , allowing for the first time to locate the threshold between polynomial and superpolynomial expected times at $$p = \varTheta ((\log n)/n^2)$$ p = Θ ( ( log n ) / n 2 ) . Hence the $$(1+1)$$ ( 1 + 1 )  EA on Leading-Ones is surprisingly sensitive to noise. We also show that offspring populations of size $$\lambda \ge 3.42\log n$$ λ ≥ 3.42 log n can effectively deal with much higher noise than known before. Finally, we present an example of a rugged landscape where prior noise can help to escape from local optima by blurring the landscape and allowing a hill climber to see the underlying gradient. We prove that in this particular setting noise can have a highly beneficial effect on performance.