Surrogate models for Single and Multi-Objective Stochastic Optimization: Integrating Support Vector Machines and Covariance-Matrix Adaptation-ES

Transcription

1 Covariance Matrix Adaptation-Evolution Strategy Surrogate models for Single and Multi-Objective Stochastic Optimization: Integrating and Covariance-Matrix Adaptation-ES Ilya Loshchilov, Marc Schoenauer, Michèle Sebag TAO CNRS INRIA Univ. Paris-Sud May 23rd, 2011 Michèle Sebag Surrogate optimization: SVM for CMA 1/ 47

2 Motivations Find Argmin {F : X IR} Context: ill-posed optimization problems Function F (fitness function) on X IR d Gradient not available or not useful F available as an oracle (black box) continuous Build {x 1,x 2,...} Argmin(F) Black-box approaches + Applicable + Robust comparison-based approaches are invariant High computational costs: number of function evaluations Michèle Sebag Surrogate optimization: SVM for CMA 2/ 47

3 Surrogate optimization Principle Gather E = {(x i,f(x i ))} Build ˆF from E training set learn surrogate model Use surrogate model ˆF for some time: Optimization: use ˆF instead of true F in std algo Filtering: select promising x i based on ˆF in population-based algo. Compute F(x i ) for some x i Update ˆF Iterate Michèle Sebag Surrogate optimization: SVM for CMA 3/ 47

4 Surrogate optimization, cont Issues Learning Hypothesis space (polynoms, neural nets, Gaussian processes,...) Selection of training set (prune, update,...) What is the learning target? Interaction of Learning & Optimization modules Schedule (when to relearn) How to use ˆF to support optimization search How to use search results to support learning ˆF This talk Using Covariance-Matrix Estimation within Support Vector Machines Using SVM for multi-objective optimization Michèle Sebag Surrogate optimization: SVM for CMA 4/ 47

5 Content 1 Covariance Matrix Adaptation-Evolution Strategy Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization 2 Statistical Machine Learning Linear classifiers The kernel trick 3 Previous Work Mixing Rank-SVM and Local Information Experiments 4 Background Dominance-based Surrogate Experimental Validation Michèle Sebag Surrogate optimization: SVM for CMA 5/ 47

6 Stochastic Search Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization A black box search template to minimize f : R n R Initialize distribution parameters θ, set sample size λ N While not terminate 1 Sample distribution P (x θ) x 1,...,x λ R n 2 Evaluate x 1,...,x λ on f 3 Update parameters θ F θ (θ,x 1,...,x λ,f(x 1 ),...,f(x λ )) Covers Deterministic algorithms, Evolutionary Algorithms, PSO, DE P implicitly defined by the variation operators Estimation of Distribution Algorithms Michèle Sebag Surrogate optimization: SVM for CMA 6/ 47

7 The (µ, λ) Evolution Strategy Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization Gaussian Mutations x i m+σn i (0,C) for i = 1,...,λ as perturbations of m where x i,m R n, σ R +, and C R n n where the mean vector m R n represents the favorite solution the so-called step-size σ R + controls the step length the covariance matrix C R n n determines the shape of the distribution ellipsoid How to update m, σ, and C? Michèle Sebag Surrogate optimization: SVM for CMA 7/ 47

8 History Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization The one-fifth rule Rechenberg, 73 One single parameter σ for the whole population Measure empirical success rate Increase σ if too large, decrease σ if too small Often wrong in non-smooth landscapes Self-adaptive mutations Schwefel, 81 Each individual carries its own mutation parameter Log-normal mutation of mutation parameters (Normal) mutation of individual Adaptation is slow for full covariance case from 1 to n2 n 2 Michèle Sebag Surrogate optimization: SVM for CMA 8/ 47

9 Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization Cumulative Step-Size Adaptation (CSA) x i = m+σy i m m+σy w Measure the length of the evolution path the pathway of the mean vector m in the generation sequence decrease σ increase σ loosely speaking steps are perpendicular under random selection (in expectation) perpendicular in the desired situation (to be most efficient) Michèle Sebag Surrogate optimization: SVM for CMA 9/ 47

10 Covariance Matrix Adaptation Rank-One Update Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization m m+σy w, y w = µ i=1 w iy i:λ, y i N i (0,C) initial distribution, C = I new distribution: C 0.8 C+0.2 y w y T w ruling principle: the adaptation increases the probability of successful steps, y w, to appear again Michèle Sebag Surrogate optimization: SVM for CMA 10/ 47

11 Covariance Matrix Adaptation Rank-One Update Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization m m+σy w, y w = µ i=1 w iy i:λ, y i N i (0,C) y w, movement of the population mean m (disregarding σ) new distribution: C 0.8 C+0.2 y w y T w ruling principle: the adaptation increases the probability of successful steps, y w, to appear again Michèle Sebag Surrogate optimization: SVM for CMA 10/ 47

12 Covariance Matrix Adaptation Rank-One Update Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization m m+σy w, y w = µ i=1 w iy i:λ, y i N i (0,C) mixture of distribution C and step y w, C 0.8 C+0.2 y w y T w new distribution: C 0.8 C+0.2 y w y T w ruling principle: the adaptation increases the probability of successful steps, y w, to appear again Michèle Sebag Surrogate optimization: SVM for CMA 10/ 47

13 Covariance Matrix Adaptation Rank-One Update Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization m m+σy w, y w = µ i=1 w iy i:λ, y i N i (0,C) new distribution (disregarding σ) new distribution: C 0.8 C+0.2 y w y T w ruling principle: the adaptation increases the probability of successful steps, y w, to appear again Michèle Sebag Surrogate optimization: SVM for CMA 10/ 47

14 Covariance Matrix Adaptation Rank-One Update Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization m m+σy w, y w = µ i=1 w iy i:λ, y i N i (0,C) movement of the population mean m new distribution: C 0.8 C+0.2 y w y T w ruling principle: the adaptation increases the probability of successful steps, y w, to appear again Michèle Sebag Surrogate optimization: SVM for CMA 10/ 47

15 Covariance Matrix Adaptation Rank-One Update Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization m m+σy w, y w = µ i=1 w iy i:λ, y i N i (0,C) mixture of distribution C and step y w, C 0.8 C+0.2 y w y T w new distribution: C 0.8 C+0.2 y w y T w ruling principle: the adaptation increases the probability of successful steps, y w, to appear again Michèle Sebag Surrogate optimization: SVM for CMA 10/ 47

16 Covariance Matrix Adaptation Rank-One Update Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization m m+σy w, y w = µ i=1 w iy i:λ, y i N i (0,C) new distribution: C 0.8 C+0.2 y w y T w ruling principle: the adaptation increases the probability of successful steps, y w, to appear again Michèle Sebag Surrogate optimization: SVM for CMA 10/ 47

17 Rank-µ Update Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization x i = m + σy i, y i N i(0,c), m m + σy w y w = µ i=1 wi y i:λ x i = m + σ y i, y i N (0,C) Cµ = µ 1 yi:λ y T i:λ C (1 1) C + 1 Cµ mnew m + 1 µ yi:λ sampling of λ = 150 solutions where C = I and σ = 1 calculating C from µ = 50 points, w 1 = = w µ = 1 µ new distribution Remark: the old (sample) distribution shape has a great influence on the new distribution iterations needed Michèle Sebag Surrogate optimization: SVM for CMA 11/ 47

18 Covariance Matrix Adaptation-Evolution Strategy Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization Invariance: Guarantee for Generalization Invariance properties of CMA-ES Invariance to order preserving transformations in function space like all comparison-based algorithms Translation and rotation invariance to rigid transformations of the search space CMA-ES is almost parameterless Tuning of a small set of functions Hansen & Ostermeier 2001 Default values generalize to whole classes Exception: population size for multi-modal functions but try the Restart-CMA-ES Auger & Hansen, 2005 Michèle Sebag Surrogate optimization: SVM for CMA 12/ 47

19 State-of-the-art Results Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization BBOB Black-Box Optimization Benchmarking ACM-GECCO workshop, in 2009 and 2010 Set of 25 benchmark functions, dimensions 2 to 40 With known difficulties (ill-conditioning, non-separability,... ) Noisy and non-noisy versions Competitors include BFGS (Matlab version), Fletcher-Powell, DFO (Derivative-Free Optimization, Powell 04) Differential Evolution Particle Swarm Optimization and many more Michèle Sebag Surrogate optimization: SVM for CMA 13/ 47

20 Contents Statistical Machine Learning Linear classifiers The kernel trick 1 Covariance Matrix Adaptation-Evolution Strategy Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization 2 Statistical Machine Learning Linear classifiers The kernel trick 3 Previous Work Mixing Rank-SVM and Local Information Experiments 4 Background Dominance-based Surrogate Experimental Validation Michèle Sebag Surrogate optimization: SVM for CMA 14/ 47

21 Supervised Machine Learning Statistical Machine Learning Linear classifiers The kernel trick Context Universe instance x i Oracle y i Input: Training set E = {(x i,y i ),i = 1...n,x i X,y i Y} Output: Hypothesis h : X Y Criterion: Quality of h Michèle Sebag Surrogate optimization: SVM for CMA 15/ 47

22 Supervised Machine Learning, 2 Statistical Machine Learning Linear classifiers The kernel trick Definitions E = {(x i,y i ),x i X,y i Y,i = 1...n} Classification : Y finite Regression : Y IR Hypothesis space H : X Y failure/ok time to failure Tasks Select H model selection Assess h H expected accuracy IE[h(x) y] Find h in H minimizing the error cost in expectation h = Arg min{ie[l(h(x) y),h H} Michèle Sebag Surrogate optimization: SVM for CMA 16/ 47

23 Dilemma Statistical Machine Learning Linear classifiers The kernel trick Fitting the data Bias variance tradeoff Michèle Sebag Surrogate optimization: SVM for CMA 17/ 47

24 Statistical Machine Learning Statistical Machine Learning Linear classifiers The kernel trick Minimize expected loss Minimize IE[l(h(x), y)] Principle If h is well-behaved on the training set, if the training set is representative and if h is regular, then h is well-behaved in expectation. n i=1 E[F] F(x i) +c(f,n) n Michèle Sebag Surrogate optimization: SVM for CMA 18/ 47

25 Statistical Machine Learning Linear classifiers The kernel trick Linear classification; the noiseless case H : X IR d IR prediction = sgn(h(x)) h(x) = < w,x > +b L 1 L 2 w L 3 b Michèle Sebag Surrogate optimization: SVM for CMA 19/ 47

26 Statistical Machine Learning Linear classifiers The kernel trick Linear classification; the noiseless case, 2 Example Constraint y i (< w,x i > +b) margin 0 Maximize minimum margin 2/ w support vector w b +1 b b 0-1 Formalisation { Minimize 1 2 w 2 subject to i, y i (< w,x i > +b) 1 2/ w i Michèle Sebag Surrogate optimization: SVM for CMA 20/ 47

27 Statistical Machine Learning Linear classifiers The kernel trick Linear classification; the noiseless case, 3 Primal form { Minimize 1 2 w 2 subject to i, y i (< w,x i > +b) 1 Using Lagrange multipliers: Minimize w,b max α { 1 2 w 2 Dual form Maximize α } n α i [y i (w x i b) 1] i=1 n α i 1 α i α j y i y j x i,x j 2 i=1 Optimization: quadratic programming i,j w = α i y i x i subject to α i 0,i = 1...n Michèle Sebag Surrogate optimization: SVM for CMA 21/ 47

28 Statistical Machine Learning Linear classifiers The kernel trick Linear classification; the noisy case Allow constraint violation; consider slack variables Primal form { Minimize 1 2 w 2 +C n i=1 ξ i subject to i, y i (< w,x i > +b) 1 ξ i, 0 ξ i Lagrange multipliers: { 1 n 2 w 2 +C ξ i Dual form Solution i=1 Maximize α Minimize w,b,ξ max α,β } n n α i [y i (w x i b) 1+ξ i ] β i ξ i i=1 n α i 1 α i α j y i y j x i,x j 2 i=1 i,j i=1 subject to 0 α i C,i = 1...n, α i y i = 0 support vectors h(x) = w,x = α i y i x i,x Michèle Sebag Surrogate optimization: SVM for CMA 22/ 47

29 The kernel trick Statistical Machine Learning Linear classifiers The kernel trick Intuition X Ω Φ : x = (x 1,x 2 ) (x 2 1, 2x 1.x 2,x 2 2) Principle: choose Φ, K such that Φ(x),Φ(x ) = K(x,x ) Michèle Sebag Surrogate optimization: SVM for CMA 23/ 47

30 The kernel trick, 2 Statistical Machine Learning Linear classifiers The kernel trick SVM only considers the scalar product PROS h(x) = i α iy i x i,x linear case h(x) = i α iy i K(x i,x) mboxkerneltrick A rich hypothesis space No computational overhead: no explicit mapping on the feature space Open problem: kernel design Michèle Sebag Surrogate optimization: SVM for CMA 24/ 47

31 The kernel trick, 3 Statistical Machine Learning Linear classifiers The kernel trick Kernels Polynomial: k(x i,x j ) = ( x i,x j +1) d Gaussian or Radial Basis Function: k(x i,x j ) = exp( xi xj 2 2σ 2 ) Hyperbolic tangent: k(x i,x j ) = tanh(k x i,x j +c) Examples for Polynomial (left) and Gaussian (right) Kernels: Michèle Sebag Surrogate optimization: SVM for CMA 25/ 47

32 Rank-based SVM Statistical Machine Learning Linear classifiers The kernel trick Learning to order things On training set E = {x i,i = 1...n} expert gives preferences: (x ik x jk ), k = 1...K underconstrained regression Order constraints Primal form { Minimize 1 2 w 2 +C K k=1 ξ k subject to k, w,x ik w,x jk 1 ξ k Michèle Sebag Surrogate optimization: SVM for CMA 26/ 47

33 Contents Previous Work Mixing Rank-SVM and Local Information Experiments 1 Covariance Matrix Adaptation-Evolution Strategy Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization 2 Statistical Machine Learning Linear classifiers The kernel trick 3 Previous Work Mixing Rank-SVM and Local Information Experiments 4 Background Dominance-based Surrogate Experimental Validation Michèle Sebag Surrogate optimization: SVM for CMA 27/ 47

34 Surrogate Models for CMA-ES Previous Work Mixing Rank-SVM and Local Information Experiments lmm-cma-es Build a full quadratic meta-model around current point Weighted by Mahalanobis distance from covariance matric Speed-up: a factor of 2-3 for n 4 Complexity: from O(n 4 ) to O(n 6 ) (intractable for n>16) Rank-invariance is lost S. Kern et al. (2006). "Local Meta-Models for Optimization Using Evolution Strategies" Z. Bouzarkouna et al. (2010). Investigating the lmm-cma-es for Large Population Sizes Michèle Sebag Surrogate optimization: SVM for CMA 28/ 47

35 Previous Work Mixing Rank-SVM and Local Information Experiments Surrogate Models for CMA-ES, cont Using Rank-SVM Builds a global model using Rank-SVM x i x j iff F(x i ) < F(x j ) Kernel and parameters highly problem-dependent Note: no use of information from current state of CMA T. Runarsson (2006). "Ordinal Regression in Evolutionary Computation" ACM Algorithm Use C from CMA-ES as Gaussian kernel I. Loschilov et al. (2010). "Comparison-based optimizers need comparison-based surrogates Michèle Sebag Surrogate optimization: SVM for CMA 29/ 47

36 Model Learning Non-separable Ellipsoid problem Previous Work Mixing Rank-SVM and Local Information Experiments X 2 0 X X X 1 Rank-SVM regression in original coordinate system Rank-SVM regression in transformed coordinate system given by current covariance matrix C and mean m: x = C 1 2(x m) Michèle Sebag Surrogate optimization: SVM for CMA 30/ 47

37 Using the Surrogate Model Previous Work Mixing Rank-SVM and Local Information Experiments Optimization: Significant Speed-Up... if global and accurate model Filtering: "Guaranteed" Speed-Up Prescreen (λ ) Evaluate (λ ) Retain with rank, Retain with rank λ, ~λ ~ Probability Density Rank Michèle Sebag Surrogate optimization: SVM for CMA 31/ 47

38 ACM-ES Optimization Loop Previous Work Mixing Rank-SVM and Local Information Experiments from the current covariance matrix A. Select training points B. Build a surrogate model 1. Select best k training points. 2. The change of coordinates, defined and the current mean value, reads [4]: X 2 0 X Add new λ training points and update parameters of CMA-ES. D. Select most promising children Prescreen (λ ) Evaluate (λ ) Probability Density Retain with rank, Retain with rank λ, ~ λ Rank -1 ~ X1-1 Rank-based Surrogate Model X1 3. Build a surrogate model using Rank SVM. C. Generate pre-children 4. Generate pre-children and rank them according to surrogate fitness function. Michèle Sebag Surrogate optimization: SVM for CMA 32/ 47

39 Parameters Previous Work Mixing Rank-SVM and Local Information Experiments SVM Learning Number of training points: N training = 30 d for all problems, except Rosenbrock and Rastrigin, where N training = 70 (d) Number of iterations: N iter = d Kernel function: RBF function with σ equal to the average distance of the training points The cost of constraint violation: C i = 10 6 (N training i) 2.0 Offspring Selection Number of test points: N test = 500 Number of evaluated offsprings: λ = λ 3 Offspring selection pressure parameters: σ 2 sel0 = 2σ2 sel1 = 0.8 Michèle Sebag Surrogate optimization: SVM for CMA 33/ 47

40 Results Speed-up Previous Work Mixing Rank-SVM and Local Information Experiments Speedup Schwefel 2 Schwefel 1/4 Rosenbrock 1 Noisy Sphere 1 Ackley Ellipsoid Problem Dimension Function n λ λ e ACM-ES spu CMA-ES Schwefel /4 Schwefel Rosenbrock (0.95) (0.90) (0.75) (0.9) NoisySphere (0.95) Ackley Ellipsoid Rastrigin (0.3) (0.75) Michèle Sebag Surrogate optimization: SVM for CMA 34/ 47

41 Results Learning Time Previous Work Mixing Rank-SVM and Local Information Experiments Cost of model learning/testing increases quasi-linearly with d on Sphere function: 10 Learning time (ms) Slope= Problem Dimension 10 3 Michèle Sebag Surrogate optimization: SVM for CMA 35/ 47

42 ACM-ES: conclusion Previous Work Mixing Rank-SVM and Local Information Experiments ACM-ES From 2 to 4 times faster on Uni-Modal Problems Invariance to rank-preserving transformations preserved Computation complexity is O(n) Available online at Open Issues Extention to multi-modal optimization On-line adaptation of selection pressure and surrogate model complexity Michèle Sebag Surrogate optimization: SVM for CMA 36/ 47

43 Contents Background Dominance-based Surrogate Experimental Validation 1 Covariance Matrix Adaptation-Evolution Strategy Evolution Strategies CMA-ES The state-of-the-art of (Stochastic) Optimization 2 Statistical Machine Learning Linear classifiers The kernel trick 3 Previous Work Mixing Rank-SVM and Local Information Experiments 4 Background Dominance-based Surrogate Experimental Validation Michèle Sebag Surrogate optimization: SVM for CMA 37/ 47

44 Background Dominance-based Surrogate Experimental Validation Multi-objective CMA-ES (MO-CMA-ES) MO-CMA-ES = µ mo independent (1+1)-CMA-ES. Each (1+1)-CMA samples new offspring. The size of the temporary population is 2µ mo. Only µ mo best solutions should be chosen for new population after the hypervolume-based non-dominated sorting. Update of CMA individuals takes place. Dominated Pareto Objective 2 Michèle Sebag Surrogate optimization: SVM for CMA 38/ 47

45 Background Dominance-based Surrogate Experimental Validation A Multi-Objective Surrogate Model Rationale Rationale: find a unique function F(x) that defines the aggregated quality of the solution x in multi-objective case. Idea originally proposed using a mixture of One-Class SVM and regression-svm a a I. Loshchilov, M. Schoenauer, M. Sebag (GECCO 2010). "A Mono Surrogate for Multiobjective Optimization" Dominated Pareto New Pareto p+e 2e p-e Objective 2 X2 Objective 1 FSVM p+e p-e Michèle Sebag Surrogate optimization: SVM for CMA 39/ 47 X1

46 Unsing the Surrogate Model Background Dominance-based Surrogate Experimental Validation Filtering Generate N inform pre-children For each pre-children A and the nearest parent B calculate Gain(A,B) = F svm (A) F svm (B) New children is the point with the maximum value of Gain true Pareto SVM Pareto X2 X1 Michèle Sebag Surrogate optimization: SVM for CMA 40/ 47

47 Dominance-Based Surrogate Using Rank-SVM Which ordered pairs? Background Dominance-based Surrogate Experimental Validation Considering all possible relations may be too expensive. Primary constraints: x and its nearest dominated point Secondary constraints: any 2 points not belonging to the same front (according to non-dominated sorting) Objective 2 a b c e f d FSVM > - constraints: Primary Secondary All primary constraints, and a limited number of secondary constraints Objective 1 Michèle Sebag Surrogate optimization: SVM for CMA 41/ 47

48 Dominance-Based Surrogate (2) Background Dominance-based Surrogate Experimental Validation Construction of the surrogate model Initialize archive Ω active as the set of Primary constraints, and Ω passive as the set of Secondary constraints. Learn the model for 1000 Ω active iterations. Add the most violated passive contraint from Ω passive to Ω active and optimize the model for 10 Ω active iterations. Repeat the last step 0.1 Ω active times. Michèle Sebag Surrogate optimization: SVM for CMA 42/ 47

49 Experimental Validation Parameters Background Dominance-based Surrogate Experimental Validation Surrogate Models ASM - aggregated surrogate model based on One-Class SVM and Regression SVM RASM - proposed Rank-based SVM SVM Learning Number of training points: at most N training = 1000 points Number of iterations: 1000 Ω active + Ω active 2 2N 2 training Kernel function: RBF function with σ equal to the average distance of the training points The cost of constraint violation: C = 1000 Offspring Selection Number of pre-children: p = 2 and p = 10 Michèle Sebag Surrogate optimization: SVM for CMA 43/ 47

50 Experimental Validation Comparative Results Background Dominance-based Surrogate Experimental Validation ASM and Rank-based ASM applied on top of NSGA-II (with hypervolume secondary criterion) and MO-CMA-ES, on ZDT and IHR functions. N = How many more true evaluations than best performer Michèle Sebag Surrogate optimization: SVM for CMA 44/ 47

51 Discussion Background Dominance-based Surrogate Experimental Validation Results on ZDT and IHR problems Rank-SVM versions are 1.5 times faster for p = for p = 10 before the algorithm reaches nearly-optimal Pareto points premature convergence of approximation of optimal µ-distribution: the surrogate model only enforces convergence toward Pareto front, but does not care about diversity. Michèle Sebag Surrogate optimization: SVM for CMA 45/ 47

52 Background Dominance-based Surrogate Experimental Validation Dominance-based Surrogate: Conclusion The proposed aggregated surrogate model is invariant to preserving transformation of the objective functions. The speed-up is significant, but limited to the convergence to the optimal Pareto front. Objective 2 a b g c e f d > - constraints: Primary Secondary = - constraints: Primary Secondary FSVM Thanks to the flexibility of SVM, "any" kind of preference could be taken into account: extreme points, "=" relations, Hypervolume Contribution, Decision Maker - defined relations. Objective 1 Michèle Sebag Surrogate optimization: SVM for CMA 46/ 47

53 Machine Learning for Optimization: Discussion Learning about the landscape easy Using available samples Using prior knowledge / constraints...using it to speed-up the search Dilemma Exploration vs Exploitation Multi-modal optimization very doable; more difficult Michèle Sebag Surrogate optimization: SVM for CMA 47/ 47