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The Markov Chain Monte Carlo Method

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The Markov Chain Monte Carlo Method Isabelle Stanton May 8, 2008 Theory Lunch Monte Carlo vs Las Vegas Las Vegas Algorithms are randomized and always give the correct ... – PowerPoint PPT presentation

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Title: The Markov Chain Monte Carlo Method


1
The Markov Chain Monte Carlo Method
  • Isabelle Stanton
  • May 8, 2008 Theory Lunch

2
Monte Carlo vs Las Vegas
  • Las Vegas Algorithms are randomized and always
    give the correct results but gamble with
    computation time
  • Quicksort
  • Monte Carlo algorithms have fixed running time
    but may be wrong
  • Simulated Annealing
  • Estimating volume

3
Markov Chains
  • a memoryless stochastic process, eg, flipping a
    coin

1/6
1/6
3
2
4
1
1/6
1/6
6
5
1/6
1/6
4
Other Examples of Markov Chains
  • Shuffling cards
  • Flipping a coin
  • PageRank Model
  • Particle systems focus of MCMC work

5
General Idea
  • Model the system using a Markov Chain
  • Use a Monte Carlo Algorithm to perform some
    computation task

6
Applications
  • Approximate Counting - of solutions to 3-SAT or
    Knapsack
  • Statistical Physics when do phase transitions
    occur?
  • Combinatorial optimization simulated annealing
    type of algorithms
  • We'll focus on counting

7
Monte Carlo Counting
  • How do you estimate the volume
  • of a complex solid?
  • Render with environment maps
  • efficiently?
  • Estimate an integral numerically?

8
(Picnic) Knapsack
weighs 5
weighs 4
weighs 10
What is a solution?
How many solutions are there?
weighs 2
weighs 4
Holds 20
9
Counting Knapsack Solutions
  • Item weights a (a0,...an)
  • Knapsack size a real number b
  • Estimate the number of 0,1 vectors, x, that
    satisfy ax b
  • Let N denote the number of solutions

10
Na?ve Solution
  • Randomly generate x
  • Calculate ax
  • If ax b return 2n
  • else return 0
  • This will return N in expectation
  • 0(2n-N) N2n / 2n

11
Is this fast?
  • Counterexample
  • a (1, ... 1) and b n/3
  • Any solution has less than n/3 1's
  • There are (n choose n/3)2n/3 solutions

12
no
  • Pr(sample x, x n/3) lt (n choose n/3)2-2n/3
  • In expectation, need to generate 2n/3 x's before
    we get a single solution!
  • Any polynomial number of trials will grossly
    underestimate N

13
Knapsack with MCMC
  • Let Mknap be a markov chain withstate space O(b)
    x ax b
  • This will allow us to sample a solution

14
Various Mknap
a(0,.5,.5) b 1.5
a(0,1,1) b 1.5
111
011
101
110
110
101
011
110
101
001
010
100
001
010
100
001
010
100
000
000
000
15
Mknap Transitions
  • Transitions
  • With probability 1/2, x transitions to x
  • Otherwise, select an i u.a.r.
  • from 0 to n-1 and flip
  • the ith bit of x.
  • If x' is a
  • solution,
  • transition there.

111
0.5
0.5
011
101
110
110
101
110
101
1/6
1/6
1/6
1/6
0.5
001
010
100
001
010
100
001
010
100
0.5
0.5
1/6
1/6
1/6
000
000
000
0.5
a(0,1,1) b 1.5
16
Connected?
  • Is Mknap connected?
  • Yes. To get from x to x' go through 0.

17
Ergodicity
  • What is the stationary distribution of Knapsack?
  • Sample each solution with prob 1/N
  • A MC is ergodic if the probability distribution
    over the states converges to the stationary
    distribution of the system, regardless of the
    starting configuration
  • Is Mknap ergodic? Yes.

18
Algorithm Idea
  • Start at 0 and simulate Mknap for enough steps
    that the distribution over the states is close to
    uniform
  • Why does uniformity matter?
  • Does this fix the problem yet?

19
The trick
  • Assume that a0 a1 ... an (0,1,2,,n-1,n)
  • Let b0 0 and bi minb, Siaj
  • O(bi-1) O(bi) - why?
  • O(bi) (n1)O(bi-1) - why?

Change any element of O(bi) to one of O(bi-1) by
switching the rightmost 1 to a 0
20
How does that help?
  • O(b) O(bn) O(bn)/O(bn-1) x
    O(bn-1)/O(bn-2) x ... x
    O(b1)/O(b0) x O(b0)
  • We can estimate each of these ratios by doing a
    walk on O(bi) and computing the fraction of
    samples in O(bi-1)?
  • Good estimate since
  • O(bi-1) O(bi) (n1)O(bi-1)

21
Analysis
  • Ignoring bias, the expectation of each trial is
    O(bi-1)/O(bi)
  • We perform t 17e-2n2 steps
  • Focus on Var(X)/E(X)2 in analyzing efficiency
    for MCMC methods

22
Analysis
  • If Z is the product of the trials,
    EZ ? O(bi-1)/O(bi)
  • Magic Statistics Steps
  • Var(Z)/(EZ)2 e2/16
  • By Chebyshev's
  • Pr(1-e/2)O(b) Z (1e/2)O(b) 3/4

23
Analysis
  • We used nt 17e-2n3 steps
  • This is a FPRAS (Fully Polynomial Randomized
    Approximation Scheme)?
  • Except... what assumption did I make?

24
Mixing Time
  • Assumption We are close to the uniform
    distribution in 17e-2n2 steps
  • This is known as the mixing time
  • It is unknown if this distribution mixes in
    polynomial time

25
Mixing Time
  • What does mix in polynomial time?
  • Dice 1 transition
  • Shuffling cards 7 shuffles
  • ferromagnetic Ising model at high temperature
    O(nlog n)?
  • What doesn't?
  • ferromagnetic Ising model at low temperature
    starts to form magnets

26
Wes Weimer Memorial Conclusion Slide
  • The markov chain monte carlo
  • method models the problem
  • as a Markov Chain and then
  • uses random walks
  • Mixing time is important
  • P problems are hard
  • Wes likes trespassing
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