1.
d-VMP:
Distributed Variational Message Passing
Andrés R. Masegosa1, Ana M. Martínez2,
Helge Langseth1, Thomas D. Nielsen2, Antonio Salmerón3,
Darío Ramos-López3, Anders L. Madsen2,4
1Department of Computer Science, Aalborg University, Denmark
2 Department of Computer and Information Science,
The Norwegian University of Science and Technology, Norway
3Department of Mathematics, University of Almería, Spain
4 Hugin Expert A/S, Aalborg, Denmark
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 1

2.
Outline
1 Motivation
2 Variational Message Passing
3 d-VMP
4 Experimental results
5 Conclusions
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 2

3.
Outline
1 Motivation
2 Variational Message Passing
3 d-VMP
4 Experimental results
5 Conclusions
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 3

4.
Motivation
Large
Imbalance
? values
Complex distributions
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 4

5.
Motivation
Large
Imbalance
? values
Complex distributions
Attribute range
Density
0 1 2 3 4 5 6
0123456
●
●
●
●
SVI 1%
SVI 5%
SVI 10%
VMP 1%
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 5

6.
Motivation
Goal: learn a generative model for a finantial dataset to monitor the
customers and make predictions for a single customer.
Xi Hi
θα
i = 1, . . . , N
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 6

7.
Popular existing approach: SVI
Stochastic Variational Inference: iteratively updates the model
parameters based on subsampled data batches.
No estimation of all local hidden variables of the model.
No generation of lower bound.
Poor fit if batch of data is not representative from all data.
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 7

8.
Our contribution:
d-VMP: a distributed message passing scheme.
Defined for a broader class of models (than SVI).
Better and faster convergence results compared to SVI.
Posterior over all local latent variables and lower bound.
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 8

9.
Outline
1 Motivation
2 Variational Message Passing
3 d-VMP
4 Experimental results
5 Conclusions
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 9

10.
Models:
Bayesian learning on iid. data using conjugate exponential BN models:
ln p(X) = ln hX + sX · η − AX (η)
Xi Hi
θα
i = 1, . . . , N
We want to calculate p(θ, H|D).
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 10

11.
Variational Inference:
Approximate p(θ, H|D) (often intractable) by finding tractable
posterior distributions q ∈ Q by minimizing:
min
q(θ,H)∈Q
KL(q(θ, H)|p(θ, H|D)),
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 11

12.
Variational Inference:
Approximate p(θ, H|D) (often intractable) by finding tractable
posterior distributions q ∈ Q by minimizing:
min
q(θ,H)∈Q
KL(q(θ, H)|p(θ, H|D)),
In the mean field variational approach, Q is assumed to fully
factorize:
q(θ, H) =
M
k=1
q(θk)
N
i=1
J
j=1
q(Hi,j ),
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 12

13.
Variational Inference:
Variational Inference exploits:
ln P(D)
constant
= L(q(θ, H))
Maximize
+ KL(q(θ, H)|p(θ, H|D))
Minimize
,
Iterative coordinate ascent of the variational distributions.
Updates in the variational distribution of a variable only
involves variables in its Markov blanket.
Coordinate ascent algorithm formulated as a message passing
scheme.
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 13

14.
Variational Message Passing, VMP:
Message from parent to child: moment parameters
(expectation of the sufficient statistics).
Message from child to parent: natural parameters
(based on the messages received from the co-parents).
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 14

15.
Outline
1 Motivation
2 Variational Message Passing
3 d-VMP
4 Experimental results
5 Conclusions
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 15

16.
Distributed optimization of the lower bound:
Master
θα
q(t)(θ)
X
H
θ1
i = 1, . . . , N
Slave 1
q(t)(H1)
X
H
θ2
i = 1, . . . , N
Slave 2
q(t)(H2)
X
H
θ3
i = 1, . . . , N
Slave 3
q(t)(H3)
q(t)(θ) is broadcasted to all the slave nodes.
maxq∈Q L(q(θ, H))
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 16

17.
Distributed optimization of the lower bound:
Master
θα
q(t)(θ)
X
H
θ1
i = 1, . . . , N
Slave 1
q(t)(H1)
X
H
θ2
i = 1, . . . , N
Slave 2
q(t)(H2)
X
H
θ3
i = 1, . . . , N
Slave 3
q(t)(H3)
q(t+1)(H) = arg maxq(H) L(q(H), q(t)(θ))
maxq∈Q L(q(θ, H))
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 17

18.
Distributed optimization of the lower bound:
Master
θα
q(t)(θ)
X
H
θ1
i = 1, . . . , N
Slave 1
q(t)(H1)
X
H
θ2
i = 1, . . . , N
Slave 2
q(t)(H2)
X
H
θ3
i = 1, . . . , N
Slave 3
q(t)(H3)
q(t+1)(Hn) = arg maxq(Hn) Ln(q(Hn), q(t)(θ))
maxq∈Q L(q(θ, H))
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 18

19.
Distributed optimization of the lower bound:
Master
θα
q(t)(θ)
X
H
θ1
i = 1, . . . , N
Slave 1
q(t)(H1)
X
H
θ2
i = 1, . . . , N
Slave 2
q(t)(H2)
X
H
θ3
i = 1, . . . , N
Slave 3
q(t)(H3)
q(t+1)
(θ) = arg max
q(θ)
L(q(t)
(H), q(θ))
May be coupled
maxq∈Q L(q(θ, H))
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 19

20.
Candidate solutions:
Resort to a generalized mean-field approximation as SVI: does
not factorize over the global parameters.
Prohibitive for models with a large number of global (coupled)
parameters, e.g. linear regression.
Our proposal: VMP as a distributed projected natural
gradient ascent algorithm (PGNA).
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 20

21.
d-VMP as a projected natural gradient ascent
Insight 1: VMP can be expressed as a projected natural
gradient ascent algorithm.
η
(t+1)
X = η
(t)
X + ρX,t[ ˆ ηL(η(t)
)]+
X (1)
[·] is the projection operator.
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 21

22.
d-VMP as a projected natural gradient ascent
Insight 2: The natural gradient of the lower bound can be
expressed as follows:
ˆ ηθ
L = mPa(θ)→θ + mHi →θ
The gradient can be computed in parallel.
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 22

23.
d-VMP as a projected natural gradient ascent
Insight 3: Global parameters are “coupled” only if they belong
to each other’s Markov blanket.
Define a disjoint partition of the global parameters:
R = {J1, . . . , JS }
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 23

24.
d-VMP as a projected natural gradient ascent
d-VMP is based on performing independent global updates
over the global parameters of each partition:
η
(t+1)
Jr
= η
(t)
Jr
+ ρr,t[ ˆ ηL(η(t)
)]+
Jr
ρr,t is the learning rate. If |Jr | = 1 then ρr,t = 1.
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 24

25.
dVMP as a distributed PNGA algorithm:
Master
θα
η
(t+1)
Jr
X
H
θ1
i = 1, . . . , N
Slave 1
η
(t+1)
1
X
H
θ2
i = 1, . . . , N
Slave 2
η
(t+1)
2
X
H
θ3
i = 1, . . . , N
Slave 3
η
(t+1)
3
η
(t)
Jr
+ ρr,t[ˆηL(η(t)
)]+
Jr
For all Jr ∈ R
maxq∈Q L(q(θ, H))
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 25

26.
Outline
1 Motivation
2 Variational Message Passing
3 d-VMP
4 Experimental results
5 Conclusions
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 26

27.
Model fit to the data
Xij HijYi
Hi θ
α
j = 1, . . . , J
i = 1, . . . , N
Attribute range
Density
0 1 2 3 4 5 6
0123456
●
●
●
●
SVI 1%
SVI 5%
SVI 10%
VMP 1%
Representative sample of 55K clients (N) and 33 attributes (J).
“Unrolled” model of more than 3.5M nodes (75% latent variables).
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 27

28.
Model fit to the data
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 28

29.
Model fit to the data
0 500 1000 1500 2000
−1.5e+07−1.0e+07−5.0e+060.0e+00
Time (seconds)
Globallowerbound
Alg. BS(data%)/LR
SVI 1%/0.55
SVI 1%/0.75
SVI 1%/0.99
SVI 5%/0.55
SVI 5%/0.75
SVI 5%/0.99
SVI 10%/0.55
SVI 10%/0.75
SVI 10%/0.99
d−VMP
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 29

30.
Test marginal log-likelihood
BS (% data) LR Log-Likel.
SVI
1 %
0.55 -180902.87
0.75 -298564.03
0.99 -426979.52
5 %
0.55 -177302.24
0.75 -333264.16
0.99 -628105.70
10 %
0.55 -347035.22
0.75 -397525.45
0.99 -538087.13
d-VMP 1.0 67265.34
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 30

31.
Mixtures of learnt posteriors for one attribute
●
●
●
●
SVI 1%
SVI 5%
SVI 10%
VMP 1%
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 31

32.
Scalability settings
Generated data set of 42 million samples per client and 12
variables.
“Unrolled” model of more than 1 billion (109) nodes
(75% latent variables).
AMIDST Toolbox with Apache Flink.
Amazon Web Services (AWS) as distributed computing
environment.
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 32

33.
Scalability results
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 33

34.
Outline
1 Motivation
2 Variational Message Passing
3 d-VMP
4 Experimental results
5 Conclusions
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 34

35.
Conclusions
Variational methods can be scaled using distributed
computation instead of sampling techniques.
Bayesian learning in model with more than 1 billion nodes
(75% of hidden).
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 35

36.
Thank you for your attention
Questions?
You can download our open source Java toolbox:
amidsttoolbox.com
Acknowledgments: This project has received funding from the European Union’s
Seventh Framework Programme for research, technological development and
demonstration under grant agreement no 619209
Int. Conf. on Probabilistic Graphical Models, Lugano, Sept. 6–9, 2016 36