Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Multiobjective optimization
with evolutionary algorithms and metaheuristics
in distributed computing environnement
Sébastien Verel
LISIC
Université du Littoral Côte d’Opale
Equipe OSMOSE
[email protected]
http://www.lisic.univ-littoral.fr/~verel
Séminaire SERMA,
February, 10, 2014
1/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Optimization
Inputs
Search space: Set of all feasible solutions,
X
Objective function: Quality criterium
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Contexte
Black box Scenario
We have only {(x0 , f (x0 )), (x1 , f (x1 )), ...} 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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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) = argminM Error(Data, M)
Shape optimization:
Design (shape, parameters of design)
using a model and a numerical simulator
4/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Fitness landscapes
Definition
Fitness landscape (X , N , f )
X is the search space
N : X → 2X is a
neighborhood relation
f : X → IR is a
objective function
8/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Local Optima Network (LON)
Join work
Gabriela Ochoa (univ. Stirling),
Marco Tomassini (univ. Lausanne),
Fabio Daolio (univ. Lausanne)
10/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Energy surface and inherent networks (Doye, 2002)
a Model of 2D energy surface
b Contour plot, partition of
the configuration space into
basins of attraction
surrounding minima
c 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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Local optima network
[GECCO’08 - IEEE TEC’10]
0.76
0.65
0.185
fit=0.7657
fit=0.7133
0.29
0.4
Nodes :
local optima
Edges :
transition probabilities
0.27
0.05
0.055
fit=0.7046
0.33
12/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
ILS Performance vs LON Metrics
[GECCO’12]
1
Multiple linear regression on all possible predictors:
perf = β0 + β1 k + β2 log(nv ) + β2 lo + · · · + β10 knn + 2
Step-wise backward elimination of each predictor in turn.
Table : Summary statistics of the final linear regression model. Multiple
R-squared: 0.8494, Adjusted R-squared: 0.8471.
Predictor
(Intercept)
lo
zout
y2
knn
βbi
10.3838
0.0439
−0.0306
−7.2831
−0.7457
Std. Error
0.58512
0.00434
0.00831
1.63038
0.40501
p-value
9.24 · 10−47
1.67 · 10−20
2.81 · 10−04
1.18 · 10−05
6.67 · 10−02
15/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Multiobjective optimization
Multiobjective optimization problem
X : set of feasible solutions in the decision space
M > 2 objective functions f = (f1 , f2 , . . . , fM ) (to maximize)
Z = f (X ) ⊆ IRM : set of feasible outcome vectors in the
objective space
x2
f2
Decision space
x1
Objective space f1
17/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Definition
Pareto dominance relation
A solution x ∈ X dominates a solution x 0 ∈ X (x 0 ≺ x) iff
∀i ∈ {1, 2, . . . , M}, fi (x 0 ) 6 fi (x)
∃j ∈ {1, 2, . . . , M} such that fj (x 0 ) < fj (x)
x2
f2
nondominated
vector
dominated
nondominated
solution
Decision space
vector
x1
Objective space f1
18/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Pareto set, Pareto front
x2
f2
Pareto front
Pareto
optimal
solution
Pareto optimal
set
Decision space
x1
Objective space f1
Goal
Find the Pareto Optimal Set,
or a good approximation of the Pareto Optimal Set
19/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
f2
supported
solution
f2
Accept
No
accept
Objective space f1
Objective space f1
22/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Scalar approaches
• multiple scalarized aggregations of the objective functions
f2
supported
solution
Different aggregations
Weighted sum:
X
f (x) =
λi fi (x)
i=1..m
Weighted Tchebycheff:
f (x) = max {λi (ri∗ − fi (x))}
Objective space f1
i=1..m
...
23/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
A Pareto-based approach: Pareto Local Search
• Archive solutions using Dominance relation
• Iteratively improve this archive by exploring the neighborhood
f2
f2
Accept neighbor
No Accept
current
archive
f2
current
archive
Objective space f1
f2
Objective space f1
Accept
current
archive
Objective space f1
current
archive
Objective space f1
24/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Summary of problem features
k
m
ρ
npo
hv
avgd
maxd
nconnec
lconnec
kconnec
nplo
ladapt
corhv
corlhv
low-level features
number of epistatic interactions
number of objective functions
objective correlation
high-level features
number of Pareto optimal solutions
hypervolume
average distance between Pareto sol.
maximum distance between Pareto sol.
number of connected components
size of the largest connected component
minimum distance to be connected
number of Pareto local optima
length of a Pareto-based adaptive walk
correlation length of solution hypervolume
correlation length of local hypervolume
unknown (black-box)
known
known (estimated?)
enumeration (bounds)
enumeration
enumeration
enumeration
enumeration
enumeration
enumeration
enumeration (estim.)
estimation
estimation
estimation
. . . correlation analysis?
26/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
Objective 2
0.8
Objective 2
Objective 2
MNK-landscapes with tunable objective correlation
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.3
0.4
0.5
0.6
Objective 1
0.7
0.8
Conflicting objectives
ρ = −0.9
• Large Pareto set
• Few Pareto sol.
0.2
0.2
0.3
0.4
0.5
0.6
Objective 1
0.7
0.8
Independent objectives
ρ = 0.0
0.2
0.2
0.3
0.4
0.5
0.6
Objective 1
0.7
0.8
Correlated objectives
ρ = 0.9
• Small Pareto set
• All supported sol.
27/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Summary of results and guidelines for local search design
Landscape features
Problem properties
N
K
M
ρ
Cardinality of the
Pareto optimal set
Proportion of
supported solutions
Connectedness of the
Pareto optimal set
Number of
Pareto local optima
Suggestion for the
design of local search
+
−
+
−
limited-size archive
unbounded archive
scalar efficient
scalar not efficient
ρ≥0
+ two-phase efficient
ρ<0
− two-phase not efficient
+ Pareto not efficient
− Pareto efficient
Following project...
28/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Join work
K. Tanaka, H. Aguirre,
Shinshu university,
Nagano, Japon
A. Liefooghe,
Inria Lille - Nord Europe
29/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Performance vs. problem features
30/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Performance prediction
Predicting the performance of EMO algorithm?
Multiple linear regression model(s)
log(ert) = β0 + β1 · k + β2 · m + β3 · ρ + β4 · npo + . . . + β14 · corlhv + e
Towards a portfolio selection approach (in progress)
31/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Distributed Localized Bi-objective Search
Join work
Bilel Derbel, Arnaud Liefooghe, Jérémie Humeau,
Université Lille 1, INRIA Lille Nord Europe
32/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
f2
f2
Objective space f1
Objective space f1
34/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Expected dynamic
35/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Observed dynamic
0.7
0.7
0.65
0.65
t=112
f2
0.8
0.75
f2
0.8
0.75
0.6
0.6
0.55
0.55
0.5
0.5
t=16
t=32
t=0
0.45
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
f1
0.45
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
f1
Tested on benchmark
Effective in terms of:
approximation quality,
computational time,
scalability
36/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Distributed Adaptive Metaheuristic Selection (DAMS)
[GECCO, 2011]
Join work
Bilel Derbel,
Université Lille 1, INRIA Lille Nord Europe
38/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Execution of metaheuristics in a distributed environment
3 metaheuristics {M1 , M2 , M3 }
4 nodes of computation
Which algorithm can we design?
M1
M2
M3
39/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Execution of metaheuristics in a distributed environment
3 metaheuristics {M1 , M2 , M3 }
4 nodes of computation
Which algorithm can we design?
M1
M3
M3
M3
M3
M3
M3
M3
M3
M3
M3
M3
M3
M2
M3
39/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Execution of metaheuristics in a distributed environment
3 metaheuristics {M1 , M2 , M3 }
4 nodes of computation
Which algorithm can we design?
M1
M3
M3
M3
M3
M3
M3
M3
M2
M3
M2
M3
M2
M2
M3
39/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Execution of metaheuristics in a distributed environment
3 metaheuristics {M1 , M2 , M3 }
4 nodes of computation
M1
M2
M3
40/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Execution of metaheuristics in a distributed environment
3 metaheuristics {M1 , M2 , M3 }
4 nodes of computation
M1
M3
M3
M1
M2
M1
M1
M3
M3
M1
M2
M1
M1
M2
M3
Goal of control: Optimal strategy
The strategy that gives the best overall performances:
At first, execution of M3
then, M1 and M2 , and then, M1 .
40/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Control: Local strategy
mn attachements with n nodes and m metaheuristics
8
5
3
4
4
2
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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 (ki , rewardi )
k ← best meta from ki ’s
if rnd(0, 1) < pmut then
k ← random meta
end if
Pnew ← metak (P)
reward ← Reward(P, Pnew )
P ← Pnew
end while
42/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Select Best and Mutate strategy (SBM)
For each nodes:
8
5
3
4
2
4
Select ’best’ one (rate 1 − pmut )
8
5
3
4
2
4
Select random one (rate pmut )
8
5
3
k ← initMeta()
P ← initPop()
reward ← 0
while Not stop do
Migration of populations
Send (k, reward)
Receive ∀i (ki , rewardi )
4
4
2
k ← best meta from ki ’s
if rnd(0, 1) < pmut then
k ← random meta
end if
Pnew ← metak (P)
reward ← Reward(P, Pnew )
P ← Pnew
end while
42/44
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
Dynamics of SBM-DAMS
Average frequency and fitness
1
fitness
5 bits
3 bits
1 bit
bit-flip
Frequency
0.8
0.6
0.4
0.2
0
0
500
1000
1500
Rounds
2000
2500
3000
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
Contexte
Fitness landscapes
Multiobjective Optimization
Distributed MO
DAMS
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