Introduction Preferences Budgeted optimization Batch criteria Budgeted Bayesian Multiobjective Optimization David Gaudrie 1 , Rodolphe Le Riche 2 , Victor Picheny 3 , ıt Enaux 1 , Vincent Herbert 1 Benoˆ 1 Groupe PSA 2 CNRS LIMOS, ´ erieure des Mines de Saint-´ Ecole Nationale Sup´ Etienne 3 Prowler.io Etienne, November 28 th 2019 ees de la Chaire Oquaido, Saint-´ Journ´
Introduction Preferences Budgeted optimization Batch criteria Context Optimization of parametric systems.
Introduction Preferences Budgeted optimization Batch criteria Context Optimization of parametric systems. Multiobjective optimization problems: x ∈ X ⊂ R d ( f 1 ( x ) , . . . , f m ( x )) . min
Introduction Preferences Budgeted optimization Batch criteria Context Optimization of parametric systems. Multiobjective optimization problems: x ∈ X ⊂ R d ( f 1 ( x ) , . . . , f m ( x )) . min Complex systems and physics ⇒ expensive simulations ⇒ restricted budget ( ≈ 100 evaluations). How to obtain optimal and relevant solutions in spite of the extremely parsimonious use of the computer code?
Introduction Preferences Budgeted optimization Batch criteria Context Optimization of parametric systems. Multiobjective optimization problems: x ∈ X ⊂ R d ( f 1 ( x ) , . . . , f m ( x )) . min Complex systems and physics ⇒ expensive simulations ⇒ restricted budget ( ≈ 100 evaluations). How to obtain optimal and relevant solutions in spite of the extremely parsimonious use of the computer code? Bayesian Multiobjective Optimization
Introduction Preferences Budgeted optimization Batch criteria Bayesian Multiobjective Optimization Extension of EGO [Jones et al., 1998] to multiple objectives.
Introduction Preferences Budgeted optimization Batch criteria Bayesian Multiobjective Optimization Extension of EGO [Jones et al., 1998] to multiple objectives.
Introduction Preferences Budgeted optimization Batch criteria EHI: A Bayesian Multiobjective Infill Criterion Expected Hypervolume Improvement (EHI) [Emmerich et al., 2006] aims at uncovering the entire P Y in few calls to f ( · ).
Introduction Preferences Budgeted optimization Batch criteria EHI: A Bayesian Multiobjective Infill Criterion Expected Hypervolume Improvement (EHI) [Emmerich et al., 2006] aims at uncovering the entire P Y in few calls to f ( · ). EHI( x ; R ) = E [ I H ( � P Y ∪ { Y ( x ) } ; R )]: expected growth of the hypervolume indicator [Zitzler, 1999].
Introduction Preferences Budgeted optimization Batch criteria EHI: A Bayesian Multiobjective Infill Criterion Initial DoE 1.1 0.4 f 2 −1.0 −1.7 f 1 −2.2 −1.2 0.0 1.0
Introduction Preferences Budgeted optimization Batch criteria EHI: A Bayesian Multiobjective Infill Criterion After 1 iteration 1.1 0.4 f 2 1 −1.0 −1.7 f 1 −2.2 −1.2 0.0 1.0
Introduction Preferences Budgeted optimization Batch criteria EHI: A Bayesian Multiobjective Infill Criterion After 5 iterations 1.1 0.4 3 f 2 1 −1.0 5 2 4 −1.7 f 1 −2.2 −1.2 0.0 1.0
Introduction Preferences Budgeted optimization Batch criteria EHI: A Bayesian Multiobjective Infill Criterion After 20 iterations 1.1 0.4 20 3 7 f 2 14 1 10 17 −1.0 11 5 12 2 15 8 4 6 −1.7 18 9 19 13 16 f 1 −2.2 −1.2 0.0 1.0
Introduction Preferences Budgeted optimization Batch criteria EHI: A Bayesian Multiobjective Infill Criterion After 40 iterations 1.1 0.4 20 3 37 39 22 7 40 24 f 2 14 36 1 10 23 17 −1.0 28 11 5 12 2 15 27 34 21 8 4 30 33 6 −1.7 18 9 2526 31 35 19 29 13 32 38 16 f 1 −2.2 −1.2 0.0 1.0
Introduction Preferences Budgeted optimization Batch criteria Targeting: Motivation Good convergence towards P Y , but
Introduction Preferences Budgeted optimization Batch criteria Targeting: Motivation Good convergence towards P Y , but The size of P Y grows exponentially with m ⇒ it might not be possible to approximate it accurately under a restricted budget.
Introduction Preferences Budgeted optimization Batch criteria Targeting: Motivation Good convergence towards P Y , but The size of P Y grows exponentially with m ⇒ it might not be possible to approximate it accurately under a restricted budget. All Pareto-optimal solutions may not satisfy the decision maker (especially with large m ).
Introduction Preferences Budgeted optimization Batch criteria Targeting: Motivation Good convergence towards P Y , but The size of P Y grows exponentially with m ⇒ it might not be possible to approximate it accurately under a restricted budget. All Pareto-optimal solutions may not satisfy the decision maker (especially with large m ). ⇒ Prioritize and enhance convergence towards preferred parts of the Pareto front.
Introduction Preferences Budgeted optimization Batch criteria Preferred regions Determined through a user-provided reference point R ∈ R m to be attained/improved. 1.0 1.0 N N R 0.7 0.7 R f 2 f 2 R 0.3 0.3 R I I 0.0 0.0 f 1 f 1 0.0 0.3 0.7 1.0 0.0 0.3 0.7 1.0
Introduction Preferences Budgeted optimization Batch criteria Preferred regions Determined through a user-provided reference point R ∈ R m to be attained/improved. 1.0 1.0 N N R 0.7 0.7 R f 2 f 2 R 0.3 0.3 R I I 0.0 0.0 f 1 f 1 0.0 0.3 0.7 1.0 0.0 0.3 0.7 1.0 No preference expressed ⇒ employ the center of the Pareto front, a well-balanced solution , as a default preference.
Introduction Preferences Budgeted optimization Batch criteria Center of the Pareto front Definition [Gaudrie et al., 2018b] The center of a Pareto front C is the closest point in Euclidean distance to P Y on the Ideal-Nadir ( IN ) line L .
Introduction Preferences Budgeted optimization Batch criteria Center of the Pareto front Definition [Gaudrie et al., 2018b] The center of a Pareto front C is the closest point in Euclidean distance to P Y on the Ideal-Nadir ( IN ) line L .
Introduction Preferences Budgeted optimization Batch criteria Center of the Pareto front Properties [Gaudrie et al., 2018a] Invariance under an affine transformation of the objective space when L intersects P Y or when m = 2. a Low-sensitivity to the Ideal and to the Nadir point. In Game Theory: particular Kala¨ ı-Smorodinsky [Kalai and Smorodinsky, 1975] solution b (disagreement point ≡ N ). a exceptions may occur when m ≥ 3. b in the case of a convex objective space.
Introduction Preferences Budgeted optimization Batch criteria Center of the Pareto front Empirical front: might lead to weak estimates.
Introduction Preferences Budgeted optimization Batch criteria Center of the Pareto front Empirical front: might lead to weak estimates. Estimation using Gaussian Processes
Introduction Preferences Budgeted optimization Batch criteria Center of the Pareto front Empirical front: might lead to weak estimates. Estimation using Gaussian Processes Simulate n sim GPs conditioned by D t ⇒ simulated Pareto fronts ⇒ plausible values for I and N .
Introduction Preferences Budgeted optimization Batch criteria Center of the Pareto front Empirical front: might lead to weak estimates. Estimation using Gaussian Processes Simulate n sim GPs conditioned by D t ⇒ simulated Pareto fronts ⇒ plausible values for I and N . Choice of x ’s where Y ( · ) is simulated is critical!
Introduction Preferences Budgeted optimization Batch criteria Center of the Pareto front Empirical front: might lead to weak estimates. Estimation using Gaussian Processes Simulate n sim GPs conditioned by D t ⇒ simulated Pareto fronts ⇒ plausible values for I and N . Choice of x ’s where Y ( · ) is simulated is critical! Choose x ’s according to probability to lead to I or N ⇒ GP simulations driven towards extreme parts of P Y .
Introduction Preferences Budgeted optimization Batch criteria Center of the Pareto front Empirical front: might lead to weak estimates. Estimation using Gaussian Processes Simulate n sim GPs conditioned by D t ⇒ simulated Pareto fronts ⇒ plausible values for I and N . Choice of x ’s where Y ( · ) is simulated is critical! Choose x ’s according to probability to lead to I or N ⇒ GP simulations driven towards extreme parts of P Y . Estimated center � C : projection of closest point in empirical Pareto front � P Y on estimated Ideal-Nadir line � L .
Introduction Preferences Budgeted optimization Batch criteria Targeting in Bayesian Multiobjective Optimization Weighted EHI [Auger et al., 2009]: externally supplied function w ( y ) and weighted integration. Truncated EHI [Yang et al., 2016]: truncate normal distribution to a user-supplied box [ A , B ]. 1.1 1.1 0.4 0.4 f 2 f 2 B −1.0 −1.0 A −1.7 −1.7 f 1 f 1 −2.2 −1.2 0.0 1.0 −2.2 −1.2 0.0 1.0
Introduction Preferences Budgeted optimization Batch criteria Targeting in Bayesian Multiobjective Optimization EHI: hypervolume computed up to R ∈ R m .
Introduction Preferences Budgeted optimization Batch criteria Targeting in Bayesian Multiobjective Optimization EHI: hypervolume computed up to R ∈ R m . Defines an improvement region I R := { z ∈ R m : z � R } where solutions are sought.
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