Simulated annealing and Gibbs sampler

14 Pages·2013·0.38 MB·English

by  Leonid Zhukov

Simulated annealing and Gibbs sampler
Leonid Zhukov
SchoolofAppliedMathematicsandInformationScience
National Research University Higher School of Economics
12.12.2013
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Simulated annealing
Global optimization method (vs greedy strategies - local minimum)
Works for both continues and discrite optimization problems
Intuition from thermodynamics, physical annealing process
Algorithm:
1 generate trial point and evaluate function at that location.
2 accept new location if it reduces "energy"(improves solution)
3 accept some new location even not improving the solution
4 probability accepting non-improving location decreases with lowering
"temperture"
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Simulated annealing
Compute change of energy for a k-step ∆E = E −E :
k k k−1
If ∆E ≤0, accept the step
k
if ∆Ek >0, accept the step with probability P(∆Ek)=e−∆Ek/Tk
Cooling schedule:
T =αT /k
k 0
T =αT /logk
k 0
T =T /αk
k 0
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Traveling salesman problem
Travelling salesman problem: given a list of cities and the distances
between each pair of cities, what is the shortest possible route that visits
each city exactly once and returns to the origin city? (NP-hard problem in
combinatorial optimization)
Simulated annealing algorithm:
Define energy cost function:
(cid:88)N (cid:113)
E = (x −x2)+(y −y )2
i+1 i i+1 i
i
Select initial route (sequence of city labeles)
Iteratively imporve the route by trying local changes:
swap a pair of cities + reverse the section between them
always accept lower enegergy swap, sometimes higher enegery
reduce temperature with cooling schedule
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Traveling salesman problem
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MCMC connection
MCMC constructs Markov chain that generates random samples
distributed according to P(x)
Metropolis algorithm uses symmetric "candidate"distribution
Q(x,x∗) = Q(x∗,x)
Probabiity of "forward move"of the chain:
(cid:20) P(x∗)(cid:21)
α(x,x∗) = min 1,
P(x)
Consider Boltzman (Gibbs) distribution
P(x) = 1e−ET(x)
Z
Probability of move
(cid:20) (cid:21)
α(x,x∗) = min 1,e−E(x∗)T−E(x) = min(cid:104)1,e−∆TE(cid:105)
∆E ≤ 0 - always move, ∆E > 0 move with probability exp(−∆E)
T
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The Gibbs sampler
Goal: get random samples from joint multivariate density p(x ,...x ) if
1 n
it is not known explicitely or hard to sample from
Given conditional univatiate distributions p(x |x ...x ), p(x |x ...x ),
1 2 n 2 1 n
..p(x |x ...x )
n 1 n−1
Bayesian inference, posterior distributions
Approximate joint distribution (histogram), compute averages
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Bivariate case
joint bivariate distribution
p(x,y)
marginal distribution
(cid:82)
p(x) = p(x,y)dy
(cid:82)
p(y) = p(x,y)dx
conditional probability
p(x|y) = p(x,y)/p(y)
p(y|x) = p(x,y)/p(x)
marginal from conditional distribution
(cid:82)
p(x) = p(x|y)p(y)dy = E [p(x|y)]
p(y)
(cid:82)
p(y) = p(y|x)p(x)dx = E [p(y|x)]
p(x)
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Gibbs sampler: bivariate case
Given: p(x|y),p(y|x)
Choose y , t = 0
0
do "sampler scan"
x ∼ p(x|y = y )
t t
y ∼ p(y|x = x )
t+1 t
repeat k-times, Gibbs sequence (x ,y ),(x ,y )...(x ,y )
0 0 1 1 k k
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Gibbs sampler
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