1.
Performance of Evolutionary Algorithms on NK
Landscapes with Nearest Neighbor Interactions
and Tunable Overlap
Martin Pelikan, Kumara Sastry, David E. Goldberg,
Martin V. Butz, and Mark Hauschild
Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)
University of Missouri, St. Louis, MO
http://medal.cs.umsl.edu/
pelikan@cs.umsl.edu
Download MEDAL Report No. 2009002
http://medal.cs.umsl.edu/files/2009002.pdf
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
2.
Motivation
Testing evolutionary algorithms
Adversarial problems on the boundary of design envelope.
Random instances of important classes of problems.
Real-world problems.
This work bridges and extends two prior studies on random
problems
Random additively decomposable problems (rADPs)
(Pelikan et al., 2006).
NK landscapes (superset of rADPs)
(Pelikan et al., 2007).
This study
Propose the class of polynomially solvable NK landscapes with
nearest neighbor interactions and tunable overlap.
Generate large number of instances of proposed problem class.
Test evolutionary algorithms on the generated instances.
Analyze the results.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
3.
Outline
1. Additively decomposable problems
NK landscapes.
Random additively decomposable problems (rADPs).
2. NK with nearest neighbors and tunable overlap.
3. Experiments.
4. Conclusions and future work.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
4.
Additively Decomposable Problems (ADPs)
Additively decomposable problem (ADP)
Fitness deﬁned as
m
f (X1 , X2 , . . . , Xn ) = fi (Si ),
i=1
n is the number of bits (variables),
m is the number of subproblems,
Si is the subset of variables in ith subproblem.
ADPs play crucial role in design and analysis of GAs & EDAs.
All problems in this work are ADPs.
Two prior studies on ADPs serve as starting points
Unrestricted NK landscapes.
Restricted random ADPs (rADPs).
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
5.
NK Landscape
NK landscape
Proposed by Kauﬀman (1989).
Model of rugged landscape and popular test function.
An NK landscape is deﬁned by
Number of bits, n.
Number of neighbors per bit, k.
Set of k neighbors Π(Xi ) for i-th bit, Xi .
Subfunction fi deﬁning contribution of Xi and Π(Xi ).
The objective function fnk to maximize is then deﬁned as
n−1
fnk (X0 , X1 , . . . , Xn−1 ) = fi (Xi , Π(Xi )).
i=0
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
6.
NK Landscape
Exmaple for n = 9 and k = 2:
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
7.
Restricted Random ADPs (rADPs) of Bounded Order
Order-k rADPs with and without overlap
Each subproblem contains k bits.
Separable problems contain non-overlapping subproblems:
Tight linkage: Shuﬄed:
There may be overlap in o bits between neighboring
subproblems (may also be shuﬄed):
Tight linkage: Shuﬄed:
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
8.
Properties of NK Landscapes and rADPs
Common properties
Additive decomposability.
Subproblems are complex (look-up tables).
High multimodality, complex structure.
Overlap further increases problem diﬃculty.
Challenge for most genetic algorithms and local search.
NK landscapes
NP-completeness (can’t solve worst case in polynomial time).
rADPs
Using prior knowledge of problem structure, we can exactly
solve rADPs in polynomial time (dynamic programming) in
O(2k n) evaluations.
Multivariate EDAs can solve shuﬄed EDAs polynomially fast.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
9.
NK Landscapes with Nearest Neighbors & Tunable Overlap
NK Landscapes with Nearest Neighbors and Tunable Overlap
Neighbors of each bit are restricted to the following k bits.
For simplicity, the neighborhoods don’t wrap around.
Some subproblems may be excluded to provide a mechanism
for tuning the size of overlap.
Use parameter step ∈ {1, 2, . . . , k + 1}.
Only subproblems at positions i, i mod step = 0 contribute.
Bit positions shuﬄed randomly to eliminate tight linkage.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
10.
NK Landscapes with Nearest Neighbors & Tunable Overlap
High overlap (k = 2, step = 1):
Sequential Shuﬄed
Note
step = 1 maximizes the amount of overlap between subproblems.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
11.
NK Landscapes with Nearest Neighbors & Tunable Overlap
Low overlap (k = 2, step = 2):
Sequential Shuﬄed
Note
step parameter allows tuning of the size of overlap.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
12.
NK Landscapes with Nearest Neighbors & Tunable Overlap
No overlap (k = 2, step = 3):
Sequential Shuﬄed
Note
step = k + 1 implies separability (subproblems are independent).
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
13.
NK Landscapes with Nearest Neighbors & Tunable Overlap
Why?
Nearest neighbors enable polynomial solvability
Deshuﬄe the string.
Use dynamic programming.
Parameter step enables tunining the overlap between
subproblems:
For standard NK landscapes, step = 1.
With larger values of step, the amount of overlap between
consequent subproblems is reduced.
For step = k + 1, the problem becomes separable (the
subproblems are fully independent).
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
14.
Problem Instances
Parameters
n = 20 to 120.
k = 2 to 5.
step = 1 to k + 1 for each k.
Variety of instances
For each (n, k, step), generate 10,000 random instances.
Overall 1,800,000 unique problem instances.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
15.
Compared Algorithms
Basic algorithms
Hierarchical Bayesian optimization algorithm (hBOA).
Genetic algorithm with uniform crossover (GAU).
Genetic algorithm with twopoint crossover (G2P).
Local search
Single-bit-ﬂip hill climbing (DHC) on each solution.
Improves performance of all methods.
Niching
Restricted tournament replacement (niching).
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
16.
Num
Results: Flips Until Optimum; hBOA; k = 2 and k = 5
2
10
20 40 60 80 100 20 40 60 80 100
Problem size Problem size
4
10 5
10 4
k=4, step=1 k=2, step=1 10
k=5, step=1 k=3, step=1
Number of flips (hBOA)
Number of flips (hBOA)
k=2, step=2 k=3, step=2
Number of flips (hBOA)
k=4, step=2 k=5, step=2
k=4, step=3 k=2, step=3 k=5, step=3 k=3, step=3
k=4, step=4 4 k=5, step=4 k=3, step=4
k=4, step=5 10 k=5, step=5
3
10 k=5, step=6 3
10
3
10
2 2
10 10
20 4020 60 80 40
100 60 80 100 20 40 20 60 80 100
40 60
Problem size Problem size Problem size Problem size
5
10
k=4, step=1 k=5, step=1
4Growth appears to be polynomial w.r.t. problem size, n.
umber of flips (hBOA)
umber of flips (hBOA)
10 k=4, step=2 k=5, step=2
Performance best with no overlap. for hBOA. k=5, step=3
Figurestep=3
k=4, 1: Average number of ﬂips
k=4, step=4 4 k=5, step=4
k=4, step=5 10
Besides n, performance depends on both k and step. step=5
k=5,
k=5, step=6
the eﬀects of k on performance of all compared algorithms, ﬁgure
10
3
6 sh
umber of DHC ﬂips with k for hBOA and GA on problems of size n = 3
M. are K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild was incapable 10 solving many inst
DA Pelikan,not included, because UMDA NK Landscapes with Nearest Neighbors and Tunable Overlap
of
17.
Results: Comparison w.r.t. Flips
DHC steps (ﬂips) until optimum
n k step hBOA GA (uniform) GA (twopoint)
120 5 1 37,155 141,108 220,318
120 5 2 40,151 212,635 353,748
120 5 3 37,480 249,217 443,570
120 5 4 27,411 195,673 310,894
120 5 5 15,589 100,378 145,406
120 5 6 9,607 35,101 47,576
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
18.
Results: Comparison w.r.t. Evaluations
Number of evaluations until optimum
n k step hBOA GA (uniform) GA (twopoint)
120 5 1 7,414 16,519 34,696
120 5 2 9,011 25,032 56,059
120 5 3 9,988 30,285 72,359
120 5 4 8,606 24,016 51,521
120 5 5 7,307 13,749 26,807
120 5 6 7,328 6,004 10,949
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
19.
0.75 0.75
Number
Num
Number
Nu
Results: Flips Until Optimum; hBOA vs. GA; k = 5
0.5 0.5
20 40 20
60 80 40
100 60 80 100 20 40 60 80 100
Problem size Problem size Problem size
k=4, step=1 7 7 k=5, step=1
Number of flips (GA, uniform) /
k=5, step=1 6
k=4, step=2 6 k=5, step=2
Number of flips (GA, twopoint) /
k=5, step=2
Number of flips (hBOA)
k=4, step=3 5 5 k=5, step=3
k=5, step=3
Number of flips (hBOA)
k=4, step=4 4 4 k=5, step=4
k=4, step=5 k=5, step=4 k=5, step=5
3 k=5, step=5 3 k=5, step=6
k=5, step=6
2 2
1 1
20 40 20 60 80 40
100 60 80 100 20 40 60 80 100
Problem size Problem size Problem size
hBOA outperforms both versions of GA.
rRatiowithDiﬀerences grow faster than with twopoint crossover and hBOA.
GA of the number of ﬂips for GA polynomially with n.
uniform crossover and hBOA.
Besides n, diﬀerences depend on both k and step.
f DHC ﬂips until optimum
=GA and step ∈ GA6}; since UMDA was not capable of solving many o
5, (uniform) {1, (twopoint)
s in141,108 time, the results for UMDA are not included. The ﬁgure sho
practical 220,318
ofM.DHC K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap sm
Pelikan, ﬂips until optimum for diﬀerent percentages of instances with
20.
Results: Correlations Between Algorithms
step = 1 (high overlap):
step = 6 (separable):
GA versions more similar than hBOA with GA.
Correlations stronger for problems with more overlap/less
structure.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
21.
Problem Diﬃculty: Signal-to-Noise and Signal Variance
Signal and noise
Signal: The diﬀerence between ﬁtness of the best and the 2nd
best solutions to a subproblem.
Noise: Models contributions of other subproblems.
Signal-to-noise ratio
Decision making done by GA is stochastic.
The larger the signal-to-noise ratio, the easier the decision
making.
Signal variance
Sequential vs. parallel convergence.
How much do contributions of diﬀerent subproblems diﬀer?
One way to model this is to look at the variance of the signal.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
22.
hBOA (a) hBOA (b) GA (uniform) (uniform)
(b) GA (c) GA (twopoint) (
(c) GA
Results: Flips Until Optimum; hBOA vs. GA; k = 5
re 13:Figure 13: of overlap of overlap for n = 1205 and k = 5 (step varies with o
Inﬂuence Inﬂuence for n = 120 and k = (step varies with overlap).
step = 1 (high overlap) step = 6 (separable)
1.075 1.075 1.075 1.075
GA (twpoint) GA (twpoint) GA (twpoint) GA (tw
GA (uniform) GA (uniform) GA (uniform) GA (u
Average number of flips
Average number of flips
Average number of flips
1.05 1.05 hBOA hBOA 1.05 1.05 hBOA hBOA
(divided by mean)
(divided by mean)
(divided by mean)
(divided by mean)
1.025 1.025 1.025 1.025
1 1 1 1
0.975 0.975 0.975 0.975
0.95 0.95 0.95 0.95
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.5 0.6 0.7 0.8 0.9 1
Signal to noise percentilenoise percentile (% smallest)
Signal to (% smallest) Signal to noise percentilenoise percentile (% s
Signal to (% smallest)
(a) step = 1(a) step = 1 (b) step = 6(b) step = 6
For separable problems, noise clearly matters.
For problems with overlap, noise appears insigniﬁcant.
:Figure 14: of signal-to-noise ratio on the number ofnumber of ﬂips for n = 120
Inﬂuence Inﬂuence of signal-to-noise ratio on the ﬂips for n = 120 and k =
cknowledgments M.V. Butz, M. Hauschild
edgments D.E. Goldberg,
M. Pelikan, K. Sastry, NK Landscapes with Nearest Neighbors and Tunable Overlap
23.
Results: Flips Until Optimum; hBOA vs. GA; k = 5
step = 1 (high overlap) step = 6 (separable)
1.1 1.1 1.1 1.1
GA (twopoint) GA (twopoint) GA (twopoint) (twopoi
GA
GA (uniform) GA (uniform) GA (uniform)GA (uniform
Average number of flips
1.075 1.075
Average number of flips
Average number of flips
1.075 1.075
hBOA hBOA hBOA hBOA
(divided by mean)
(divided by mean)
(divided by mean)
(divided by mean)
1.05 1.05 1.05 1.05
1.025 1.025 1.025 1.025
1 1 1 1
0.975 0.975 0.975 0.975
0.95 0.95 0.95 0.95
0.1 0.2 0.3 0.40.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.8 0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Signal variance percentile (% smallest) (% smallest)
Signal variance percentile Signal variance percentile (%percentile (% small
Signal variance smallest)
(a) step = (a) step = 1
1 (b) step = 6 step = 6
(b)
For separable problems, signal variance clearly matters.
For problems with overlap, signal variance appears
e 15: Inﬂuence Inﬂuence of signal variance on the of ﬂips for n = 120 n = 120 and
Figure 15: of signal variance on the number number of ﬂips for and k = 5.
insigniﬁcant.
eferences
es
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
24.
Conclusions and Future Work
Summary and conclusions
Considered subset of NK landscapes as class of random test
problems with tunable subproblem size and overlap.
All proposed instances solvable in polynomial time.
Generated a broad range of problem instances.
Analyzed results using hybrids of GEAs.
Future work
Use generated problems to test other algorithms.
Relate performance to other measures of problem diﬃculty.
Develop/test new tools for understanding of problem diﬃculty.
Wrap subproblems around.
Use other distributions for generating look-up tables.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
25.
Acknowledgments
Acknowledgments
NSF; NSF CAREER grant ECS-0547013.
U.S. Air Force, AFOSR; FA9550-06-1-0096.
University of Missouri; High Performance Computing
Collaboratory sponsored by Information Technology Services;
Research Award; Research Board.
M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
Be the first to comment