The document discusses the Wang-Landau algorithm, which is used to estimate the density of states of a physical system. It summarizes the original Wang-Landau algorithm and introduces an improved version that guarantees achieving a "flat histogram" - where each bin in the histogram has approximately equal frequency - in a finite number of steps. This is achieved by only decreasing the schedule parameter γ when the flat histogram criterion is met, rather than at every step. The document proves that this algorithm will converge to the flat histogram given some assumptions about the state space and proposal distribution. It also shows that an alternative update rule does not guarantee convergence in finite time.
1. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
On the convergence properties of the
Wang-Landau Algorithm
Robin J. Ryder
CEREMADE, Universit´ Paris Dauphine
e
and CREST
ENSAE – 31 January 2012
joint work with Pierre Jacob (National University of Singapore)
and Pierre Del Moral (INRIA & Universit´ de Bordeaux)
e
R. Ryder (Dauphine) & CREST Wang-Landau 1/ 40
2. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Outline
1 Wang–Landau algorithm
2 Flat Histogram in finite time
3 Parallel Wang–Landau: asymptotic behaviour
R. Ryder (Dauphine) & CREST Wang-Landau 2/ 40
3. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Motivation
0.5
0.4
0.3
density
0.2
0.1
0.0
−5 0 5 10 15
X
Figure: A mixture of well-separated normal distributions.
R. Ryder (Dauphine) & CREST Wang-Landau 3/ 40
4. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Motivation
0.5
0.4
0.3
density
0.2
0.1
0.0
−5 0 5 10 15
X
Figure: A mixture of well-separated normal distributions.
R. Ryder (Dauphine) & CREST Wang-Landau 4/ 40
5. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Motivation
0.5
0.4
0.3
density
0.2
0.1
0.0
−5 0 5 10 15
X
Figure: A mixture of well-separated normal distributions.
R. Ryder (Dauphine) & CREST Wang-Landau 5/ 40
6. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Setting
Partition the state space
d
X = Xi
i=1
Desired frequencies
φ = (φ1 , . . . , φd )
Penalized distribution
π(x)
πθ (x) ∝
θ(J(x))
where J(x) such that x ∈ XJ(x) .
R. Ryder (Dauphine) & CREST Wang-Landau 6/ 40
7. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
First algorithm
Algorithm 1 Wang-Landau with deterministic schedule (γt )
1: Init θ0 > 0, X0 ∈ X .
2: for t = 1 to T do
3: Sample Xt from Kθt−1 (Xt−1 , ·), MH kernel targeting πθt−1 .
4: Update the penalties:
log θt (i) ← log θt−1 (i) + γt (1 Xi (Xt ) − φi )
I
5: end for
R. Ryder (Dauphine) & CREST Wang-Landau 7/ 40
8. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Video
R. Ryder (Dauphine) & CREST Wang-Landau 8/ 40
9. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Video
R. Ryder (Dauphine) & CREST Wang-Landau 8/ 40
10. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Flat Histogram
Issues with the first version
Choice of γ has a huge impact on the results.
Flat Histogram
Define the counters:
t
νt (i) := 1 Xi (Xn )
I
n=1
Flat Histogram (FH) is reached when:
νt (i)
max − φi < c
i∈{1,...,d} t
R. Ryder (Dauphine) & CREST Wang-Landau 9/ 40
11. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Flat Histogram
Idea
Instead of decreasing γt at each time step t, decrease only when
the Flat Histogram criterion is reached.
In practice
Denote by κt the number of FH criteria reached up to time t.
Use γκt instead of γt at time t.
If FH is reached at time t, reset νt (i) to 0 for all i.
R. Ryder (Dauphine) & CREST Wang-Landau 10/ 40
12. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Wang–Landau with Flat Histogram
Algorithm 2 Wang-Landau with stochastic schedule (γκt )
1: Init θ0 > 0, X0 ∈ X .
2: Init κ0 ← 0.
3: for t = 1 to T do
4: Sample Xt from Kθt−1 (Xt−1 , ·), MH kernel targeting πθt−1 .
5: If (FH) then κt ← κt−1 + 1, otherwise κt ← κt−1 .
6: Update the penalties:
log θt (i) ← log θt−1 (i) + γκt (1 Xi (Xt ) − φi )
I
7: end for
R. Ryder (Dauphine) & CREST Wang-Landau 11/ 40
13. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
FH is met in finite time
To be sure that eventually, for any c > 0:
νt (i)
max − φi < c
i∈{1,...,d} t
we want to prove:
νt (i) P
∀i ∈ {1, . . . , d} − − φi
−→
t t→∞
for any fixed γ > 0.
R. Ryder (Dauphine) & CREST Wang-Landau 12/ 40
14. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Various updates
Right update
log θt (i) ← log θt−1 (i) + γ (1 Xi (Xt ) − φi )
I (1)
Using this update, FH is met in finite time.
Wrong update
θt (i) ← θt−1 (i) [1 + γ (1 Xi (Xt ) − φi )]
I
⇔
log θt (i) ← log θt−1 (i) + log [1 + γ (1 Xi (Xt ) − φi )]
I (2)
Here FH is not necessarily met in finite time.
1
Special case: ∀i φi = d .
R. Ryder (Dauphine) & CREST Wang-Landau 13/ 40
15. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Assumptions
In this talk, there are only two bins: d = 2. Additionally:
Assumption
The bins are not empty with respect to µ and π:
∀i ∈ {1, 2} µ(Xi ) > 0 and π(Xi ) > 0
Assumption
The state space X is compact.
R. Ryder (Dauphine) & CREST Wang-Landau 14/ 40
16. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Assumptions
Assumption
The proposal distribution Q(x, y ) is such that:
∃qmin > 0 ∀x ∈ X ∀y ∈ X Q(x, y ) > qmin
Assumption
The MH acceptance ratio is bounded from both sides:
π(y ) Q(y , x)
∃m > 0 ∃M > 0 ∀x ∈ X ∀y ∈ X m< <M
π(x) Q(x, y )
R. Ryder (Dauphine) & CREST Wang-Landau 15/ 40
17. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Theorem
Theorem
Consider the sequence of penalties θt introduced in the WL
algorithm. We define:
θt (1)
Zt = log = log θt (1) − log θt (2)
θt (2)
Then:
Zt L1
−− 0
−→
t t→∞
and consequently, with update (1) (FH) is reached in finite time
for any precision threshold c, whereas this is not guaranteed for
update (2).
R. Ryder (Dauphine) & CREST Wang-Landau 16/ 40
18. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Consequence
Using update (1), and starting from Z0 = 0:
X
Zt = log θt (1) 1 XX
IX
X
− log θt (2) 1 XX
IX
R. Ryder (Dauphine) & CREST Wang-Landau 17/ 40
19. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Consequence
Using update (1), and starting from Z0 = 0:
t X
Zt = log θ0 (1) + γ I IX
n=1 (1 X1 (Xn ) − φ1 ) 1 XX
t X
− log θ0 (2) + γ IX
n=1 (1 X2 (Xn ) − φ2 ) 1 XX
I
R. Ryder (Dauphine) & CREST Wang-Landau 17/ 40
20. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Consequence
Using update (1), and starting from Z0 = 0:
X
Zt = γ(νt (1) − tφ1 ) 1 XX
IX
X
−γ(νt (2) − tφ2 ) 1 XX
IX
R. Ryder (Dauphine) & CREST Wang-Landau 17/ 40
21. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Consequence
Using update (1), and starting from Z0 = 0:
X
Zt = γ(νt (1) − tφ1 ) 1 XX
IX
X
−γ(t − νt (1) − t(1 − φ1 )) 1 XX
IX
R. Ryder (Dauphine) & CREST Wang-Landau 17/ 40
22. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Consequence
Using update (1), and starting from Z0 = 0:
X
Zt = γ(νt (1) − tφ1 ) 1 XX
IX
X
+γ(νt (1) − tφ1 )) 1 XX
IX
R. Ryder (Dauphine) & CREST Wang-Landau 17/ 40
23. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Consequence
Using update (1), and starting from Z0 = 0:
X
Zt = 2γ(νt (1) − tφ1 ) 1 XX
IX
X
1 XX
IX
R. Ryder (Dauphine) & CREST Wang-Landau 17/ 40
24. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Consequence
Using update (1), and starting from Z0 = 0:
νt (1) X
Zt
t = 2γ t − φ1 1 XX
IX
X
1 XX
IX
R. Ryder (Dauphine) & CREST Wang-Landau 17/ 40
25. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Consequence
Using update (1), and starting from Z0 = 0:
νt (1) L X
Zt
t = 2γ t − φ1 − − 0 1 XX
−1→ IX
t→∞
νt (1) L X
⇒ t −1→
− − φ1 1 XX
IX
t→∞
R. Ryder (Dauphine) & CREST Wang-Landau 17/ 40
26. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Consequence
Using update (1), and starting from Z0 = 0:
νt (1) L X
Zt
t = 2γ t − φ1 − − 0 1 XX
−1→ IX
t→∞
νt (1) L X
⇒ t −1→
− − φ1 1 XX
IX
t→∞
Using update (2), these simplifications do not occur: νt (i)/t
converges, but not to φi in general.
R. Ryder (Dauphine) & CREST Wang-Landau 17/ 40
27. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Proof
To prove that
Zt L1
− − 0,
−→
t t→∞
first notice that
Zt+1 − Zt = 2γ (νt+1 (1) − νt (1) − φ1 )
can only take two different values, which we note +a and −b.
R. Ryder (Dauphine) & CREST Wang-Landau 18/ 40
28. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Proof
Introduce Ut , the increment of Zt :
Zt+1 = Zt + Ut
Lemma
With the introduced processes Zt and Ut , there exists > 0 such
¯ ¯
that for all η > 0, there exists Z hi such that, if Zt ≥ Z hi , we have
the following two inequalities:
P[Ut+1 = −b|Ut = +a, Zt ] >
P[Ut+1 = −b|Ut = −b, Zt ] > 1 − η.
(and the symmetric corollary)
R. Ryder (Dauphine) & CREST Wang-Landau 19/ 40
29. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Proof
q
q
q
q
q
q
q
q q
q
q
q q
q
q q q
q
q
q q
q
q q q
q q
q
q
Zs q
q
q
q q q
q q
~ ~
q
q
q q
Zs+T
q
q
q
q
q q q
q q
q
q
Zs+T
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
5 10 15 20 25 30 35
time
Figure: We prove that Zt returns below a given horizontal bar whenever
it goes above it, and it does so in finite time. This implies Zt /t → 0.
R. Ryder (Dauphine) & CREST Wang-Landau 20/ 40
30. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Proof
Introduce a crossing time s: Zs−1 ≤ Z hi and Zs ≥ Z hi .
˜
Introduce a new process Zt :
˜ ˜ ˜
Zs = Zs and Zt+1 = Zt + Vt
˜
We can choose Vt such that Zt ≥ Zt and
Lemma
(Vt ) is a Markov chain over the space {+a, −b} with transition
matrix
1−
η 1−η
where the first state corresponds to +a and the second state to
−b.
R. Ryder (Dauphine) & CREST Wang-Landau 21/ 40
31. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Proof
q
q
q
q
q
q
q
q q
q
q
q q
q
q q q
q
q
q q
q
q q q
q q
q
q
Zs q
q
q
q q q
q q
~ ~
q
q
q q
Zs+T
q
q
q
q
q q q
q q
q
q
Zs+T
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
5 10 15 20 25 30 35
time
Figure: We prove that Zt returns below a given horizontal bar whenever
it goes above it, and it does so in finite time. This implies Zt /t → 0.
R. Ryder (Dauphine) & CREST Wang-Landau 22/ 40
32. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Case d = 2
This shows that Zt /t → 0, hence (FH) is met in finite time.
For the case d > 2, we shall use only update 1.
R. Ryder (Dauphine) & CREST Wang-Landau 23/ 40
33. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Case d > 2
To prove that (FH) is met in finite time in the case with d > 2, we
need an extra assumption:
Assumption
The desired frequencies are all rational numbers:
∀i, φi ∈ Q
R. Ryder (Dauphine) & CREST Wang-Landau 24/ 40
34. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Irreducible Markov chain
Under the assumption of rationality, we have the following lemma:
Lemma
Let Θ be the following subset of Rd :
d
d d
Θ = {z ∈ R : ∃(n1 , . . . , nd ) ∈ N zi = ni − φi nj }
j=1
Then denoting by λ the product of the Lebesgue measure µ on X
and of the counting measure on Θ, (Xt , log θt ) is λ-irreducible.
Proof: B´zout’s lemma.
e
R. Ryder (Dauphine) & CREST Wang-Landau 25/ 40
35. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
A limit exists
Since (Xt , log θt ) is λ-irreducible, the proportion of visits to any
λ-measurable set of X × Θ converges to a limit in [0, 1]. Hence
the vector (ν(i)/t) converges to some limit pi . We now need to
prove that this limit corresponds to (FH).
R. Ryder (Dauphine) & CREST Wang-Landau 26/ 40
36. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Reductio ad absurdum
Suppose (reductio ad absurdum) that p = φ. Then there exist i
and j such that
pi − φi < pj − φj
and hence
Ztj,i = −νt (i) + νt (j) + t(φi − φj ) ∼ t(−pi + φi + pj − φj ) → ∞
˜
As before, we can construct Ut and Ut and show that Ut decreases
on average, which contradicts the assumption that Ztj,i → ∞.
R. Ryder (Dauphine) & CREST Wang-Landau 27/ 40
37. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Conclusion for d > 2
Conclusion: using update 1, (FH) is met in finite time for any
number of bins d.
R. Ryder (Dauphine) & CREST Wang-Landau 28/ 40
38. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Illustration
1.0 1.0
Proportions of visits to each bin
Proportions of visits to each bin
0.5
0.8 0.8
0.4
0.6 0.6
density
0.3
0.4 0.4
0.2
0.1 0.2 0.2
0.0 0.0 0.0
−4 −2 0 2 4 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000
X iterations iterations
(a) Histogram of the gen- (b) Convergence of the (c) Convergence of the
erated sample proportions of visits to proportions of visits to
each bin, using the right each bin, using the wrong
update update
R. Ryder (Dauphine) & CREST Wang-Landau 29/ 40
39. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Parallel chains
(1) (N)
N chains (Xt , . . . , Xt ) instead of one.
targeting the same biased distribution πθt at iteration t,
sharing the same estimated bias θt at iteration t.
Old update of the penalties
X
log θt (i) ← log θt−1 (i) + γ (1 Xi (Xt ) − φi ) 1 XX
I IX
(L. Bornn, P. Jacob, P. Del Moral, A. Doucet)
R. Ryder (Dauphine) & CREST Wang-Landau 30/ 40
40. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Parallel chains
(1) (N)
N chains (Xt , . . . , Xt ) instead of one.
targeting the same biased distribution πθt at iteration t,
sharing the same estimated bias θt at iteration t.
New update of the penalties
N (k) X
log θt (i) ← log θt−1 (i) + γ 1
N k=1 1 Xi (Xt )
I − φi 1 XX
IX
(L. Bornn, P. Jacob, P. Del Moral, A. Doucet)
R. Ryder (Dauphine) & CREST Wang-Landau 30/ 40
41. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Infinite number of chains
Suppose φi = 1/d. We can use either update and choose the
“wrong” update
N
1 (k) 1
θt (i) ← θt−1 (i) 1 + γ 1 Xi (Xt ) −
I
N d
k=1
(k) iid
Note that if (Xt )N ∼ πθt then
k=1
N −1
1 (k) a.s. ψi ψk
1 Xi (Xt )
I −−→
−− πθt (x)dx =
N N→∞ Xi θt−1 (i) θ(k)
k=1 k
R. Ryder (Dauphine) & CREST Wang-Landau 31/ 40
42. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
This corresponds to update
1
θt (i) ← θt−1 (i) 1 + γ πθt (x)dx −
X d
i
−1
ψi ψk 1
⇔ θt (i) ← θt−1 (i) 1 + γ −
θt−1 (i) θ(k) d
k
Normalize:
θt (i)
θt (i) ←
j θt (j)
Then the penalties (θt )t≥0 should converge towards ψ.
R. Ryder (Dauphine) & CREST Wang-Landau 32/ 40
43. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
ψi ψk −1 XX
θt−1 (i) 1+γ θt−1 (i) ( k θ(k) ) 1
−d I
1 XX
θt (i) ← XX
ψj ψk −1 1
d
j=1 θt−1 (j) 1+γ θt−1 (j) ( k θ(k) −d) I
1 XX
R. Ryder (Dauphine) & CREST Wang-Landau 33/ 40
44. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
ψi XX
θt−1 (i) 1+γ −1 1
θt−1 (i) Mψ (θt−1 )− d I
1 XX
θt (i) ← ψj XX
d
j=1 θt−1 (j) 1+γ −1 1
θt−1 (j) Mψ (θt−1 )− d I
1 XX
R. Ryder (Dauphine) & CREST Wang-Landau 33/ 40
45. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
XX
−1
I
θt−1 (i)(1− γ ) + ψi γMψ (θt−1 ) 1 XX
d
θt (i) ← XX
d γ
j=1 θt−1 (j)(1− d )
−1
I
+ ψj γMψ (θt−1 ) 1 XX
R. Ryder (Dauphine) & CREST Wang-Landau 33/ 40
46. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
XX
−1
I
θt−1 (i)(1− γ ) + ψi γMψ (θt−1 ) 1 XX
d
θt (i) ← XX
−1
I
(1− γ )+γMψ (θt−1 ) 1 XX
d
R. Ryder (Dauphine) & CREST Wang-Landau 33/ 40
47. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
1− γ γM −1 (θt−1 )
ψ
θt (i) ← θt−1 (i) 1− γ +γM −1 (θ
d
+ ψi 1− γ +γM −1 (θ
d ψ t−1 ) d ψ t−1 )
R. Ryder (Dauphine) & CREST Wang-Landau 33/ 40
49. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Hence the update can be written as a convex combination
θt (i) ← θt−1 (i) × (1 − α(θt−1 )) + ψi × α(θt−1 )
where
−1
γMψ (θt−1 )
α(θt−1 ) = γ −1
1− d + γMψ (θt−1 )
It can easily be shown that
∀t ≥ 0 0 < αmin ≤ α(θt ) ≤ αmax < 1
provided that ∀i, θ0 (i) > 0, ψ(i) > 0.
iid
Of course the assumptions N → ∞ and ∼ sampling are strong.
What happens in practice?
R. Ryder (Dauphine) & CREST Wang-Landau 34/ 40
50. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Animation
Toy example
X = {1, 2, 3, 4, 5}, π = (π1 , . . . , π5 )
We take the partition: Xi = {i} and launch the Wang–Landau
algorithm for N = 1, 10, 100 and γ = 1.
In this case we know the value of ψi : ψi = πi , so we can compute
the deterministic sequence of penalties corresponding to N = ∞.
Animation
R. Ryder (Dauphine) & CREST Wang-Landau 35/ 40
51. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Animation
N=∞
R. Ryder (Dauphine) & CREST Wang-Landau 36/ 40
52. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Animation
N = 100
R. Ryder (Dauphine) & CREST Wang-Landau 36/ 40
53. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Animation
N = 10
R. Ryder (Dauphine) & CREST Wang-Landau 36/ 40
54. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Animation
N=1
R. Ryder (Dauphine) & CREST Wang-Landau 36/ 40
55. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Animation
“
R. Ryder (Dauphine) & CREST Wang-Landau 37/ 40
56. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
New take on the Parallel Wang–Landau
Call fψ the function such that:
θt ← fψ (θt−1 )
As we saw:
fψ (θ) = θ(1 − α(θ)) + ψα(θ))
where
−1
γMψ (θ)
α(θ) = γ −1
1− d + γMψ (θ)
R. Ryder (Dauphine) & CREST Wang-Landau 38/ 40
57. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
New take on the Parallel Wang–Landau (in progress)
In the ideal case we have
n
θn = fψ ◦ . . . ◦ fψ (θ0 ) = fψ (θ0 )
×n
If we add perturbations to ψ we also add perturbations to θ, as
follows:
ε
θ1 = fψ (θ0 ) = fψ (θ0 ) + V0
R. Ryder (Dauphine) & CREST Wang-Landau 39/ 40
58. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
New take on the Parallel Wang–Landau (in progress)
In the ideal case we have
n
θn = fψ ◦ . . . ◦ fψ (θ0 ) = fψ (θ0 )
×n
If we add perturbations to ψ we also add perturbations to θ, as
follows:
ε
θ1 = fψ (θ0 ) = fψ (θ0 ) + V0
ε ε 2
θ2 = fψ (θ1 ) = fψ (fψ (θ0 ) + V0 ) = fψ (θ0 ) + V1
...
R. Ryder (Dauphine) & CREST Wang-Landau 39/ 40
59. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
New take on the Parallel Wang–Landau (in progress)
In the ideal case we have
n
θn = fψ ◦ . . . ◦ fψ (θ0 ) = fψ (θ0 )
×n
If we add perturbations to ψ we also add perturbations to θ, as
follows:
ε
θ1 = fψ (θ0 ) = fψ (θ0 ) + V0
ε ε 2
θ2 = fψ (θ1 ) = fψ (fψ (θ0 ) + V0 ) = fψ (θ0 ) + V1
...
The properties of fψ should help control the noise terms
(Vn )n≥0 . . .
R. Ryder (Dauphine) & CREST Wang-Landau 39/ 40
60. Wang–Landau algorithm
Flat Histogram in finite time
Parallel Wang–Landau: asymptotic behaviour
Bibliography
Atchad´, Y. and Liu, J. (2010). The Wang-Landau algorithm
e
in general state spaces: applications and convergence analysis.
Statistica Sinica, 20:209–233.
Wang, F. and Landau, D. (2001). Determining the density of
states for classical statistical models: A random walk
algorithm to produce a flat histogram. Physical Review E,
64(5):56101.
An Adaptive Interacting Wang-Landau Algorithm for
Automatic Density Exploration, L. Bornn, P. Jacob, P. Del
Moral, A. Doucet, on arXiv.
The Wang-Landau algorithm reaches the Flat Histogram
criterion in finite time, P. Jacob & RR, on arXiv.
R. Ryder (Dauphine) & CREST Wang-Landau 40/ 40