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

We 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.