( 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 O NE M AX Mutation rate is r / n where r is a constant. The expected runtime is (Giessen, Witt(2016)) + e r � 1 � 2 · n ln ln λ r · n ln n ( 1 ± o ( 1 )) ln λ λ r = 1 gives the asymptotically best runtime for λ not too large. Jing Yang (LIX) Dagsthul May 11, 2017 2 / 9
( 1 + λ ) EA with Dynamic Parameter on O NE M AX k denotes the fitness distance n − O M ( x ) . λ satisfies ln ( λ ) ≤ √ n . n ln ( en / k ) , 1 ln λ Mutation rate is p = max { n } . The expected runtime is (Badkobeh, Lehre, Sudholt(2014)) � log λ + n log n n � O λ Jing Yang (LIX) Dagsthul May 11, 2017 3 / 9
Self-adaptive ( 1 + λ ) EA Let population size satisfies λ = ω ( 1 ) and λ = n O ( 1 ) . Mutation rate r / n ∈ [ 2/ n , 1/4 ] . Perform both r /2 and 2 r in each iteration. The mutation rate is adjusted by one of the following rules: greedy selection random decision Jing Yang (LIX) Dagsthul May 11, 2017 4 / 9
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 ← r init ; for t ← 1, 2, . . . do Create x i by flipping each bit in a copy of x independently with probability r t / ( 2 n ) if 1 ≤ i ≤ λ /2 and with probability 2 r t / n if λ /2 < i ≤ λ ; x ∗ ← arg min x i f ( x i ) ; if f ( x ∗ ) ≤ f ( x ) then x ← x ∗ ; Perform one of the following two actions with prob. 1/2: Replace r t with the rate that x ∗ has been created with. Replace r t with either r t /2 or 2 r t , each with probability 1/2. r t ← min { max { 2, r t } , n /4 } . Jing Yang (LIX) Dagsthul May 11, 2017 5 / 9
Far Region Let c 1 ( k ) = ( 2 ln ( en / k )) − 1 and c 2 ( k ) = 4 n 2 / ( n − 2 k ) 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 [ c 1 ( k ) ln n , c 2 ( k ) ln n ] . If r ∈ [ c 1 ( k ) ln λ , c 2 ( k ) ln ( λ ) /200 ] for n /2 > k ≥ 2 n /5 and r ∈ [ c 1 ( k ) ln λ , ln ( λ ) /2 ] for n / ln λ ≤ k < 2 n /5, then E ( ∆ ) ≥ 0.05 ln ( λ ) / ln ( en / k ) . ( 1 + λ ) EA needs O ( n / log λ ) generations in expectation to reach a O M -value of k ≤ n / ln λ . Jing Yang (LIX) Dagsthul May 11, 2017 6 / 9
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 √ � � 1 λ k E ( ∆ ) ≥ ( 1 − o ( 1 )) min 2, . 8 n The expected time to cross this region is O ( n / log λ ) Jing Yang (LIX) Dagsthul May 11, 2017 7 / 9
Near Region Near region is the region where k ≤ n / λ . If 3 ≤ r t ≤ ln ( λ ) /2 in this region, the probability that r t + 1 = r t /2 is at least 0.51. The expected number of generations until the optimum is reached is O ( n log ( n ) / λ + log n ) . Jing Yang (LIX) Dagsthul May 11, 2017 8 / 9
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 = n O ( 1 ) , the bound is dominated by � log λ + n log ( n ) � n O λ Jing Yang (LIX) Dagsthul May 11, 2017 9 / 9
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