Most surrogate approaches to multi-objective optimization
build a surrogate model for each objective. These surrogates
can be used inside a classical Evolutionary Multiobjective
Optimization Algorithm (EMOA) in lieu of the actual objectives, without modifying the underlying EMOA; or to filter
out points that the models predict to be uninteresting. In
contrast, the proposed approach aims at building a global
surrogate model defined on the decision space and tightly
characterizing the current Pareto set and the dominated region, in order to speed up the evolution progress toward
the true Pareto set. This surrogate model is specified by
combining a One-class Support Vector Machine (SVMs) to
characterize the dominated points, and a Regression SVM to
clamp the Pareto front on a single value. The resulting surrogate model is then used within state-of-the-art EMOAs to
pre-screen the individuals generated by application of standard variation operators. Empirical validation on classical
MOO benchmark problems shows a significant reduction of
the number of evaluations of the actual objective functions.
Digital magic. A small project for controlling smart light bulbs.
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization
1. A Pareto-Compliant Surrogate Approach
for Multiobjective Optimization
Ilya Loshchilov, Marc Schoenauer and Michéle Sebag
TAO, INRIA, Univesité Paris-Sud XI
2. Main Idea → Pareto Surrogate
WE HAVE:
• Current Pareto and dominated points
WE WANT TO FIND F: decision space →
• F < 0 for dominated points
• F ≈ 0 for current Pareto points
WE EXPECT:
• F > 0 for new Pareto points
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 2
3. Outline
Support Vector Machine (SVM)
A Pareto-Compliant Approach for MOO
Experiments
Discussion and on-going work
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 3
4. Support Vector Machine (1): Classification
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 4
5. Support Vector Machine (2): Regression
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 5
6. Support Vector Machine (3): One-Class SVM
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 6
7. Outline
Support Vector Machine (SVM)
A Pareto-Compliant Approach for MOO
Experiments
Discussion and on-going work
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 7
8. Pareto Support Vector Machine (Pareto-SVM)
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 8
9. SVM-informed EMOA
How we can use F?
For optimization
• If we are sure that the model F is accurate enough far
from the current Pareto points
For filtering (select the most promising children according to F):
• F(children) > F(parent) → dangerous because of
approximation noise (epsilon)
• F(children) > F(nearest point from population) is more
robust criterion
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 9
10. SVM-informed EMOA: Filtering
Filtering procedure:
Generate pre-children
For each pre-children and the nearest parent
calculate
New children is the point with the maximum value of
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 10
11. Outline
Support Vector Machine (SVM)
A Pareto-Compliant Approach for MOO
Experiments
Discussion and on-going work
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 11
12. Pareto-SVM model validation
Generate 20.000 points at a given distance from a (known)
nearly-optimal Pareto front
Apply Non-dominated sorting to rank non-dominated fronts
, where
Choose and as the training data,
where – current Pareto points, – dominated points
Update Pareto-SVM model
Compare on different (use as the Test data)
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 12
13. Pareto-SVM model for ZDT1 and IHR1 problems
A, b,c,d
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 13
14. SVM-informed S-NSGA-2 and MO-CMA-ES on ZDT's and IHR's
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 14
15. Strong and Weak Points
+ Speedup of 2-2.5 on ZDT and IHR for first 15 000 evaluations
+ CPU cost of one learning ~ 0.5-1.0 sec. (complete run<15mn)
- High selection pressure may lead to premature convergence;
- The formulation of the SVM optimization problem still requires the
user decision*.
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 15
16. Issue: A more robust Pareto-SVM formulation
Original Formulation:
In feature space:
- Pareto set → cluster around single line
- Dominated points → one side of this line
Issues:
- Which side?
- Preserve symmetry ( and )
Solutions(?):
- Manually choose
- New Formulation:
D: additional parameter → harder learning problem
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 16
17. Perspective (1): Quality indicator-based Mono surrogate
Regression of indicator values (e.g. hypervolume)
Issues: - value for dominated points?
- value for extremal points?
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 17
18. Perspective (2): Rank-SVM (Ordinal Regression SVM)
Ordinal Regression SVM: surrogate constrained by ordered pairs
MO-case: derive constraints from Pareto dominance
If X dominates Y, add constraint (F(X)<F(Y))
Learning time
- 'quadratic' with number of constraints
- quasi-linear with dimension
Which constraints?
Results for single objective problems(*):
Cost of learning/testing with n-1 constraints
increases quasi-linearly with dimension.
(*) - "Comparison-Based Optimizers Need Comparison-Based
Surrogates” I.Loshchilov, M. Schoenauer, M. Sebag.
In PPSN-XI (2010).
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 18
19. Summary
Contributions
Single-valued surrogate to capture Pareto
dominance
Several alternative formulations
Using surrogate as filter limits possible speedup
Further work
Comparison with multi-surrogate
Rank-based SVM
A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 19