Far Region Let c 1 ( k ) = ( 2 ln ( en / k )) 1 and c 2 ( k ) = 4 n - - PowerPoint PPT Presentation

(1 + λ) Evolutionary Algorithm with

Self-Adjusting Mutation Rate

Jing Yang joint work with Benjamin Doerr, Christian Gießen and Carsten Witt

Ecole Polytechnique

May 11, 2017

Jing Yang (LIX) Dagsthul May 11, 2017 1 / 9

(1 + λ) EA with Static Parameter on ONEMAX

Mutation rate is r/n where r is a constant. The expected runtime is (Giessen, Witt(2016)) (1 ± o(1)) 1 2 · n ln ln λ ln λ + er r · n ln n λ

• r = 1 gives the asymptotically best runtime for λ not too large.

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(1 + λ) EA with Dynamic Parameter on ONEMAX

k denotes the fitness distance n − OM(x). λ satisfies ln(λ) ≤ √n. Mutation rate is p = max{

ln λ n ln(en/k), 1 n}.

The expected runtime is (Badkobeh, Lehre, Sudholt(2014)) O

• n

log λ + n log n λ

• Jing Yang (LIX)

Dagsthul May 11, 2017 3 / 9

Self-adaptive (1 + λ) EA

Let population size satisfies λ = ω(1) and λ = nO(1). Mutation rate r/n ∈ [2/n, 1/4]. Perform both r/2 and 2r in each iteration. The mutation rate is adjusted by one of the following rules:

greedy selection random decision

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Structure of Self-adpative (1 + λ) EA

Algorithm 1 (1 + λ) EA with two-rate standard bit mutation Select x uniformly at random from {0, 1}n and set r ← rinit; for t ← 1, 2, . . . do Create xi by flipping each bit in a copy of x independently with probability rt/(2n) if 1 ≤ i ≤ λ/2 and with probability 2rt/n if λ/2 < i ≤ λ; x∗ ← arg minxi f (xi); if f (x∗) ≤ f (x) then x ← x∗; Perform one of the following two actions with prob. 1/2: Replace rt with the rate that x∗ has been created with. Replace rt with either rt/2 or 2rt, each with probability 1/2. rt ← min{max{2, rt}, n/4}.

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Far Region

Let c1(k) = (2 ln(en/k))−1 and c2(k) = 4n2/(n − 2k)2. We use ∆ := ∆(λ/2, k, r) to denote the fitness gain after selection among the best of λ/2 offspring. Far region is the region where k ≥ n/ ln λ. The rate r is attracted to the interval [c1(k) ln n, c2(k) ln n]. If r ∈ [c1(k) ln λ, c2(k) ln(λ)/200] for n/2 > k ≥ 2n/5 and r ∈ [c1(k) ln λ, ln(λ)/2] for n/ ln λ ≤ k < 2n/5, then E(∆) ≥ 0.05 ln(λ)/ ln(en/k). (1 + λ) EA needs O(n/ log λ) generations in expectation to reach a OM-value of k ≤ n/ ln λ.

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Middle Region

Middle region is the region where n/λ ≤ k ≤ n/ ln λ. If 1 ≤ r ≤ ln(λ)/2 in middle region, the expected fitness gain E(∆) ≥ (1 − o(1)) min

• 1

2, √ λk 8n

• .

The expected time to cross this region is O(n/ log λ)

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Near Region

Near region is the region where k ≤ n/λ. If 3 ≤ rt ≤ ln(λ)/2 in this region, the probability that rt+1 = rt/2 is at least 0.51. The expected number of generations until the optimum is reached is O(n log(n)/λ + log n).

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Conlcusion

Far region: O(n/ log λ) Middle region: O(n/ log λ) Near region: O(n log(n)/λ + log n) In total: O(n/ log λ + n log(n)/λ + log n) Use assumption λ = nO(1), the bound is dominated by O

• n

log λ + n log(n) λ

• Jing Yang (LIX)

Dagsthul May 11, 2017 9 / 9