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Covariance Matrix Adaptation Evolution Strategy (CMA-ES)
- 1. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . . . Covariance Matrix Adaptation Evolution Strategies(CMA-ES) Hossein Abedi Evolutionary Computation Autumn 2014 Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 1 / 19
- 2. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Overview 1. Introduction 2. Selection and Recombination 3. Adaptation of covariance matrix 4. Step size control 5. Experiments 6. Conclusion Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 2 / 19
- 3. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Introduction Idea Introduced by Hansen and Ostermeier in 2001 The idea: Figure : Movement toward a minimum through 3 generations Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 3 / 19
- 4. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Selection and Recombination Generating the children New points are sampled normally distributed: Xi M + Ni (0,C ), for i=1,..., Figure : Different shapes of C as a hyperelipsoid in 2D Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 4 / 19
- 5. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Selection and Recombination Selection and Recombination The mean vector M 2 ℜn is calculated Σas the weighted average of the best candidate solutions: M= i=1 wiXi : Where: Σ i=1 wi = 1 w1 ⩾ w2 ⩾ ::: ⩾ w 0 f (X1:) ⩽ f (X2:) ⩽ ::: ⩽ f (X:) eff = ( jjwjj1 jjwjj2 )2 = Σ 1 i=1 w2 i Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 5 / 19
- 6. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Adaptation of covariance matrix Estimating the covariance matrix from scratch For the sake of simplicity set (g) = 1 Estimating distribution within the population: C(g+1) emp = 1 1 Σ i=1(X(g+1) i 1 Σ j=1 Xj )(X(g+1) i 1 Σ j=1 Xj )T Estimating distribution of sampled steps: C(g+1) = 1 Σ i=1(X(g+1) i M(g))(X(g+1) i M(g))T Where: The sampled steps are: X(g+1) M(g) i Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 6 / 19
- 7. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Adaptation of covariance matrix Estimating the covariance matrix Estimating distribution of the most successful steps: C(g+1) = 1 Σ i=1 wi (X(g+1) i : M(g))(X(g+1) i : M(g))T Estimation of Multivariate Normal Algorithm(ENMA): C(g+1) = 1 Σ i=1(X(g+1) i : M(g+1) enma )(X(g+1) i : M(g+1) enma )T Where: M(g+1) enma = 1 Σ j=1 X(g+1) j : Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 7 / 19
- 8. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Adaptation of covariance matrix Estimating the covariance matrix Comparison: Figure : Covariance matrix estimation on f (x1; x2) = Σ2 i=1 xi Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 8 / 19
- 9. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Adaptation of covariance matrix Rank update Smaller means faster but less global search To give recent generations a higher weight, consider a leraning rate c and the equation below: C(g+1) = (1 c)C(g) + c 1 C(g+1) ((g))2 Where: 1 c is called the time back horizon Figure : Example of exponential smoothing Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 9 / 19
- 10. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Adaptation of covariance matrix Rank update C(g+1) = (1 c)C(g) + c 1 Σ i=1 wiOP(X(g+1) i : M(g) (g) ) Where: OP(y) = yyT Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 10 / 19
- 11. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Adaptation of covariance matrix Rank one update Evolution Path (Pc 2 ℜn): sum of consecutive steps: M(g+1)M(g) (g) + M(g)M(g1) (g1) + ::: Figure : Evolution path N(0; I )y1 + N(0; I )y2 + ::: + N(0; I )yg N(0; Σg i=1 yi yT i ) Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 11 / 19
- 12. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Adaptation of covariance matrix Rank one update Using exponential smoothing: P(g+1) c = (1 cc )P(g) c + √ cc(2 cc )eff M(g+1)M(g) (g) Wher√e: cc(2 cc )eff is a scaling factor such that :P(g+1) c N(0; C) So rank one update with sign is : C(g+1) = (1 c1)C(g) + c1OP(P(g+1) c ) Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 12 / 19
- 13. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Adaptation of covariance matrix Cumulation C(g+1) = (1 c1 c)C(g) + c1(y(g+1) c )(P(g+1) c )T + :::c 1 Σ i=1 wiOP(X(g+1) i : M(g) (g) ) Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 13 / 19
- 14. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Step size control Step size control Using the evolution path for adapting the stepsize Figure : Different evolution path senarios for 6 consecutive mean vectors (g+1) = (g) exp ( c d ( jjp(g+1) jj EjjN(0;I )jj 1)) Where: p(g+1) = (1 c)p(g) + √ c(2 c)eff (C(g))1 2 M(g+1)M(g) (g) Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 14 / 19
- 15. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Experiments Test on seperable and non rotated 0 200 400 600 800 1000 1200 1400 1600 1800 2000 14 12 10 8 6 4 2 0 0.01*function evauations fmin CLPSO CMA−ES Figure : Results on Ackley test function Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 15 / 19
- 16. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Experiments Test on CEC2015(shifted,rotated and non-seperable) 0 10 20 30 40 50 60 70 80 90 100 24 22 20 18 16 14 12 10 % of function evaluation log(fmin) CLPSO CMA−ES Figure : Results on function 2 CEC2015 Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 16 / 19
- 17. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Experiments Test on CEC2015(shifted,rotated and non-seperable) 0 10 20 30 40 50 60 70 80 90 100 518 516 514 512 510 508 506 504 502 500 % of function evaluations fmin CMA−ES CLPSO Figure : Results on function 5 CEC2015 Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 17 / 19
- 18. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Experiments Test on CEC2015(shifted,rotated and non-seperable) 0 10 20 30 40 50 60 70 80 90 100 612 611 610 609 608 607 606 605 604 603 % of function evaluation fmin CLPSO CMA−ES Figure : Results on function 6 CEC2015 Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 18 / 19
- 19. .. . . .. . . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . . .. . .. . .. . Conclusion Conclusion Applicable to problems in which many variables are correlated Good local search Hossein Abedi (Evolutionary Computation) CMA-ES Autumn 2014 19 / 19