Extending Exploratory Landscape Analysis for Multi-Objective and Multimodal Problems Pascal Kerschke & Mike Preuß Information Systems and Statistics Group, University of M¨ unster, Germany Thursday, October 13th, 2016 October 13th, 2016 1 / 37
Agenda introducing Exploratory Landscape Analysis 1 existing ELA features 2 ELA for multimodal problems 3 ELA for multi-objective problems 4 flacco library 5 October 13th, 2016 2 / 37
Introducing Exploratory Landscape Analysis October 13th, 2016 3 / 37
Introduction algorithm selection problem 1 � find the individually best suited algorithm for an unseen optimization problem 1Rice, J. (1976). The Algorithm Selection Problem . In Advances in Computers (pp. 65-118). October 13th, 2016 4 / 37
Introduction algorithm selection problem 1 � find the individually best suited algorithm for an unseen optimization problem 1Rice, J. (1976). The Algorithm Selection Problem . In Advances in Computers (pp. 65-118). October 13th, 2016 4 / 37
Introduction Exploratory Landscape Analysis (ELA): we aim at finding the right algorithm but also at improving problem or algorithm/problem dependency understanding basic idea (exploratory!): we start with very simple features without clear purpose match existing high-level features (expert knowledge) with our ELA features currently: mostly continuous (black-box) (global) optimization, but also in other domains (e.g. TSP) October 13th, 2016 5 / 37
Introduction Convexity Search space Global structure homogeneity y-Distribution Multimodality Plateaus Levelset Meta Model Separability Variable scaling Local Search Global to local Basin size optima contrast homogeneity Curvature Mersmann, O., Preuss, M. & Trautmann, H. (2010). Benchmarking Evolutionary Algorithms: Towards Exploratory Landscape Analysis . In Proceedings of PPSN XI (pp. 71 - 80). October 13th, 2016 6 / 37
Introduction we do not know functional relationships when designing features but we can match them to high-level characteristics (multimodality, funnel structure, etc.) of optimization problems this enables recognizing important problem properties quickly based on initial design of samples x i 1 , . . . , x iD and their corresponding fitness value y i , i = 1, . . . , n given an evaluated initial design (initial population?), most ELA features are for free there are already several different feature sets October 13th, 2016 7 / 37
Further ELA features October 13th, 2016 8 / 37
Further ELA Features General Cell Mapping Features Kerschke, P., Preuss, M., Hern´ andez, C., Sch¨ utze, O., Sun, J.-Q., Grimme, C., Rudolph, G., Bischl, B. & Trautmann, H. (2014). Cell Mapping Techniques for Exploratory Landscape Analysis . In Proceedings of EVOLVE 2014 (pp. 115 - 131). Barrier Tree Features Hern´ andez, C., Sch¨ utze, O., Emmerich, M. T. M., & Xiong, F. R. (2014). Barrier Tree for Continuous Landscapes by Means of Generalized Cell Mapping . In Proceedings of EVOLVE 2014. Cell ID (1st Dimension) 1 2 3 4 5 6 7 8 38 (root) ● 1 5 ● ● 0 39 − 33 38 (root) 39 e Cell Coordinate (2nd Dimension) 0 9 . Cell ID (2nd Dimension) f ( 1 ● 4 x 0 , − 25 y e 0 ) . 7 ● ● 33 25 1 3 0 − e 0 ● . 5 5 ● 6 1 ● ● 2 0 − 10 11 ● e 10 0 ● . 3 11 1 1 ● ● ● 0 − 1 5 6 y e 0 1 . 2 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 x − − − − − − − − e e e e e e e e 2 9 1 4 6 9 1 4 . . . . . . . . 6 1 3 4 5 6 8 9 ● 1 Cell Coordinate (1st Dimension) October 13th, 2016 9 / 37
Further ELA Features Cell Mapping Features Kerschke, P., Preuss, M., Hern´ andez, C., Sch¨ utze, O., Sun, J.-Q., Grimme, C., Rudolph, G., Bischl, B. & Trautmann, H. (2014). Cell Mapping Techniques for Exploratory Landscape Analysis . In Proceedings of EVOLVE 2014 (pp. 115 - 131). Information Content Features Mu˜ noz, M. A., Kirley, M., Halgamuge, S. K. (2015). Exploratory Landscape Analysis of Continuous Space Optimization Problems using Information Content . In IEEE Transactions on Evolutionary Computation (pp. 74 - 87). Information Content Plot ● H ( ε ) 0.7 M ( ε ) 0.6 ● H max ε s 0.5 H ( ε ) & M ( ε ) M 0 0.4 ε ratio 0.3 0.2 0.500 * M0 0.1 0.0 −4 −2 0 2 4 log 10 ( ε ) October 13th, 2016 10 / 37
Further ELA Features Dispersion Features Lunacek, M. & Whitley, D. (2006). The Dispersion Metric and the CMA Evolution Strategy . In Proceedings of GECCO 2006 (pp. 477 - 484). Nearest Better Clustering Features Kerschke, P., Preuss, M., Wessing, S. & Trautmann H. (2015). Detecting Funnel Structures by Means of Exploratory Landscape Analysis . In Proceedings of GECCO 2015 (pp. 265 - 272). Length Scale Features Morgan, R. & Gallagher M. (2015). Analyzing and Characterising Optimization Problems Using Length Scale . In Soft Computing (pp. 1 - 18). Ruggedness Features Malan, K. M. & Engelbrecht, A. P. (2013). Ruggedness, Funnels and Gradients in Fitness Landscapes and the Effect on PSO Performance . In Proceedings of CEC 2013 (pp. 963 - 970). Hill Climbing Features Abell, T., Malitsky, Y. & Tierney, K. (2013). Features for Exploiting Black-Box Optimization Problem Structure . In Proceedings of LION 2013 (pp. 30 - 36). October 13th, 2016 11 / 37
ELA for Multimodal Problems October 13th, 2016 12 / 37
Multimodal Problems? October 13th, 2016 13 / 37
Multimodal Optimization different aims possible currently most important (competitions): multiglobal = find all search space points that are globally optimal two main algorithmic approaches: parallel, large populations sequential, coordinated restarts several components that may be used: archives, clustering methods, methods for obtaining well distributed samples ELA could be helpful for selecting components/methods October 13th, 2016 14 / 37
Funnel Detection Example: (recent research) → Funnel Detection October 13th, 2016 15 / 37
Funnel Detection funnel: local optima are located near to each other and pile up to an “upside-down mountain” knowledge about underlying global structure, i.e. funnels, helps selecting the right algorithm (a) funnel (b) non-funnel (“random”) October 13th, 2016 16 / 37
Funnel Detection different algorithm candidates for either category but there is a wide variety within classes funnel and non-funnel (a) funnel (b) non-funnel (“random”) October 13th, 2016 17 / 37
Funnel Detection detailed results in our GECCO paper 2 used MPM2 3 to generate a set of 4,000 training instances initial designs of size 50 × D observations (small!) trained four classifiers (random forest, rpart, kknn and ksvm) experimentally driven reduction of the full feature set (300+ features) to 8 features validated results on BBOB and subset of problems from CEC-2013 niching competition 2Kerschke, P., Preuss, M., Wessing, S. & Trautmann H. (2016). Low-Budget Exploratory Landscape Analysis on Multiple Peaks Models. In Proceedings of GECCO 2016 (pp. 229-236) 3multiple peaks model 2 generator, available in python ( optproblems0.9 , Wessing, S.) and R ( smoof , Bossek, J.) October 13th, 2016 18 / 37
Funnel Detection 10 9 8 7 CV−Iteration 6 5 4 3 2 1 m r o o i 2 t d 2 2 i p 2 t r c r _ r r a e . _ _ s _ r j c j j _ d s j d d r d a e d e a a a s . t n . . e n . t e . t t l i c c b p i f l . a a p n e _ m r m l b r _ p e e i n n s m t i t n s n _ . n c _ i i i . n s b _ _ d c i _ w b l n w a . a n _ u n _ i q t l d n e . a i . m a l a . t u a e t _ q e m t a e m . a m l _ e t _ a e _ a l m a e e l l _ e a l e October 13th, 2016 19 / 37
ELA for Multi-Objective Problems October 13th, 2016 20 / 37
ELA for Multi-Objective Problems source: lmarti.github.io October 13th, 2016 21 / 37
ELA for Multi-Objective Problems in single-objective optimization, ELA has shown to be useful for describing the problem landscape based on a small initial design currently, there exist almost no landscape features for continuous multi-objective optimization problems let’s convert the single-objective high-level properties to the multi-objective case ⇒ multimodality in mixed-sphere problems 4 4Kerschke, P., Wang, H., Preuss, M., Grimme, C., Deutz, A., Trautmann, H., & Emmerich, M. (2016). Towards Analyzing Multimodality of Multiobjective Landscapes . In Proceedings of PPSN 2016 (pp. 962-972) October 13th, 2016 22 / 37
Mixed-Sphere Problems X 2 X 2 X 1 X 1 October 13th, 2016 23 / 37
Mixed-Sphere Problems X 2 X 2 X 1 X 2 X 1 X 1 October 13th, 2016 24 / 37
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