Inference in Probabilistic Graphical Models by …2019/04/12 · Inference in Probabilistic...

Inference in Probabilistic Graphical Models by GraphNeural Networks

KiJung Yoon, Renjie Liao, Yuwen Xiong, Lisa Zhang, Ethan Fetaya,Raquel Urtasun, Richard Zemel, Xaq Pitkow

Presenter: Arshdeep Sekhon

KiJung Yoon, Renjie Liao, Yuwen Xiong, Lisa Zhang, Ethan Fetaya, Raquel Urtasun, Richard Zemel, Xaq PitkowInference in Probabilistic Graphical Models by Graph Neural NetworksPresenter: Arshdeep Sekhon 1 / 22

Outline

1 Introduction

2 Proposed Method

3 Experiments

Probabilistic Graphical Models(PGMs)

Given x ∈ RD , joint probability p(x)

Simplify joint p(x) into a factorization based on conditionalindependence defined by a graph structure.

Factor Graphs

Factorization of the joint probability distribution for more efficientcomputationsbipartite graph: two types of nodes, edges connect different nodetypesGiven a factorization:g(X1,X2,X3) = f1(X1)f2(X1,X2)f3(X1,X2)f4(X2,X3)

Tasks for PGMs: Inference

Inference Task: Given a graphical model p(x), find marginal probabilitypi (xi )pi (xi ) =

∑x/xi

Tasks: MAP

Maximum A Posteriori(MAP) Inference: x∗ = argmaxxp(x), Finding themost probable state

Belief Propagation

Belief propagation operates on these factor graphs by constructingmessages µi→α and µα→i that are passed between variable(i) andfactor(α) nodes:

µα→i (xi ) =∑xα\xi

ψα(xα)∏

j∈Nα\i

µj→α(xj) (1)

µi→α(xi ) =∏

β∈Ni\α

µβ→i (xi ) (2)

the estimated marginal joint probability of a factor α, namely Bα(xα), isgiven by

Bα(xα) =1

Zψα(xα)

∏i∈Nα

µi→α(xi ) (3)

Belief Propagation

Issues:

Exact Inference on tree graphs, but not on graphs with cycles

Update Steps may not have closed form solutions

Special Case: Binary Markov Random Field

variables x ∈ {+1,−1}|V |

p(x) = 1Z exp (b · x + x · J · x) (4)

singleton factor: ψi (xi ) = ebixi

pairwise factors: ψi ,j(xi , xj) = eJijxixj

Goal: find p(xi )

J is a symmetric matrix

Belief Propagation on Binary Markov Fields

Belief propagation updates messages µij from i to j according to

µij(xj) =∑xi

eJijxixj+bixi∏

k∈Ni\j

µki (xi ) (5)

estimated marginals by pi (xi ) = 1Z e

bixi∏

k∈Niµki (xi )

Proposed GNN architecture

mt+1i→j =M(hti ,h

tj , εij) (6)

mt+1i =

∑j∈Ni

mt+1j→i (7)

ht+1i = U(hti ,m

t+1i ) (8)

y = σ(R(h

(T )i )

Proposed Model: Message-GNN

Convert all messages µi→j into a node in a GNN hi→j

Two GNN nodes v and w are connected if they correspond tomessages µi→j and µj→k

message from vi to vj is computed by mt+1i→j =M(

∑k∈Ni\j h

tk→i , eij).

update its hidden state by ht+1i→j = U(hti→j ,m

t+1i→j ).

Readout: pi (xi ) = R(∑

j∈Nih(T )j→i ).KiJung Yoon, Renjie Liao, Yuwen Xiong, Lisa Zhang, Ethan Fetaya, Raquel Urtasun, Richard Zemel, Xaq PitkowInference in Probabilistic Graphical Models by Graph Neural NetworksPresenter: Arshdeep Sekhon 12 / 22

Proposed Model: node-GNN

No representation for factor nodes

information about interactions in εij

GNN for inference and MAP

minimize cross entropy loss L(p, p) = −∑

i qi log pi (xi )

For MAP: delta function qi = δxi ,x∗iFor Marginal Inference: qi enumeration of ground truth

Experimental Design

generalization under 4 conditions

to unseen graphs of the same structure (I, II),

and to completely different random graphs (III, IV).

These graphs may be the same size (I, III) or larger (II, IV).

structured randomn = 9 I IIIn = 16 II IV

Experimental Set up

train on 100 graphical models of 13 classical types

Sample Jij = Jji ∼ N (0, 1)

sample biases bi ∼ N (0, (1/4)2)

Within Set Generalization

test graphs had the same size and structure as training graphsbut the values of singleton and edge potentials differedmost notable performance difference between loopy graphs

Out of Set Generalization

Train on same graphs

Test on bigger graphs

Metric: the average Kullback-Leibler divergence 〈DKL[pi (xi )‖pi (xi )]〉across the entire set of test graphs with the small and large numberof nodes.

Out of Set Generalization: different structure

connected random Erdos Renyi graphs Gn,q,changed connectivity by increasing the edge probability from q = 0.1(sparse) to 0.9 (dense)

Convergence of Inference Dynamics

How node states change over time

||htv − ht−1v ||`2

0 100.0

time step

MAP Estimation

x∗ = argmaxxp(x)

Conclusions

limited testing: binary markov random field models only

relatively small graphs

A combination of NNs approximation power to incorporate non linearstructure of inference problems

Inference in Probabilistic Graphical Models by …2019/04/12 · Inference in Probabilistic...

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