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Introduction Aging Beyond Restarts: Ideas Results Conclusions Aging Beyond Restarts Thomas Jansen University College Cork joint work with Christine Zarges Technische Universitt Dortmund J./Zarges: Aging beyond restarts. In GECCO 2010.

  1. Introduction Aging Beyond Restarts: Ideas Results Conclusions Known Results • on the maximal lifespan τ (Horoba/J./Zarges (GECCO 2009)) • if τ is too large, static pure aging is ineffective • if τ is too small, the RSH becomes unsuccessful • the ‘correct’ lifespan depends on the optimization problem • the range of ‘correct’ lifespans can be extremely small • on different aging strategies (J./Zarges (ICARIS 2009, TCS)) • static pure aging more effective than evolutionary aging in escaping local optima • evolutionary aging more robust than static pure aging on plateaus • advantages of both can be combined in phenotypic aging 4/13

  2. Introduction Aging Beyond Restarts: Ideas Results Conclusions Known Results • on the maximal lifespan τ (Horoba/J./Zarges (GECCO 2009)) • if τ is too large, static pure aging is ineffective • if τ is too small, the RSH becomes unsuccessful • the ‘correct’ lifespan depends on the optimization problem • the range of ‘correct’ lifespans can be extremely small • on different aging strategies (J./Zarges (ICARIS 2009, TCS)) • static pure aging more effective than evolutionary aging in escaping local optima • evolutionary aging more robust than static pure aging on plateaus • advantages of both can be combined in phenotypic aging • general observation • in all published cases: restarts can replace aging 4/13

  3. Introduction Aging Beyond Restarts: Ideas Results Conclusions Known Results • on the maximal lifespan τ (Horoba/J./Zarges (GECCO 2009)) • if τ is too large, static pure aging is ineffective • if τ is too small, the RSH becomes unsuccessful • the ‘correct’ lifespan depends on the optimization problem • the range of ‘correct’ lifespans can be extremely small • on different aging strategies (J./Zarges (ICARIS 2009, TCS)) • static pure aging more effective than evolutionary aging in escaping local optima • evolutionary aging more robust than static pure aging on plateaus • advantages of both can be combined in phenotypic aging • general observation • in all published cases: restarts can replace aging • restarts conceptually simpler, easier to implement, computationally cheaper 4/13

  4. Introduction Aging Beyond Restarts: Ideas Results Conclusions Known Results • on the maximal lifespan τ (Horoba/J./Zarges (GECCO 2009)) • if τ is too large, static pure aging is ineffective • if τ is too small, the RSH becomes unsuccessful • the ‘correct’ lifespan depends on the optimization problem • the range of ‘correct’ lifespans can be extremely small • on different aging strategies (J./Zarges (ICARIS 2009, TCS)) • static pure aging more effective than evolutionary aging in escaping local optima • evolutionary aging more robust than static pure aging on plateaus • advantages of both can be combined in phenotypic aging • general observation • in all published cases: restarts can replace aging • restarts conceptually simpler, easier to implement, computationally cheaper Open Question What can aging do beyond restarts? 4/13

  5. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic 5/13

  6. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) 5/13

  7. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 5/13

  8. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 5/13

  9. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 5/13

  10. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 4. Determine age: If f ( z ) > max { f ( x ) , f ( y ) } then z .age = 0 , else z .age = max { x .age , y .age } . 5/13

  11. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 4. Determine age: If f ( z ) > max { f ( x ) , f ( y ) } then z .age = 0 , else z .age = max { x .age , y .age } . 5. Dying of age: Remove all x ∈ C with x .age > τ . 5/13

  12. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 4. Determine age: If f ( z ) > max { f ( x ) , f ( y ) } then z .age = 0 , else z .age = max { x .age , y .age } . 5. Dying of age: Remove all x ∈ C with x .age > τ . 6. Selection: If | C | < µ If z .age ≤ τ then C := C ∪ { z } . While | C | < µ select x ∈ { 0 , 1 } n u. a. r., x .age = 0 , C := C ∪ { x } . 5/13

  13. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 4. Determine age: If f ( z ) > max { f ( x ) , f ( y ) } then z .age = 0 , else z .age = max { x .age , y .age } . 5. Dying of age: Remove all x ∈ C with x .age > τ . 6. Selection: If | C | < µ If z .age ≤ τ then C := C ∪ { z } . While | C | < µ select x ∈ { 0 , 1 } n u. a. r., x .age = 0 , C := C ∪ { x } . Else 5/13

  14. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 4. Determine age: If f ( z ) > max { f ( x ) , f ( y ) } then z .age = 0 , else z .age = max { x .age , y .age } . 5. Dying of age: Remove all x ∈ C with x .age > τ . 6. Selection: If | C | < µ If z .age ≤ τ then C := C ∪ { z } . While | C | < µ select x ∈ { 0 , 1 } n u. a. r., x .age = 0 , C := C ∪ { x } . Else If z .age ≤ τ and f ( z ) ≥ min { f ( x ) | x ∈ C } then 5/13

  15. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 4. Determine age: If f ( z ) > max { f ( x ) , f ( y ) } then z .age = 0 , else z .age = max { x .age , y .age } . 5. Dying of age: Remove all x ∈ C with x .age > τ . 6. Selection: If | C | < µ If z .age ≤ τ then C := C ∪ { z } . While | C | < µ select x ∈ { 0 , 1 } n u. a. r., x .age = 0 , C := C ∪ { x } . Else If z .age ≤ τ and f ( z ) ≥ min { f ( x ) | x ∈ C } then D := { x ∈ C | f ( x ) = min { f ( x ′ ) | x ′ ∈ C }} 5/13

  16. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 4. Determine age: If f ( z ) > max { f ( x ) , f ( y ) } then z .age = 0 , else z .age = max { x .age , y .age } . 5. Dying of age: Remove all x ∈ C with x .age > τ . 6. Selection: If | C | < µ If z .age ≤ τ then C := C ∪ { z } . While | C | < µ select x ∈ { 0 , 1 } n u. a. r., x .age = 0 , C := C ∪ { x } . Else If z .age ≤ τ and f ( z ) ≥ min { f ( x ) | x ∈ C } then D := { x ∈ C | f ( x ) = min { f ( x ′ ) | x ′ ∈ C }} If f ( z ) = min { f ( x ′ ) | x ′ ∈ C } then D := { x ∈ D | | x .age − z .age | = min {| x ′ .age − z .age | | x ′ ∈ D }} 5/13

  17. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 4. Determine age: If f ( z ) > max { f ( x ) , f ( y ) } then z .age = 0 , else z .age = max { x .age , y .age } . 5. Dying of age: Remove all x ∈ C with x .age > τ . 6. Selection: If | C | < µ If z .age ≤ τ then C := C ∪ { z } . While | C | < µ select x ∈ { 0 , 1 } n u. a. r., x .age = 0 , C := C ∪ { x } . Else If z .age ≤ τ and f ( z ) ≥ min { f ( x ) | x ∈ C } then D := { x ∈ C | f ( x ) = min { f ( x ′ ) | x ′ ∈ C }} If f ( z ) = min { f ( x ′ ) | x ′ ∈ C } then D := { x ∈ D | | x .age − z .age | = min {| x ′ .age − z .age | | x ′ ∈ D }} Select x ∈ D u. a. r., C := ( C \ { x } ) ∪ { z } . 5/13

  18. Introduction Aging Beyond Restarts: Ideas Results Conclusions The Randomized Search Heuristic population size µ ∈ N \ { 1 } , µ = n O (1) , lifespan τ ∈ N 0 Parameters crossover probability p c ∈ (0 , 1) , constant number of crossover points k ∈ N , k = O (1) Initialization: Select collection C of µ points x ∈ { 0 , 1 } n u. a. r., 1. set x .age = 0 for all. 2. Growing older: Increase x .age by 1 for all x ∈ C . 3. Variation: With prob. p c select x, y ∈ C u. a. r., z := mutate ( k -pt-cross ( x, y )) else (with prob. 1 − p c ) select x ∈ C u. a. r., y := x , z := mutate ( x ) . 4. Determine age: If f ( z ) > max { f ( x ) , f ( y ) } then z .age = 0 , else z .age = max { x .age , y .age } . 5. Dying of age: Remove all x ∈ C with x .age > τ . 6. Selection: If | C | < µ If z .age ≤ τ then C := C ∪ { z } . While | C | < µ select x ∈ { 0 , 1 } n u. a. r., x .age = 0 , C := C ∪ { x } . Else If z .age ≤ τ and f ( z ) ≥ min { f ( x ) | x ∈ C } then D := { x ∈ C | f ( x ) = min { f ( x ′ ) | x ′ ∈ C }} If f ( z ) = min { f ( x ′ ) | x ′ ∈ C } then D := { x ∈ D | | x .age − z .age | = min {| x ′ .age − z .age | | x ′ ∈ D }} Select x ∈ D u. a. r., C := ( C \ { x } ) ∪ { z } . 7. Continue at line 2. 5/13

  19. Introduction Aging Beyond Restarts: Ideas Results Conclusions Aging Beyond Restarts: Main Idea 6/13

  20. Introduction Aging Beyond Restarts: Ideas Results Conclusions Aging Beyond Restarts: Main Idea Observation • Aging can perform restarts. restart � = empty C completely due to age 6/13

  21. Introduction Aging Beyond Restarts: Ideas Results Conclusions Aging Beyond Restarts: Main Idea Observations • Aging can perform restarts. restart � = empty C completely due to age • Aging can perform partial restarts. partial restart � = remove some x from C due to age and insert new random points 6/13

  22. Introduction Aging Beyond Restarts: Ideas Results Conclusions Aging Beyond Restarts: Main Idea Observations • Aging can perform restarts. restart � = empty C completely due to age • Aging can perform partial restarts. partial restart � = remove some x from C due to age and insert new random points • new random points probably no good � quickly removed 6/13

  23. Introduction Aging Beyond Restarts: Ideas Results Conclusions Aging Beyond Restarts: Main Idea Observations • Aging can perform restarts. restart � = empty C completely due to age • Aging can perform partial restarts. partial restart � = remove some x from C due to age and insert new random points • new random points probably no good � quickly removed • only a few generations where new points can actually do something 6/13

  24. Introduction Aging Beyond Restarts: Ideas Results Conclusions Aging Beyond Restarts: Main Idea Observations • Aging can perform restarts. restart � = empty C completely due to age • Aging can perform partial restarts. partial restart � = remove some x from C due to age and insert new random points • new random points probably no good � quickly removed • only a few generations where new points can actually do something • “ There’s hardly anything less random than some x ∈ { 0 , 1 } n selected uniformly at random. ” 6/13

  25. Introduction Aging Beyond Restarts: Ideas Results Conclusions An Example Problem 7/13

  26. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n An Example Problem f : { 0 , 1 } n → R 0 n 7/13

  27. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n An Example Problem f : { 0 , 1 } n → R 0 n        f ( x ) =       n − OneMax ( x ) otherwise 7/13

  28. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n An Example Problem 1 n/ 4 0 3 n/ 4 n/ 4 f : { 0 , 1 } n → R 0 n        f ( x ) =  if x = 1 i 0 n − i , i ≤ n/ 4 n + i      n − OneMax ( x ) otherwise 7/13

  29. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n An Example Problem 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 f : { 0 , 1 } n → R 0 n  if x = 1 n/ 4 0 n/ 4 q , q ∈ { 0 , 1 } n/ 2 ,     2 n   OneMax ( q ) ≥ n/ 12 f ( x ) =  if x = 1 i 0 n − i , i ≤ n/ 4 n + i      n − OneMax ( x ) otherwise 7/13

  30. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n 8/13

  31. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n opt. n − OneMax ( x ) 8/13

  32. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. 8/13

  33. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. opt. 1 i 0 n − 4 8/13

  34. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. � n 2 + µn log n � opt. 1 i 0 n − 4 Θ w. high prob. 8/13

  35. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. � n 2 + µn log n � opt. 1 i 0 n − 4 Θ w. high prob. perform partial restart 8/13

  36. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. � n 2 + µn log n � opt. 1 i 0 n − 4 Θ w. high prob. perform partial restart Θ( τ ) w. prob. p 8/13

  37. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. � n 2 + µn log n � opt. 1 i 0 n − 4 Θ w. high prob. perform partial restart Θ( τ ) w. prob. p find opt. with XO 8/13

  38. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. � n 2 + µn log n � opt. 1 i 0 n − 4 Θ w. high prob. perform partial restart Θ( τ ) w. prob. p find opt. with XO 1 w. prob. q 8/13

  39. Introduction Aging Beyond Restarts: Ideas Results Conclusions 1 n Static Pure Aging on f 1 n/ 4 0 n/ 4 1 n/ 2 3 n/ 4 1 n/ 4 0 n/ 4 q n/ 2 n/ 3 1 n/ 4 0 3 n/ 4 n/ 4 0 n opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. � n 2 + µn log n � opt. 1 i 0 n − 4 Θ w. high prob. perform partial restart Θ( τ ) w. prob. p find opt. with XO 1 w. prob. q Adding things up � p − 1 · q − 1 · � τ + n 2 + µn log n �� upper bound O � p − 1 · q − 1 · � τ + n 2 + µn log n �� lower bound Ω 8/13

  40. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover 9/13

  41. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover n/ 4 local optimum x 11 · · · 11 00000000 · · · 0000000 9/13

  42. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover n/ 4 local optimum x 11 · · · 11 00000000 · · · 0000000 new random search point y 9/13

  43. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 local optimum x 11 · · · 11 00000000 · · · 0000000 new random search point y y A y B y C y D y E y F 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 9/13

  44. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 local optimum x 11 · · · 11 00000000 · · · 0000000 new random search point y y A y B y C y D y E y F 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 We need k crossover points c 1 < c 2 < · · · < c k with c 1 > n/ 2 and ≥ n/ 12 1-bits in used y -parts 9/13

  45. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 local optimum x 11 · · · 11 00000000 · · · 0000000 new random search point y y A y B y C y D y E y F 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 We need k crossover points c 1 < c 2 < · · · < c k with c 1 > n/ 2 and ≥ n/ 12 1-bits in used y -parts Observation • Prob ( n/ 2 ≤ c 1 < 7 n/ 12) = Ω(1) 9/13

  46. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 local optimum x 11 · · · 11 00000000 · · · 0000000 new random search point y y A y B y C y D y E y F 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 We need k crossover points c 1 < c 2 < · · · < c k with c 1 > n/ 2 and ≥ n/ 12 1-bits in used y -parts Observations • Prob ( n/ 2 ≤ c 1 < 7 n/ 12) = Ω(1) • Prob (3 n/ 4 ≤ c 2 < 5 n/ 6) = Ω(1) 9/13

  47. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 local optimum x 11 · · · 11 00000000 · · · 0000000 new random search point y y A y B y C y D y E y F 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 We need k crossover points c 1 < c 2 < · · · < c k with c 1 > n/ 2 and ≥ n/ 12 1-bits in used y -parts Observations • Prob ( n/ 2 ≤ c 1 < 7 n/ 12) = Ω(1) • Prob (3 n/ 4 ≤ c 2 < 5 n/ 6) = Ω(1) • ∀ i > 2: Prob ( c i > 5 n/ 6) = Ω(1) 9/13

  48. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 local optimum x 11 · · · 11 00000000 · · · 0000000 new random search point y y A y B y C y D y E y F 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 We need k crossover points c 1 < c 2 < · · · < c k with c 1 > n/ 2 and ≥ n/ 12 1-bits in used y -parts Observations • Prob ( n/ 2 ≤ c 1 < 7 n/ 12) = Ω(1) • Prob (3 n/ 4 ≤ c 2 < 5 n/ 6) = Ω(1) • ∀ i > 2: Prob ( c i > 5 n/ 6) = Ω(1) • Prob ( OneMax ( y D ) ≥ n/ 12) ≥ 1 / 2 9/13

  49. Introduction Aging Beyond Restarts: Ideas Results Conclusions Creating an Optimum with Crossover 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 local optimum x 11 · · · 11 00000000 · · · 0000000 new random search point y y A y B y C y D y E y F 7 n/ 12 3 n/ 4 5 n/ 6 n/ 4 n/ 2 We need k crossover points c 1 < c 2 < · · · < c k with c 1 > n/ 2 and ≥ n/ 12 1-bits in used y -parts Observations • Prob ( n/ 2 ≤ c 1 < 7 n/ 12) = Ω(1) • Prob (3 n/ 4 ≤ c 2 < 5 n/ 6) = Ω(1) • ∀ i > 2: Prob ( c i > 5 n/ 6) = Ω(1) • Prob ( OneMax ( y D ) ≥ n/ 12) ≥ 1 / 2 Result Prob ( k -pt-crossover ( x, y ) ∈ OPT ) = q = Θ(1) 9/13

  50. Introduction Aging Beyond Restarts: Ideas Results Conclusions Results We have 10/13

  51. Introduction Aging Beyond Restarts: Ideas Results Conclusions Results We have opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. � n 2 + µn log n � opt. 1 i 0 n − 4 Θ w. high prob. perform partial restart Θ( τ ) w. prob. p find opt. with XO 1 w. prob. Θ(1) 10/13

  52. Introduction Aging Beyond Restarts: Ideas Results Conclusions Results We have opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. � n 2 + µn log n � opt. 1 i 0 n − 4 Θ w. high prob. perform partial restart Θ( τ ) w. prob. p find opt. with XO 1 w. prob. Θ(1) Adding things up � p − 1 · � τ + n 2 + µn log n �� upper bound O � p − 1 · � τ + n 2 + µn log n �� lower bound Ω 10/13

  53. Introduction Aging Beyond Restarts: Ideas Results Conclusions Results We have opt. n − OneMax ( x ) O ( µn + n log n ) w. high prob. � n 2 + µn log n � opt. 1 i 0 n − 4 Θ w. high prob. perform partial restart Θ( τ ) w. prob. p find opt. with XO 1 w. prob. Θ(1) Adding things up � p − 1 · � τ + n 2 + µn log n �� upper bound O � p − 1 · � τ + n 2 + µn log n �� lower bound Ω � µ · � τ + n 2 + µn log n �� Theorem upper bound O for any poly. µ , const. p c , const. k , any τ = ω ( µn log µ ) � τ + n 2 + µn log n � lower bound Ω for any poly. µ , const. p c , const. k , any τ ( 2 O ( n ) ) 10/13

  54. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Setup � µ · � τ + n 2 + µn log n �� Theorem upper bound O for any poly. µ , const. p c , const. k , any τ = ω ( µn log µ ) � τ + n 2 + µn log n � lower bound Ω for any poly. µ , const. p c , const. k , any τ ( 2 O ( n ) ) Perform Experiments with 11/13

  55. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Setup � µ · � τ + n 2 + µn log n �� Theorem upper bound O for any poly. µ , const. p c , const. k , any τ = ω ( µn log µ ) � τ + n 2 + µn log n � lower bound Ω for any poly. µ , const. p c , const. k , any τ ( 2 O ( n ) ) Perform Experiments with • crossover probability p c = . 5 11/13

  56. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Setup � µ · � τ + n 2 + µn log n �� Theorem upper bound O for any poly. µ , const. p c , const. k , any τ = ω ( µn log µ ) � τ + n 2 + µn log n � lower bound Ω for any poly. µ , const. p c , const. k , any τ ( 2 O ( n ) ) Perform Experiments with • crossover probability p c = . 5 • 1-point crossover ( k = 1 ) 11/13

  57. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Setup � µ · � τ + n 2 + µn log n �� Theorem upper bound O for any poly. µ , const. p c , const. k , any τ = ω ( µn log µ ) � τ + n 2 + µn log n � lower bound Ω for any poly. µ , const. p c , const. k , any τ ( 2 O ( n ) ) Perform Experiments with • crossover probability p c = . 5 • 1-point crossover ( k = 1 ) • lifespan τ = 6 µn ⌊ log µ ⌋ ⌊ log n ⌋ 11/13

  58. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Setup � µ · � τ + n 2 + µn log n �� Theorem upper bound O for any poly. µ , const. p c , const. k , any τ = ω ( µn log µ ) � τ + n 2 + µn log n � lower bound Ω for any poly. µ , const. p c , const. k , any τ ( 2 O ( n ) ) Perform Experiments with • crossover probability p c = . 5 • 1-point crossover ( k = 1 ) • lifespan τ = 6 µn ⌊ log µ ⌋ ⌊ log n ⌋ • problem size n ∈ { 10 , 20 , 30 , . . . , 1000 } 11/13

  59. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Setup � µ · � τ + n 2 + µn log n �� Theorem upper bound O for any poly. µ , const. p c , const. k , any τ = ω ( µn log µ ) � τ + n 2 + µn log n � lower bound Ω for any poly. µ , const. p c , const. k , any τ ( 2 O ( n ) ) Perform Experiments with • crossover probability p c = . 5 • 1-point crossover ( k = 1 ) • lifespan τ = 6 µn ⌊ log µ ⌋ ⌊ log n ⌋ • problem size n ∈ { 10 , 20 , 30 , . . . , 1000 } • population size µ ∈ { 2 , ⌊√ n ⌋ , ⌊ n/ ⌊ log n ⌋⌋ , n, n ⌊ log n ⌋} 11/13

  60. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Setup � µ · � τ + n 2 + µn log n �� Theorem upper bound O for any poly. µ , const. p c , const. k , any τ = ω ( µn log µ ) � τ + n 2 + µn log n � lower bound Ω for any poly. µ , const. p c , const. k , any τ ( 2 O ( n ) ) Perform Experiments with • crossover probability p c = . 5 • 1-point crossover ( k = 1 ) • lifespan τ = 6 µn ⌊ log µ ⌋ ⌊ log n ⌋ • problem size n ∈ { 10 , 20 , 30 , . . . , 1000 } • population size µ ∈ { 2 , ⌊√ n ⌋ , ⌊ n/ ⌊ log n ⌋⌋ , n, n ⌊ log n ⌋} • 100 independent runs per setting 11/13

  61. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results # f -evals 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  62. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  63. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals µ = n 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  64. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals µ = n µ = n/ ⌊ log n ⌋ 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  65. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals µ = n µ = n/ ⌊ log n ⌋ µ = √ n 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  66. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals µ = n µ = n/ ⌊ log n ⌋ µ = √ n µ = 2 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  67. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals µ = n µ = n/ ⌊ log n ⌋ µ = √ n µ = 2 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  68. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals µ = n µ = n/ ⌊ log n ⌋ µ = √ n µ = 2 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  69. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals µ = n µ = n/ ⌊ log n ⌋ µ = √ n µ = 2 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  70. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals µ = n µ = n/ ⌊ log n ⌋ µ = √ n µ = 2 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

  71. Introduction Aging Beyond Restarts: Ideas Results Conclusions Experimental Results µ = n ⌊ log n ⌋ # f -evals µ = n µ = n/ ⌊ log n ⌋ µ = √ n µ = 2 7 · 10 7 6 · 10 7 5 · 10 7 4 · 10 7 3 · 10 7 2 · 10 7 10 7 n 100 200 300 400 500 600 700 800 900 1000 12/13

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