Evolutionary Computational Intelligence

1. Evolutionary Computational Intelligence Lecture 10a: Surrogate Assisted Ferrante Neri University of Jyväskylä

2. Computationally expensive problems • Optimization problems can be computationally expensive because of two reasons: • high cardinality decision space (usually combinatorial) • computationally expensive fitness function (e.g. design of on-line electric drives)

3. High Cardinality Decision Space • Under such conditions it should be tried, on the basis of the application, to reduce the cardinality by means of an ”a priori” analysis or an heuristic to detect a promising region of the decision space • Memetic approach (e.g. intelligent initial sampling) can be beneficial

4. Computationally expensive fitness • It might happen that the fitness function evaluation requires by itself a lot of computational effort (e.g. in online PMSM drive design each fitness evaluation requires 8 s) • In such conditions it should be found a way to reduce the numer of fitness evaluations and still reach the optimum

5. Surrogate Assisted Algorithms • Surrogate Assisted Algorithms employ approximated models of the fitness function (cheap) alternatively with the real fitness (expensive) • One of the crucial problems is the model to be employed and how to arrange such a combination

6. Global vs Local Surrogate models • There are two complementary and contrasting algorithmic philosophy: • Global Surrogate Models: attempt of finding an approximated model of the landscape over all the decision space • Local Surrogate Models: attempt of approximating locally the landscape over the neighborhood of a certain point

7. Comparison between the two philosophies • Global models assume that a wide knowledge of the decision space allows to build up an accurate model that can be employed as a cheap alternative of the real fitness • Local models assume that a huge amount of information does not help in determining an accurate model and thus it is preferable to build up models that approximate only locally the behavior of the landscape • Global models employ one very complex model, Local models employ many simple approximated functions

8. Coordination of models/real fitness • The right way to perform the coordination is very problem dependent, both deterministic and stocastic rules have been implemented • Models can be ”installed” in both evolutionary framework and local searchers

9. Surrogate Assisted Hooke-Jeeves Algorithm • Surrogate Assisted Hooke Jeeves Algorithm (SAHJA): deterministic scheme for coordinating real fitness and a linear modelobtained by least square method • Computes N+1 points and generates a local linear model for calculating the remaining N points (Cost of exploratory move is thus kept constant) • Check every directional move, by calculating the real fitness if a surrogate was prevously calculated (does not allow search directions by means of surrogate points)

10. SAHJA

11. SAHJA Results • Very promising algorithmic performance • Noise filtering

12. Evolutionary Computational Intelligence Lecture 10b: Experimentalism Ferrante Neri University of Jyväskylä

13. Goals • To propose a research protocol in order to execute a fair experimental comparison which allows us to check whether the newly proposed algorithm outperforms the methods existing in literature • In other words, if I designed a novel algorithm how can I be sure that my work outperform (for a certain problem) the state of the art?

14. Towards Performance Comparison • If I designed a novel algorithm Bhow can I prove that B outperforms the benchmark algorithm A? How can I thus have a confirmation that the novel algorithmic component is really effective? • Performance is an abstract concept not related to a specific machine. It is the capability of an algorithm of reaching a good performing solution in a certain time interval. The time trigger is the number of fitness (functional) evaluations

15. Experimental Setup • For both A and B, a certain number n of runs must be performed • The average best fitness values (e.g. at the end of each generation) must be saved • N.B. for making the trends comparable an interpolation can be necessary • Standard deviation bars can also be included

16. Two Possible kinds of outperforming Case 1: • A and B converge on different final values Case 2: • A and B converge on the same final values but with different convergence velocities

17. …Case 1 • The data define two Tolerance Intervals (TIs) • It is fixed a desired confidence level δ • The proportion γ of a set of data which falls within a given interval with a given confidence level δ are determined by: γ=1 −a/n where n is the number of available samples and ais the positive root of theequation (1 + a) − (1 − δ) · ea = 0

18. …Case 2 • A threshold value fthr is fixed. If during an experiment fbest < fthr then the algorithm ”almost converged” • For the n experiments, the number of fitness evaluations necessary to verify the inequality fbest < fthr defines a TI • The probability γ that an algorithm requires no more fitness evaluations than the most unlucky case is given by: γ =1 −d/n where n is the number of the available experiments. d is given by d = −ln(1 − δ)

19. How to conclude • In both the cases, if the tolerance intervals are not separated it is impossible to establish that B outperforms A in all the cases. In this case it is possible only to state that B outperforms A in average