615.18 -- Simulated Annealing - umich.edu

56 Pages·2008·0.42 MB·French

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Biostatistics 615/815
LLeeccttuurree 1188
So far …
“Greedy” optimization methods
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Can get trapped at local minima
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Outcome might depend on starting point
Examples:
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Golden Search
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Nelder-Mead Simplex Optimization
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E-M Algorithm
Today …
Simulated Annealing
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Markov-Chain Monte-Carlo method
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Designed to search for global minimum
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The Problem
Most minimization strategies find the
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nearest local minimum
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Generate trial point based on current estimates
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Accept new value if it improves solution
The Solution
We need a strategy to find other minima
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This means, we must sometimes select
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Annealing
One manner in which crystals are formed
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Gradual cooling of liquid …
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At high temperatures, molecules move freely
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At low temperatures, molecules are "stuck"
IIff coolliing iis sllow
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Low energy, organized crystal lattice formed
Simulated Annealing
Analogy with thermodynamics
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Incorporate a temperature parameter into the
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minimization procedure
At high temperatures, explore parameter space
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At lower temperatures, restrict exploration
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Markov Chain
The Markovian property
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Pr(Z = i | Z = i , , Z = i ) = Pr(Z = i | Z = i )
K
n n n−1 n−1 0 0 n n n−1 n−1
Transition probability
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QQ == PPrr((ZZ == jj || ZZ == ii))
ij n n−1
n-step transition
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(n) ∑ ∑
Q = Pr(Z = j | Z = i) = p p
L L
ij n 0 ii i j
1 n−1
i i
1 n−1
(all possible n-step paths i → j)
Markov Chain (In Practice)
Start with some state (Z =i)
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n
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A set of mixture parameters
Propose a change (Z =j)
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n+1
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EEddiitt miixtture parametters iin some way
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ij
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Decision is based on relative probabilities of
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Simulated Annealing Strategy
Consider decreasing series of temperatures
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For each temperature, iterate these steps:
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Propose an update and evaluate function
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AAcceptt upddattes tthhatt iimprove solluttiion
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Accept some updates that don't improve solution
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Accepptance pprobabilityy deppends on “tempperature” pparameter
If cooling is sufficiently slow, the global minimum
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