Crossover Can Provably Speed Up Evolutionary Computation Benjamin - PowerPoint PPT Presentation

Crossover Can Provably Speed Up Evolutionary Computation Benjamin Doerr, Edda Happ, Christian Klein January 29, 2008 mpi-logo

State-of-the-Art Praxis Bio-inspired-ness: Use both mutation and crossover! Practise shows that often crossover improves runtime. mpi-logo

State-of-the-Art Praxis Bio-inspired-ness: Use both mutation and crossover! Practise shows that often crossover improves runtime. ...and Theory Little proof that crossover is useful. Most theoretical analyses: only mutation. a a E.g.: Pietro S. Oliveto, J. He, X. Yao. Evolutionary Algorithms and the Vertex Cover Problem. In Proceedings of the IEEE Congress on Evolutionary Computation (CEC’07), 2007. mpi-logo

State-of-the-Art Pseudo-boolean jump Function jump function j m : { 0 , 1 } n → R m + � x i if � x i ≤ n − m   if � x i = n  j m ( x 1 , . . . , x n ) = m + n n − � x i  else  T. Jansen and I. Wegener. On the analysis of evolutionary algorithms - a proof that crossover really can help. ESA 1999, Algorithmica 2002 mpi-logo

State-of-the-Art Pseudo-boolean jump Function jump function j m : { 0 , 1 } n → R m + � x i if � x i ≤ n − m   if � x i = n  j m ( x 1 , . . . , x n ) = m + n n − � x i  else  Result Mutation only: Expected optimization time Θ( n m ). T. Jansen and I. Wegener. On the analysis of evolutionary algorithms - a proof that crossover really can help. ESA 1999, Algorithmica 2002 mpi-logo

State-of-the-Art Pseudo-boolean jump Function jump function j m : { 0 , 1 } n → R m + � x i if � x i ≤ n − m   if � x i = n  j m ( x 1 , . . . , x n ) = m + n n − � x i  else  Result Mutation only: Expected optimization time Θ( n m ). Mutation and crossover: Expected optimization time O ( n 2 log n ). T. Jansen and I. Wegener. On the analysis of evolutionary algorithms - a proof that crossover really can help. ESA 1999, Algorithmica 2002 mpi-logo

State-of-the-Art Simplified Ising Problem Given a tree T = ( V , E ). Assign x i ∈ {− 1 , +1 } to the vertices maximizing � x i x j . ( v i , v j ) ∈ E mpi-logo D. Sudholt. Crossover is provably essential for the ising model on trees. GECCO 2005

State-of-the-Art Simplified Ising Problem Given a tree T = ( V , E ). Assign x i ∈ {− 1 , +1 } to the vertices maximizing � x i x j . ( v i , v j ) ∈ E ◮ Find a monochromatic coloring! mpi-logo D. Sudholt. Crossover is provably essential for the ising model on trees. GECCO 2005

State-of-the-Art Simplified Ising Problem Given a tree T = ( V , E ). Assign x i ∈ {− 1 , +1 } to the vertices maximizing � x i x j . ( v i , v j ) ∈ E ◮ Find a monochromatic coloring! Result Only mutation: Optimization time exponential. mpi-logo D. Sudholt. Crossover is provably essential for the ising model on trees. GECCO 2005

State-of-the-Art Simplified Ising Problem Given a tree T = ( V , E ). Assign x i ∈ {− 1 , +1 } to the vertices maximizing � x i x j . ( v i , v j ) ∈ E ◮ Find a monochromatic coloring! Result Only mutation: Optimization time exponential. mpi-logo D. Sudholt. Crossover is provably essential for the ising model on trees. GECCO 2005

State-of-the-Art Simplified Ising Problem Given a tree T = ( V , E ). Assign x i ∈ {− 1 , +1 } to the vertices maximizing � x i x j . ( v i , v j ) ∈ E ◮ Find a monochromatic coloring! Result Only mutation: Optimization time exponential. Mutation, fitness sharing and “1”-point crossover: O ( n 3 ) optimization time. mpi-logo D. Sudholt. Crossover is provably essential for the ising model on trees. GECCO 2005

State-of-the-Art Simplified Ising Problem Given a tree T = ( V , E ). Assign x i ∈ {− 1 , +1 } to the vertices maximizing � x i x j . ( v i , v j ) ∈ E ◮ Find a monochromatic coloring! Result Only mutation: Optimization time exponential. Mutation, fitness sharing and “1”-point crossover: O ( n 3 ) optimization time. mpi-logo D. Sudholt. Crossover is provably essential for the ising model on trees. GECCO 2005

State-of-the-Art Summary Existing Results Some proofs that crossover helps mpi-logo

State-of-the-Art Summary Existing Results Some proofs that crossover helps Artificial problems only mpi-logo

State-of-the-Art Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique mpi-logo

State-of-the-Art Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation mpi-logo

State-of-the-Art Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation need shared fitness mpi-logo

State-of-the-Art, our Results Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation need shared fitness mpi-logo

State-of-the-Art, our Results Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation need shared fitness Our Result Rigorous analysis of the All Pairs Shortest Path problem. mpi-logo

State-of-the-Art, our Results Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation need shared fitness Our Result Rigorous analysis of the All Pairs Shortest Path problem. Mutation only: Expected optimization time Θ( n 4 ) mpi-logo

State-of-the-Art, our Results Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation need shared fitness Our Result Rigorous analysis of the All Pairs Shortest Path problem. Mutation only: Expected optimization time Θ( n 4 ) Mutation and crossover: Exp. opt. time O ( n 3 . 5 √ log n ) mpi-logo

State-of-the-Art, our Results Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation need shared fitness Our Result Rigorous analysis of the All Pairs Shortest Path problem. Mutation only: Expected optimization time Θ( n 4 ) Mutation and crossover: Exp. opt. time O ( n 3 . 5 √ log n ) Remainder of the talk mpi-logo

State-of-the-Art, our Results Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation need shared fitness Our Result Rigorous analysis of the All Pairs Shortest Path problem. Mutation only: Expected optimization time Θ( n 4 ) Mutation and crossover: Exp. opt. time O ( n 3 . 5 √ log n ) Remainder of the talk All Pairs Shortest Path problem mpi-logo

State-of-the-Art, our Results Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation need shared fitness Our Result Rigorous analysis of the All Pairs Shortest Path problem. Mutation only: Expected optimization time Θ( n 4 ) Mutation and crossover: Exp. opt. time O ( n 3 . 5 √ log n ) Remainder of the talk All Pairs Shortest Path problem A ( µ + 1) evolutionary algorithm for the APSP mpi-logo

State-of-the-Art, our Results Summary Existing Results Some proofs that crossover helps Artificial problems only Some additional critique 1 − o (1) fraction mutation need shared fitness Our Result Rigorous analysis of the All Pairs Shortest Path problem. Mutation only: Expected optimization time Θ( n 4 ) Mutation and crossover: Exp. opt. time O ( n 3 . 5 √ log n ) Remainder of the talk All Pairs Shortest Path problem A ( µ + 1) evolutionary algorithm for the APSP mpi-logo Some proof ideas.

The All Pairs Shortest Path (APSP) Problem Given Directed graph G = ( V , E ). edge weights w : E → Z ∪ {∞} No negative cycles! mpi-logo

The All Pairs Shortest Path (APSP) Problem Given Directed graph G = ( V , E ). edge weights w : E → Z ∪ {∞} No negative cycles! Problem For all pairs ( u , v ) ∈ V × V , find a shortest path (SP) from u to v . mpi-logo

The All Pairs Shortest Path (APSP) Problem Given Directed graph G = ( V , E ). edge weights w : E → Z ∪ {∞} No negative cycles! Problem For all pairs ( u , v ) ∈ V × V , find a shortest path (SP) from u to v . mpi-logo

The All Pairs Shortest Path (APSP) Problem Given Directed graph G = ( V , E ). edge weights w : E → Z ∪ {∞} No negative cycles! Problem For all pairs ( u , v ) ∈ V × V , find a shortest path (SP) from u to v . mpi-logo

The All Pairs Shortest Path (APSP) Problem Given Directed graph G = ( V , E ). edge weights w : E → Z ∪ {∞} No negative cycles! Problem For all pairs ( u , v ) ∈ V × V , find a shortest path (SP) from u to v . mpi-logo

The All Pairs Shortest Path (APSP) Problem Given Directed graph G = ( V , E ). edge weights w : E → Z ∪ {∞} No negative cycles! Problem For all pairs ( u , v ) ∈ V × V , find a shortest path (SP) from u to v . mpi-logo

The All Pairs Shortest Path (APSP) Problem Given Directed graph G = ( V , E ). edge weights w : E → Z ∪ {∞} No negative cycles! Problem For all pairs ( u , v ) ∈ V × V , find a shortest path (SP) from u to v . In total: µ = n ( n − 1) non-trivial SPs. mpi-logo