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

Abstract We analyse the performance of well-known evolutionary algorithms, the $$(1+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> EA and the $$(1+\lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$(1+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> EA on Leading-Ones runs in polynomial expected time if $$p = O((\log n)/n^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and needs superpolynomial expected time if $$p = \omega ((\log n)/n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>(</mml:mo> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , leaving a huge gap for which no results were known. We close this gap by showing that the expected optimisation time is $$\varTheta (n^2) \cdot \exp (\varTheta (\min \{pn^2, n\}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Θ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>·</mml:mo> <mml:mo>exp</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Θ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>min</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>p</mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for all $$p \le 1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , allowing for the first time to locate the threshold between polynomial and superpolynomial expected times at $$p = \varTheta ((\log n)/n^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Θ</mml:mi> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Hence the $$(1+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> EA on Leading-Ones is surprisingly sensitive to noise. We also show that offspring populations of size $$\lambda \ge 3.42\log n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3.42</mml:mn> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> 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.