On the Analysis of the Simple Genetic Algorithm Pietro Oliveto 1 - PowerPoint PPT Presentation

Introduction Proof Ideas End On the Analysis of the Simple Genetic Algorithm Pietro Oliveto 1 Carsten Witt 2 1 School of Computer Science, The University of Birmingham, UK 2 Technical University of Denmark, Kgs. Lyngby, Denmark ThRaSH 2012, 2nd May 2012 1/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End Context Runtime Analysis of Evolutionary Computation (EC) has made significant progress in the last 15 years, is believed to explain the working principles of EC, is often able to capture intuitive ideas about these working principles in rigorous theorems. 2/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End Context Runtime Analysis of Evolutionary Computation (EC) has made significant progress in the last 15 years, is believed to explain the working principles of EC, is often able to capture intuitive ideas about these working principles in rigorous theorems. One Long-Term Aim of Runtime Analysis Prove rigorous statements about the working principles of “realistic”/standard EC, e. g. Genetic Algorithms with crossover. 2/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End Context Runtime Analysis of Evolutionary Computation (EC) has made significant progress in the last 15 years, is believed to explain the working principles of EC, is often able to capture intuitive ideas about these working principles in rigorous theorems. One Long-Term Aim of Runtime Analysis Prove rigorous statements about the working principles of “realistic”/standard EC, e. g. Genetic Algorithms with crossover. Is state-of-the-art runtime analysis sufficiently far developed for this? 2/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End A Challenge Runtime Analysis of Crossover-Based EC is often avoided ( → (1+1) EA), but has done with success in several cases (e.g., Real Royal Road functions, crossover in combinatorial optimization, . . . ), is an emergent issue ( → next talk(s)), definitely gives insight into the working principles of crossover. 3/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End A Challenge Runtime Analysis of Crossover-Based EC is often avoided ( → (1+1) EA), but has done with success in several cases (e.g., Real Royal Road functions, crossover in combinatorial optimization, . . . ), is an emergent issue ( → next talk(s)), definitely gives insight into the working principles of crossover. But Do these analyses study “standard GAs” on “simple standard problems”? 3/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End A Challenge Runtime Analysis of Crossover-Based EC is often avoided ( → (1+1) EA), but has done with success in several cases (e.g., Real Royal Road functions, crossover in combinatorial optimization, . . . ), is an emergent issue ( → next talk(s)), definitely gives insight into the working principles of crossover. But Do these analyses study “standard GAs” on “simple standard problems”? Not necessarily: mostly non-standard selection or non-standard problems 3/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End A Standard GA The Simple GA (SGA) for maximization of f : { 0 , 1 } n → R 1 Create population P of µ randomly chosen individuals. 2 C := ∅ . 3 While | C | < µ do Fitness-proportional selection: Select two parents x ′ and x ′′ from P proportional to their fitness without replacement. Uniform crossover: Create an offspring x by setting each bit x i = x ′ i with probability 1 / 2 and x i = x ′′ otherwise, for 1 ≤ i ≤ n . i Standard Bit Mutation: Flip each bit x i of x with probability 1 / n . C := C ∪ { x } . 4 Set P := C and go to 2. 4/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End A Standard GA The Simple GA (SGA) for maximization of f : { 0 , 1 } n → R 1 Create population P of µ randomly chosen individuals. 2 C := ∅ . 3 While | C | < µ do Fitness-proportional selection: Select two parents x ′ and x ′′ from P proportional to their fitness without replacement. Uniform crossover: Create an offspring x by setting each bit x i = x ′ i with probability 1 / 2 and x i = x ′′ otherwise, for 1 ≤ i ≤ n . i Standard Bit Mutation: Flip each bit x i of x with probability 1 / n . C := C ∪ { x } . 4 Set P := C and go to 2. Why “Standard” studied in monographs on GAs (Goldberg 1989) theoretically analyzed in the infinite-population model (Vose 1999) 4/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End The Challenge, More Precisely Can we do a runtime analysis for the Simple GA, e. g., on OneMax ? 5/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End The Challenge, More Precisely Can we do a runtime analysis for the Simple GA, e. g., on OneMax ? “It might be simple, but it is not easy” (from an amazon.com review) 5/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End The Challenge, More Precisely Can we do a runtime analysis for the Simple GA, e. g., on OneMax ? “It might be simple, but it is not easy” (from an amazon.com review) What Would We Expect 5/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End The Challenge, More Precisely Can we do a runtime analysis for the Simple GA, e. g., on OneMax ? “It might be simple, but it is not easy” (from an amazon.com review) What Would We Expect Backward drift due to mutation close to the optimum, no positive drift due to crossover, selection too weak to keep positive fluctuations. 5/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End The Challenge, More Precisely Can we do a runtime analysis for the Simple GA, e. g., on OneMax ? “It might be simple, but it is not easy” (from an amazon.com review) What Would We Expect Backward drift due to mutation close to the optimum, no positive drift due to crossover, selection too weak to keep positive fluctuations. Main New Result Let µ ≤ n 1 / 8 − ǫ for an arbitrarily small constant ǫ > 0. ( 1 , . . . , 1 ) Then with probability 1 − 2 − Ω( n ǫ/ 9 ) , the SGA on OneMax does not create individuals with more than ( 1 + c ) n 2 or less than ( 1 − c ) n 2 one-bits, for arbitrarily small constant c > 0, within the first 2 n ǫ/ 10 generations. In particular, it does not reach the optimum then. ( 0 , . . . , 0 ) 5/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End Towards A Proof Prior Work Happ/Johannsen/Klein/Neumann (GECCO 08): (1+1) EA with fitness-proportional selection needs exponential time on OneMax Neumann/Oliveto/W. (GECCO 09): same for a variant of SGA without crossover 6/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End Towards A Proof Prior Work Happ/Johannsen/Klein/Neumann (GECCO 08): (1+1) EA with fitness-proportional selection needs exponential time on OneMax Neumann/Oliveto/W. (GECCO 09): same for a variant of SGA without crossover Difficulties When Introducing Crossover Lehre’s drift theorem for populations doesn’t allow crossover 6/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End Towards A Proof Prior Work Happ/Johannsen/Klein/Neumann (GECCO 08): (1+1) EA with fitness-proportional selection needs exponential time on OneMax Neumann/Oliveto/W. (GECCO 09): same for a variant of SGA without crossover Difficulties When Introducing Crossover Lehre’s drift theorem for populations doesn’t allow crossover Variance of offspring distribution # flipping bits due to mutation Poisson-distributed → variance O ( 1 ) 6/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End Towards A Proof Prior Work Happ/Johannsen/Klein/Neumann (GECCO 08): (1+1) EA with fitness-proportional selection needs exponential time on OneMax Neumann/Oliveto/W. (GECCO 09): same for a variant of SGA without crossover Difficulties When Introducing Crossover Lehre’s drift theorem for populations doesn’t allow crossover Variance of offspring distribution # flipping bits due to mutation Poisson-distributed → variance O ( 1 ) # of one-bits created by crossover binomially distributed according to Hamming distance of parents and 1 / 2 → variance Ω( √ n ) possible 6/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA

Introduction Proof Ideas End Towards A Proof Prior Work Happ/Johannsen/Klein/Neumann (GECCO 08): (1+1) EA with fitness-proportional selection needs exponential time on OneMax Neumann/Oliveto/W. (GECCO 09): same for a variant of SGA without crossover Difficulties When Introducing Crossover Lehre’s drift theorem for populations doesn’t allow crossover Variance of offspring distribution # flipping bits due to mutation Poisson-distributed → variance O ( 1 ) # of one-bits created by crossover binomially distributed according to Hamming distance of parents and 1 / 2 → variance Ω( √ n ) possible Classical negative (“simplified”) drift theorem needs variance O ( 1 ) . 6/10 Pietro Oliveto, Carsten Witt On the Analysis of the SGA