1. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Parallel Adaptive Wang–Landau Algorithm
Pierre E. Jacob
CEREMADE - Universit´ Paris Dauphine & CREST, funded by AXA Research
e
15 novembre 2011
joint work with Luke Bornn (UBC), Arnaud Doucet (Oxford), Pierre Del Moral
(INRIA & Universit´ de Bordeaux)
e
Pierre E. Jacob PAWL 1/ 18
2. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Outline
1 Wang–Landau algorithm
2 Improvements
Automatic Binning
Parallel Interacting Chains
Adaptive proposals
3 2D Ising model
4 Conclusion
Pierre E. Jacob PAWL 2/ 18
3. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Wang–Landau
Context
unnormalized target density π
on a state space X
A kind of adaptive MCMC algorithm
It iteratively generates a sequence Xt .
The stationary distribution is not π itself.
At each iteration a different stationary distribution is targeted.
Pierre E. Jacob PAWL 3/ 18
4. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Wang–Landau
Partition the space
The state space X is cut into d bins:
d
X = Xi and ∀i = j Xi ∩ Xj = ∅
i=1
Goal
The generated sequence spends the same time in each bin Xi ,
within each bin Xi the sequence is asymptotically distributed
according to the restriction of π to Xi .
Pierre E. Jacob PAWL 4/ 18
5. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Wang–Landau
Stationary distribution
Define the mass of π over Xi by:
ψi = π(x)dx
Xi
The stationary distribution of the WL algorithm is:
1
πψ (x) ∝ π(x) ×
ψJ(x)
where J(x) is the index such that x ∈ XJ(x)
Pierre E. Jacob PAWL 5/ 18
6. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Wang–Landau
Example with a bimodal, univariate target density: π and two πψ
corresponding to different partitions.
Original Density, with partition lines Biased by X Biased by Log Density
0
−2
−4
Log Density
−6
−8
−10
−12
−5 0 5 10 15 −5 0 5 10 15 −5 0 5 10 15
X
Pierre E. Jacob PAWL 6/ 18
7. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Wang–Landau
Plugging estimates
In practice we cannot compute ψi analytically. Instead we plug in
estimates θt (i) of ψi at iteration t, and define the distribution πθt
by:
1
πθt (x) ∝ π(x) ×
θt (J(x))
Metropolis–Hastings
The algorithm does a Metropolis–Hastings step, aiming πθt at
iteration t, generating a new point Xt .
Pierre E. Jacob PAWL 7/ 18
8. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Wang–Landau
Estimate of the bias
The update of the estimated bias θt (i) is done according to:
θt (i) ← θt−1 (i)[1 + γt (IXt ∈Xi − d −1 )]
with γt a decreasing sequence or “step size”. E.g. γt = 1/t.
Pierre E. Jacob PAWL 8/ 18
9. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Wang–Landau
Result
In the end we get:
a sequence Xt asymptotically following πψ ,
as well as estimates θt (i) of ψi .
Pierre E. Jacob PAWL 9/ 18
10. Wang–Landau algorithm
Automatic Binning
Improvements
Parallel Interacting Chains
2D Ising model
Adaptive proposals
Conclusion
Automate Binning
Easily move from one bin to another
Maintain some kind of uniformity within bins. If non-uniform, split
the bin.
Frequency
Frequency
Log density Log density
(a) Before the split (b) After the split
Pierre E. Jacob PAWL 10/ 18
11. Wang–Landau algorithm
Automatic Binning
Improvements
Parallel Interacting Chains
2D Ising model
Adaptive proposals
Conclusion
Parallel Interacting 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.
The update of the estimated bias becomes:
N
1
θt (i) ← θt−1 (i)[1 + γt ( IX (j) ∈X − d −1 )]
N t i
j=1
Pierre E. Jacob PAWL 11/ 18
12. Wang–Landau algorithm
Automatic Binning
Improvements
Parallel Interacting Chains
2D Ising model
Adaptive proposals
Conclusion
Adaptive proposals
For continuous state spaces
We can use the adaptive Random Walk proposal where the
variance σt is learned along the iterations to target an acceptance
rate.
Robbins-Monro stochastic approximation update
σt+1 = σt + ρt (2I(A > 0.234) − 1)
Or alternatively
Σt = δ × Cov (X1 , . . . , Xt )
Pierre E. Jacob PAWL 12/ 18
13. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
2D Ising model
Higdon (1998), JASA 93(442)
Target density
Consider a 2D Ising model, with posterior density
π(x|y ) ∝ exp α I[yi = xi ] + β I[xi = xj ]
i i∼j
with α = 1, β = 0.7.
The first term (likelihood) encourages states x which are
similar to the original image y .
The second term (prior) favors states x for which
neighbouring pixels are equal, like a Potts model.
Pierre E. Jacob PAWL 13/ 18
14. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
2D Ising models
(a) Original Image (b) Focused Region of Image
Pierre E. Jacob PAWL 14/ 18
15. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
2D Ising models
Iteration 300,000 Iteration 350,000 Iteration 400,000 Iteration 450,000 Iteration 500,000
40
Metropolis−Hastings
30
20
10
Pixel
On
X2
40 Off
30
Wang−Landau
20
10
10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40
X1
Figure: Spatial model example: states explored over 200,000 iterations
for Metropolis-Hastings (top) and proposed algorithm (bottom).
Pierre E. Jacob PAWL 15/ 18
16. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
2D Ising models
Metropolis−Hastings Wang−Landau
40
30
Pixel
0.4
0.6
X2
20
0.8
1.0
10
10 20 30 40 10 20 30 40
X1
Figure: Spatial model example: average state explored with
Metropolis-Hastings (left) and Wang-Landau after importance sampling
(right).
Pierre E. Jacob PAWL 16/ 18
17. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Conclusion
Automatic binning
We still have to define a range.
Parallel Chains
In practice it is more efficient to use N chains for T iterations
instead of 1 chain for N × T iterations.
Adaptive Proposals
Convergence results with fixed proposals are already challenging,
and making the proposal adaptive might add a layer of complexity.
Pierre E. Jacob PAWL 17/ 18
18. Wang–Landau algorithm
Improvements
2D Ising model
Conclusion
Bibliography
Article: An Adaptive Interacting Wang-Landau Algorithm for
Automatic Density Exploration, L. Bornn, P.E. Jacob, P. Del
Moral, A. Doucet, available on arXiv.
Software: PAWL, an R package, available on CRAN:
install.packages("PAWL")
References:
F. Wang, D. Landau, Physical Review E, 64(5):56101
Y. Atchad´, J. Liu, Statistica Sinica, 20:209-233
e
Pierre E. Jacob PAWL 18/ 18
