Download Diversity Loss in General Estimation of Distribution Algorithms

Diversity Loss
in General Estimation of Distribution
Algorithms
J. L. Shapiro
PPSN (Parallel Problem Solving From Nature) ’06
BISCuit 2nd EDA Seminar
Abstract

A class of EDAs on which universal results on the rate of
diversity loss can be derived.
 The class (SML-EDA) requires two restrictions.
 In
each generation, the new probability model is built using only
sampled from the current probability model.
 Maximum likelihood is used to set model parameters.

The algorithm will never find the optimum unless the
population size grows exponentially.
EDA

Initialize the EDA probability model
(Start with a random population of M vectors.)
Repeat
Select N vectors using a selection method
Learn a new probability model
from the selected population
Sample M vectors from the probability model
Until stopping criteria are satisfied
SML-EDA

A restricted class of EDAs needs 3 assumptions.
 1. Data in generation t can only affect data in generation t+1
through the probability model.
 2. The parameters of the estimated model are chosen using
maximum likelihood.
 3. The sample size M and the size of the population N are of a
constant ratio independent of the number of variables L.

SML-EDA (Simple, Maximum-Likelihood EDA)
 The class of EDAs for which assumptions 1 and 2 hold are
called SML-EDAs.
 This class of EDAs includes BOA, MIMIC, FDA, UMDA, etc.
Some Definitions for diversity loss

The Empirical frequency
(the component i has the value A through the population)
iA 
1
N


(
x
 i  A)

Population size

Population member
One diversity measure
1
a
a
v

(1



i
i )
i | A| a
the value is fixates,
0
random population, maximum value

The trace of the empirical co-variance matrix
1
Cij 
[  ( xi  a) ( x j  a)     ( xi  a)   ( x j  a) ]

| A| a
Expectation that 2 components
both take the value A
Expectation that they take value A
independently.
Diversity Loss
on a Flat Fitness Landscape

Theorem 1
 For EDAs in class SML-EDA on a flat landscape, the expected
value of v is reduced in each generation by the following.
1
 vt  vt 1  (1  )
N
 Because the parameters are set by ML, empirical variance is
reduced by a factor (1-1/N) from the parent population.

Universal “expected diversity loss” for all SML-EDA
t
1

 vt  v0  1  
 N
 decays with characteristic time approximately equal to the
population size for large N.
A universal bound for the minimum
population size in the needle problem

Needle in a haystack problem
 There is one special state (the needle),
which has a high fitness value,
and all others have the same low fitness value.

Probability that the needle is never sampled.
prob(TN  )  B( N , L)
The time that the needle is first sampled.

Theorem 2
2
L
 In the limit that L   such that N L | A |  0
B( N , L)  1  O(| A | L / 2 )
for any EDA in SML-EDA searching on the Needle problem.
 The population size must grow at least as fast as
for the optimum to be found.
| A |L / L

Proof of theorem 2
 Lemma 1
 Let
t* be defined as
t*  
L
2
log(| A |)  log( LNv0 )
log(1  1/ N )
If the needle has not been found after a time t > t* , the probability
that the needle will never be found is greater that 1 – ε,
where  | A | L / 2 .
prob(TN   | TN  t*)  1 | A | L / 2
 Proof of Lemma 1
 We
can write the probability about the diversity loss
with the formula that the variance is small.
 v(r )  

prob v(t ) 
 1 




 Choosing
t* to be large enough so that
vt 
 vt 


1
1
1



N N
if vt is so small, there must be fixation at every component except
possibly one component. If L-1 components are fixed, the needle
will only be sampled if they are fixed at values found in the needle.
prob(TN   | TN  t*, vt  1/ N  1/ N 2 )  1 | A | ( L 1)
We are not certain that vt appropriately small, it is just probable so.
Take  | A | L / 2
 t* is calculated by solving simultaneous equations.
 Lemma 2
 The
probability that the needle is not found after t* steps obeys
N2
prob(TN  t*)  1 
| A |L
L

log(|
A
|)

log(
LN
)
 2

 Proof
– The probability that SML-EDA does not find the needle in time t*
prob(TN  t*)  (1 | A | L )t*N
The Prob. of not finding the needle
the result is found by putting into this equation the value for t*
 So, the probability of never finding the needle can be
decomposed into
prob(TN  )  prob(TN   | TN  t*) prob(TN  t*)
 Combining Lemma 1, 2 gives the following

 ( L / 2)
2
L  L
prob(TN  )  1 | A |
 N | A |  log(| A |)  log( LN )   ...
2

 If in the limit L   , N grows sufficiently slowly that the third
term vanishes, then the probability of never finding the needle
will go to 1.
 | A |L / 2 
 Thus, if N  o 
 , as L   , the needle will never be
 L 
found.
Expected Runtime for the Needle
Problem and the Limits of Universality

Corollary 1.
 If the needle is found, the time to find it is bounded above by t*
with probability 1 | A | L / 2

Convergence conditions will not be universal, but will be
particular to the EDA.