Multiobjective optimization with evolutionary algorithms and metaheuristics in distributed computing environnement

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1 Multiobjective optimization with evolutionary algorithms and metaheuristics in distributed computing environnement Sébastien Verel LISIC Université du Littoral Côte d Opale Equipe OSMOSE verel@lisic.univ-littoral.fr Séminaire SERMA, February, 10, /44

2 Optimization Inputs Search space: Set of all feasible solutions, Objective function: Quality criterium X f : X IR Goal Find the best solution according to the criterium x = argmax f But, sometime, the set of all best solutions, good approximation of the best solution, good robust solution... 2/44

3 Contexte Black box Scenario We have only {(x 0, f (x 0 )), (x 1, f (x 1 )),...} given by an oracle No information is either not available or needed on the definition of objective function Objective function given by a computation, or a simulation Objective function can be irregular, non differentiable, non continous, etc. (Very) large search space for discrete case (combinatorial optimization), i.e. NP-complete problems 3/44

4 Contexte Typical applications Large combinatorial problems: Scheduling problems, logistic problems, DOE, problems from combinatorics (Schur numbers), etc. Calibration of models: Physic world Model(params) Simulator(params) Model(Params) = argmin M Error(Data, M) Shape optimization: Design (shape, parameters of design) using a model and a numerical simulator 4/44

5 Search algorithms Principle Enumeration of the search space A lot of ways to enumerate the search space Using random sampling: Monte Carlo technics Local search technics: 5/44

6 Search algorithms Single solution-based: Hill-climbing technics, Simulated-annealing, tabu search, Iterative Local Search, etc. Population solution-based: Genetic algorithm, Genetic programming, ant colony algorithm, etc. Design components are well-known Probability to decrease, Memory of path, of sub-space Diversity of population, etc. 6/44

7 Position of the work One Evolutionary Algorithm key point: Exploitation / Exploration tradeoff One main practicle difficulty: Choose operators, design components, value of parameters, representation of solutions Parameters setting (Lobo et al. 2007): Off-line before the run: parameter tuning, On-line during the run: parameter control. 7/44

8 Position of the work One Evolutionary Algorithm key point: Exploitation / Exploration tradeoff One main practicle difficulty: Choose operators, design components, value of parameters, representation of solutions Parameters setting (Lobo et al. 2007): Off-line before the run: parameter tuning, On-line during the run: parameter control. Main question How to combine correctly the design components according to the problem? 7/44

9 Fitness landscapes Definition Fitness landscape (X, N, f ) X is the search space N : X 2 X is a neighborhood relation f : X IR is a objective function 8/44

10 Multimodal Fitness landscapes Local optima x no neighbor solution with higher fitness value x N (x ), f (x) f (x ) Fitness Search space 9/44

11 Local Optima Network (LON) Join work Gabriela Ochoa (univ. Stirling), Marco Tomassini (univ. Lausanne), Fabio Daolio (univ. Lausanne) 10/44

12 Energy surface and inherent networks (Doye, 2002) a b c Model of 2D energy surface Contour plot, partition of the configuration space into basins of attraction surrounding minima landscape as a network Inherent network: Nodes: energy minima Edges: two nodes are connected if the energy barrier separating them is sufficiently low (transition state) 11/44

13 Local optima network [GECCO 08 - IEEE TEC 10] fit= fit= fit= Nodes : local optima Edges : transition probabilities /44

14 Structure of Local Optima Network Quadratic Assignment Problem [CEC 10 - Phys A 11] Random instance Real-like instance Structure of the Local Optima Network related to search difficulty 13/44

15 Network Metrics nv: vertices lo: path to best lv: avg path fnn: fitness corr. wii: self loops cc: clust. coef. zout: out degree y 2: disparity knn: degree corr. 14/44

16 ILS Performance vs LON Metrics [GECCO 12] 1 Multiple linear regression on all possible predictors: perf = β 0 + β 1 k + β 2 log(nv) + β 2 lo + + β 10 knn + ɛ 2 Step-wise backward elimination of each predictor in turn. Table : Summary statistics of the final linear regression model. Multiple R-squared: , Adjusted R-squared: Predictor βi Std. Error p-value (Intercept) lo zout y knn /44

17 Tuning and Local Optima Network Relevant features using complex system technics Search difficulty is related to the features of LON: Classification of instances Understanding the search difficulty Guideline of design Time complexity can be predicted: Tuning with Portefolio technics 16/44

18 Multiobjective optimization Multiobjective optimization problem X : set of feasible solutions in the decision space M 2 objective functions f = (f 1, f 2,..., f M ) (to maximize) Z = f (X ) IR M : set of feasible outcome vectors in the objective space x 2 f 2 Decision space x 1 Objective space f 1 17/44

19 Definition Pareto dominance relation A solution x X dominates a solution x X (x x) iff i {1, 2,..., M}, f i (x ) f i (x) j {1, 2,..., M} such that f j (x ) < f j (x) x 2 f 2 dominated nondominated vector nondominated solution vector Decision space x 1 Objective space f 1 18/44

20 Pareto set, Pareto front x 2 Pareto optimal solution f 2 Pareto front Pareto optimal set Decision space x 1 Objective space f 1 Goal Find the Pareto Optimal Set, or a good approximation of the Pareto Optimal Set 19/44

21 Why multiobjective optimization? Position of multiobjective optimization No a priori on the importance of the different objectives a posterioro selection by a decision marker: Selection of one Pareto optimal solution after a deep study of the possible solutions. 20/44

22 Motivations Join work Arnaud Liefooghe, Clarisse Dhaenens, Laetitia Jourdan, Jérémie Humeau, DOLPHIN Team, INRIA Lille - Nord Europe Goal Understand the dynamics of search algorithms and evaluate the time complexity Guideline to design efficient local search algorithms Portefolio: between of set of algorithms, select the best one 21/44

23 Local search algorithms Two main classes: Scalar approaches: multiple scalarized aggregations of the objective functions, only able to find a subset of Pareto solutions (supported solutions) Pareto-based approaches: directly or indirectly focus the search on the Pareto dominance relation f 2 supported solution f 2 No accept Accept Objective space f 1 Objective space f 1 22/44

24 Scalar approaches multiple scalarized aggregations of the objective functions f 2 supported solution Different aggregations Weighted sum: f (x) = λ i f i (x) i=1..m Weighted Tchebycheff: Objective space f 1... f (x) = max {λ i(ri f i (x))} i=1..m 23/44

25 A Pareto-based approach: Pareto Local Search Archive solutions using Dominance relation Iteratively improve this archive by exploring the neighborhood f 2 f 2 Accept neighbor No Accept current archive current archive Objective space f 2 f 1 f 2 Objective space f 1 Accept current archive current archive Objective space f 1 Objective space f 1 24/44

26 What makes a multiobjective optimization problem difficult? The search space size? The difficulty of underlying single-objective problems? The number of objectives? The objective correlation? The number of non-dominated solutions? The shape of the Pareto front? what are the relevant features? 25/44

27 Summary of problem features low-level features k number of epistatic interactions unknown (black-box) m number of objective functions known ρ objective correlation known (estimated?) high-level features npo number of Pareto optimal solutions enumeration (bounds) hv hypervolume enumeration avgd average distance between Pareto sol. enumeration maxd maximum distance between Pareto sol. enumeration nconnec number of connected components enumeration lconnec size of the largest connected component enumeration kconnec minimum distance to be connected enumeration nplo number of Pareto local optima enumeration (estim.) ladapt length of a Pareto-based adaptive walk estimation corhv correlation length of solution hypervolume estimation corlhv correlation length of local hypervolume estimation... correlation analysis? 26/44

28 Contexte Fitness landscapes Multiobjective Optimization Distributed MO DAMS Objective Objective 2 Objective 2 MNK-landscapes with tunable objective correlation Objective Conflicting objectives ρ = 0.9 Large Pareto set Few Pareto sol Objective Independent objectives ρ = Objective Correlated objectives ρ = 0.9 Small Pareto set All supported sol. 27/44

29 Summary of results and guidelines for local search design Landscape features Problem properties Suggestion for the N K M ρ design of local search Cardinality of the Pareto optimal set Proportion of supported solutions Connectedness of the Pareto optimal set Number of Pareto local optima ρ 0 ρ < 0 + limited-size archive unbounded archive + scalar efficient scalar not efficient + two-phase efficient two-phase not efficient + Pareto not efficient Pareto efficient Following project... 28/44

30 Join work K. Tanaka, H. Aguirre, Shinshu university, Nagano, Japon A. Liefooghe, Inria Lille - Nord Europe 29/44

31 Performance vs. problem features 30/44

32 Performance vs. problem features 0 m, ρ, npo, hv connectedness (nconnec, lconnec, kconnec) + distance between Pareto solutions (avgd, maxd) + number of local optima (nplo, ladapt) ++ ruggedness (k, corhv, corlhv) 30/44

33 Performance prediction Predicting the performance of EMO algorithm? Multiple linear regression model(s) log(ert) = β 0 + β 1 k + β 2 m + β 3 ρ + β 4 npo β 14 corlhv + e Towards a portfolio selection approach (in progress) 31/44

34 Distributed Localized Bi-objective Search Join work Bilel Derbel, Arnaud Liefooghe, Jérémie Humeau, Université Lille 1, INRIA Lille Nord Europe 32/44

35 Motivations Context High computational cost Subdivide the problem into subtasks which can be processed in parallel Most approaches: top-down, centralized need global information to adapt its design components Distributed computing, local information is often sufficient. Crossroads approach Parallel cooperative computation Decomposition-based algorithms Adaptive search 33/44

36 Principles Between fully independent and fully centralized approach One node of computation = One solution Each node optimize a localized fitness function weighted sum, Tchebichev, etc. (based on search direction) Communicate the state of the search position to neighboring nodes f 2 f 2 Objective space f 1 Objective space f 1 34/44

37 Expected dynamic 35/44

38 Observed dynamic t=112 f2 0.6 f t= f1 t= t= f1 Tested on benchmark Effective in terms of: approximation quality, computational time, scalability 36/44

39 Position of the work One EA key point: Exploitation / Exploration tradeoff One main practicle difficulty: Choose operators, design components, value of parameters, representation of solutions Parameters setting (Lobo et al. 2007): Off-line before the run: parameter tuning, On-line during the run: parameter control. Increasing number of computation ressources (GPU,...) but, add parameters Main question Control the execution of different metaheuristics in a distributed environment 37/44

40 Distributed Adaptive Metaheuristic Selection (DAMS) [GECCO, 2011] Join work Bilel Derbel, Université Lille 1, INRIA Lille Nord Europe 38/44

41 Execution of metaheuristics in a distributed environment 3 metaheuristics {M 1, M 2, M 3 } 4 nodes of computation Which algorithm can we design? M 1 M 2 M 3 39/44

42 Execution of metaheuristics in a distributed environment 3 metaheuristics {M 1, M 2, M 3 } 4 nodes of computation Which algorithm can we design? M 1 M 3 M 3 M 3 M 3 M 3 M 3 M 2 M 3 M 3 M 3 M 3 M 3 M 3 M 3 39/44

43 Execution of metaheuristics in a distributed environment 3 metaheuristics {M 1, M 2, M 3 } 4 nodes of computation Which algorithm can we design? M 1 M 3 M 3 M 3 M 3 M 3 M 3 M 2 M 3 M 3 M 2 M 3 M 2 M 3 M 2 39/44

44 Execution of metaheuristics in a distributed environment 3 metaheuristics {M 1, M 2, M 3 } 4 nodes of computation M 1 M 2 M 3 40/44

45 Execution of metaheuristics in a distributed environment 3 metaheuristics {M 1, M 2, M 3 } 4 nodes of computation M 1 M 3 M 3 M 1 M 2 M 1 M 1 M 2 M 3 M 3 M 3 M 1 M 2 M 1 M 1 Goal of control: Optimal strategy The strategy that gives the best overall performances: At first, execution of M 3 then, M 1 and M 2, and then, M 1. 40/44

46 Control: Local strategy m n attachements with n nodes and m metaheuristics Local distributed strategy: For each computational node: Measure the efficiency of neighboring metaheuristics Select a metaheuristic according to the local information Trade-off between: Exploitation of the most promizing metaheuristics Exploration of news ones 41/44

47 Select Best and Mutate strategy (SBM) For each nodes: k initmeta() P initpop() reward 0 while Not stop do Migration of populations Send (k, reward) Receive i (k i, reward i ) k best meta from k i s if rnd(0, 1) < p mut then k random meta end if P new meta k (P) reward Reward(P, P new ) P P new end while 42/44

48 Select Best and Mutate strategy (SBM) Select best one (rate 1 p mut) Select random one (rate p mut) For each nodes: k initmeta() P initpop() reward 0 while Not stop do Migration of populations Send (k, reward) Receive i (k i, reward i ) k best meta from k i s if rnd(0, 1) < p mut then k random meta end if P new meta k (P) reward Reward(P, P new ) P P new end while 42/44

49 Dynamics of SBM-DAMS Average frequency and fitness fitness 5 bits 3 bits 1 bit bit-flip Frequency Rounds First, 1/l bit-flip mutation (opposite of sequential oracle) Then 5-bits, 3-bits, 1-bit, 1/l bit-flip See sbmdams.mov 43/44

50 Conclusion One EA key point: Exploitation / Exploration tradeoff One main practicle difficulty: Choose operators, design components, value of parameters Fitness Landscapes: Features, description of the search space Off-line tuning: Model to predict the performances, Design guideline, portefolio On-line control: Machine learning technics Toward (more) automatic design 44/44