analysis of exchange ratio for exchange monte carlo method kenji nagata, sumio watanabe tokyo...
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Analysis of Exchange Ratio for Exchange Monte Carlo Method
Kenji Nagata, Sumio Watanabe
Tokyo Institute of Technology
Japan
Contents
Background Exchange Monte Carlo (EMC) method Design of EMC method
Main result Settings Symmetrized Kullback divergence Average exchange ratio
Discussion and Conclusion
Contents
Background Exchange Monte Carlo (EMC) method Design of EMC method
Main result Settings Symmetrized Kullback divergence Average exchange ratio
Discussion and Conclusion
Exchange Monte Carlo (EMC) method[Hukushima, 96]
<Markov chain Monte Carlo (MCMC) Method>
huge computational cost!
Multi Canonical algorithm Simulated tempering
Exchange Monte Carlo (EMC) method
<improvement of MCMC method>
wwnHnZ
wp exp1
)( dRw
EMC method is to generate the sample sequence from the following joint distribution,
K
kkkK twpwwp
11 |,,
Kktk ,,1: :temperatures
Target distribution ：
EMC method
<Aim> Sampling from the following target distribution!
wwntHntZ
twp exp1
|
EMC method
<Algorithm>
Carrying out the following two updates alternately.
１． [conventional MCMC sampling] Parallel sampling from each distribution by using conventional MCMC method.
２． [exchange process] The exchange of two position, and , is tried and accepted with the following probability .
Hereafter, we call “exchange ratio”.
)|( kk twp
kw 1kw
u
ru ,1min
)(ˆ)(ˆexp
||
||11
11
11kkkk
kkkk
kkkk wHwHtttwptwp
twptwpr
EMC method<conventional MCMC method>
<EMC method>
)(wp
)|( 11 twp
)|( 22 twp
)|( 33 twp
)|( 44 twp
EMC method
2, Ryxw )(w : standard normal distribution
22)( yxwH
)())(exp()(
1)|( wwntH
ntZtwp
1000n
0t 10 t 1t
),1min( ru )()()(exp 11 kkkk wHwHttnr
0)( wH
Contents
Background Exchange Monte Carlo (EMC) method Design of EMC method
Main result Settings Symmetrized Kullback divergence Average exchange ratio
Discussion and Conclusion
Design of EMC method
<Setting of temperature>
)()()(exp 11 kkkk wHwHttnr
01 t 10 t 1Kt
Kktk ,,1:
How should and be set? ktK
Temperature has close relation to the exchange ratio.
Design of EMC method
<Average exchange ratio>
Acceptance ratio of exchange process.
)(ˆ)(ˆexp
),1min(
11 kkkk wHwHttr
ru
32
21
32
t
1t
2t
3t
4t
For efficient EMC method, the average exchange ratio needs to be of and nearly constant for all temperatures. 1O
Design of EMC method
<Symmetrized Kullback divergence>
1
1
1111
11 )|(
)|(log)|(
)|(
)|(log)|(),( k
kk
kkkkk
kk
kkkkkk dw
twp
twptwpdw
twp
twptwpttI
)|()|( 11 kkkk twptwprlog][log rE
),(][log 1 kk ttIrE
: the expectation of over .
<Property>
For efficient EMC method, the symmetrized Kullback divergence needs to be nearly constant over the various temperatures.
)|( kk twp )|( 11 kk twp
Purpose
nWe analytically clarify the symmetrized Kullback divergence andthe average exchange ratio in a low temperature limit, .
•Average exchange ratio•Symmetrized Kullback divergence
Criteria for the design of EMC method
•We need previous EMC simulations in order to obtain these values.
•The accuracy of experimental values are unknown.
Purpose
Contents
Background Exchange Monte Carlo (EMC) method Design of EMC method
Main result Settings Symmetrized Kullback divergence Average exchange ratio
Discussion and Conclusion
Settings
We consider the EMC method between the following two distribution,
twpwp |1 dRw ttwpwp |2
<Average exchange ratio>
212211 )()( dwdwwpwupJ
tt )())(exp(
)(
1)|( wwntH
ntZtwp
<Symmetrized Kullback divergence>
221
22221
12
1111 loglog dw
wp
wpwpdw
wp
wpwpI
Main result
<Theorem 1>The symmetrized Kullback divergence converges to
the following value for
22
1t
tO
t
t
t
tI
In
dwwwHz z )()()(
: rational number
zRe
zIm
0
Main result
<Outline of the proof>
22221111 )()()()( dwwpwHdwwpwHtnI
t
t
sVntsds
sVntssdstndwwpwHtn
0
01
exp
exp)()(
11 )log()()( mscsdwwwHssV [Watanabe,2001]
22
1t
tO
t
t
t
t
tt
t
t
tI
<Lemma>
0s
Main result
<Theorem 2>The average exchange ratio converges to the following value for n
dwwwHz z )()()(
zRe
zIm
0
J
2
21||
1t
tO
t
tJ
: rational number
Main result
<Outline of the proof>
0 20 1)(
12
0 0 21)(
12
)()( 212211
)()( 212211)()( 212211
21
221
21
2121
2
)()(2
)()()()(
sVsVeedsds
sVsVeedsds
dwdwwpwp
dwdwwpwprdwdwwpwpJ
sttnnts
s sttnnts
wHwH
wHwHwHwH
)()(
)()(
2211
1221
wpwp
wpwpr
2
21
1t
tO
t
t
ru ,1min0tIf ,
Contents
Background Exchange Monte Carlo (EMC) method Design of EMC method
Main result Settings Symmetrized Kullback divergence Average exchange ratio
Discussion and Conclusion
Discussion
2
21
)(
)(||1
t
tO
t
tJ
Average exchange ratio :
22
1t
tO
t
t
t
tI Symmetrized Kullback divergence :
From these results, we can adjust the temperatures without carrying out the previous EMC simulations.
These results can be used as criteria for checking the convergence of EMC simulations.
Discussion
2
21
)(
)(||1
t
tO
t
tJ
Average exchange ratio :
t
1t 2t 3t 4t 5t
1
12
t
tt
2
23
t
tt
3
34
t
tt
4
45
t
tt 2t 3t 4t
11 1 k
k tt t
t
geometrical progression!!
22
1t
tO
t
t
t
tI Symmetrized Kullback divergence :
Discussion
0t 10 t 1t
nCondition of theorem
large enough!
Not applicable. Applicable.
)())(exp()(
1)|( wwntH
ntZtwp
Conclusion
We analytically clarified the symmetrized Kullback divergence and the average exchange ratio in a low temperature limit.
As a result, it is clarified that The set of temperature should be set as a geometrical
progression in order to make the average exchange ratio constant over the various temperatures.
As the future works, Verifying the theoretical results in this study
experimentally. Constructing the design of EMC method.