Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Relaxation-Aware Heuristics for ExactOptimization in Graphical Models

Fulya Trosser 1, Simon de Givry 1, George Katsirelos 2

1INRA MIA Toulouse

2INRA MIA Paris

5 December 2019

Journee AIGM 2019

1 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Overview

1 Introduction

2 PreliminariesConstraint Satisfaction ProblemWeighted Constraint Satisfaction ProblemLocal Consistency

3 ContributionsVAC-integralityVariable Ordering HeuristicRASPS for UB Computation

4 Numerical Results

5 Future Work

6 References

2 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Introduction

Exact solvers for optimization problems on graphicalmodels (CFN, MRF etc), typically use branch-and-bound.

Two main factors: quality of the bound at each node andbranching heuristics.

Virtual Arc Consistency (VAC) algorithm: high qualitybounds, but at a significant cost

Search NodeInitial UB ComputationLB ComputationBranching Heuristic

3 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Introduction

Branching heuristics ignore the information that VACproduces on the linear relaxation of the problem

Branching heuristic may make decisions that are clearlyineffective

By eliminating these ineffective decisions, we significantlyreduce the size of the branch-and-bound tree.

DIFFICULTEASY

4 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Constraint Satisfaction Problem (CSP)

A Constraint Satisfaction Problem [Cooper and Schiex, 2004]is a triple 〈X ,D,C 〉.

X : set of n variables X = {1, . . . , n}.

D: set of domains D = {Di : i ∈ X}.

C : set of constraints.

A tuple t

is a solution iff it satisfies all the constraints in C .

5 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Cost Function

A cost function is defined over the scope S of the constraint cSto which it corresponds. It associates a cost to each tuplet ∈ `(S).

c∅: the nullary cost function = constant cost.

ci : the unary cost function on variable i .

cij : the binary cost function on variables i and j .

x cx y cy x y cxy

a 0 a 1a a 0a b 1

b 2 b 0b a 0b b 3

ab

X

2ab

Y

113

6 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Weighted Constraint Satisfaction Problem

Weighted Constraint Satisfaction Problem (WCSP)

is a quadruple 〈X ,D,C ,m〉 where C is a set of cost functionsand m is the upper bound [Cooper et al., 2010].

Find a solution such that the sum

c∅ +∑i∈X

ci +∑ij∈X 2

cij

is minimized,is less than the upper bound m.

ab

X

2ab

Y

113

7 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Levels of Local Consistency

Node Consistency (NC)

A WCSP is Node Consistent if for any variable i ∈ {1, . . . , n},∃a ∈ Di such that ci (a) = 0

(Soft) Arc Consistency (SAC)

A binary WCSP is Arc Consistent if for all cxy ∈ C we have:

∀a ∈ Dx , ∃b ∈ Dy such that cxy (a, b) = 0.

ab

X

2ab

Y

113

ab

X

2ab

Y

112

c∅=0 c∅=0NC but not SAC SAC but not NC

ab

X

2ab

Y2

c∅=1SAC and NC

8 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Levels of Local Consistency

Bool(P)

Values with non-zero unary cost ⇒ REMOVEDAssignments with non-zero binary cost ⇒ FORBIDDEN

Virtual Arc Consistency

A WCSP P is virtual arc consistent (VAC) if Bool(P) is arcconsistent.

ab

X

2ab

Y

113

WCSP P

a

X

b

YBool(P)not AC!

9 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Strict Arc Consistency vs VAC-integrality

Strict AC [Savchynskyy et al., 2013]

a unique value a ∈ Di such that ci (a) = 0

∀cij ∈ C , a unique tuple {(i , a), (j , b)} which satisfiescij(a, b) = 0

VAC-integrality

a unique value a ∈ Di such that ci (a) = 0

∀cij ∈ C , at least one tuple {(i , a), (j , b)} which satisfiescij(a, b) + cj(b) = 0.

10 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Strict Arc Consistency vs VAC-integrality

Strict AC

unique a ∈ Di s.t.ci (a) = 0

∀cij ∈ C , unique{(i , a), (j , b)} s.t.cij(a, b) = 0

VAC-integrality

unique a ∈ Di s.t.ci (a) = 0

∀cij ∈ C , at least one{(i , a), (j , b)} s.t.cij(a, b) + cj(b) = 0.

2

P

i

j2

ab

ab

k2

ab

CijCikCjk

Bool(P)

i

ja

a

k

a

strict AC:{ i }strict* AC:{ i, j, k }

They all have aUNIQUE valuein their domain

in Bool(P)

11 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Variable Ordering Heuristic

It makes sense to AVOID branching on VAC-integralvariables, since the dual bound will not be improved.

So, we classify thevariables with respect totheir domain sizes inBool(P): those withsingleton domains, and therest.

Then, we give priority tothe variables with morethan one value.

DIFFICULTEASY

12 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Heuristic for UB Inspired by CombiLP

[Haller et al., 2018] develop an approach called CombiLP

Subproblem is constructed by fixing the variables thathave the same value in the incumbent and in the currentrelaxation.

Following this approach, we propose a primal heuristicwhich runs in preprocessing.

= +

13 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Relaxation-Aware Sub-Problem Search (RASPS)

X

Y Z

M Na

a a

aa

bb

b

b bcc 2

2

22

22

1 11

2 2

1

θ = 1

X

Y Z

M Na

a a

aab

b bc

X

Y Z

M Na

a a

aab

b b

X

Y Z

M Na

a a

aab

b b22

22

22

θ = 2

X

Y Z

M Na

a a

aab

b bcc 1

X

Y Z

M N

a

a a

aab

b bcc 2

2

22

221

1

X

Y Z

M N

a

a a

aab

b bcc 2

2

22

22

1

14 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

RASPS

X

Y Z

M Na

a a

aa

bb

b

b bcc 2

2

22

22

1 11

2 2

1

Figure: WCSP P. OPT = {a, a, b, c, a} with total cost 1.

X

Y Z

M Na

a a

aab

b b22

22

22

Figure: θ = 1, {a, a, b, a, a}with total cost 2.

1

X

Y Z

M N

a

a a

aab

b bcc 2

2

22

22

1

Figure: θ = 2, {a, a, b, c, a}with total cost 1.

15 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

RASPS

θi : Threshold atiteration i of VAC

ri : Ratio ofVAC-integral ACvariables

αi = ri/θi

= +

16 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Experiments

ToulBar2 [Cooper et al., 2010]: an open-source exactsolver for cost function networks that solves variouscombinatorial optimization problems.

CombiLP [Haller et al., 2018]: an open-source algorithmfor energy minimization of graphical models which usesToulBar2 as its internal combinatorial solver.

431 instances from various benchmarks including

21 instances from CPD30 instances from worms

Time limit: 3600 seconds (36000 for CPD)

17 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Numerical Results for Worms

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30

CPUtim

e(inse

conds)

Numberofsolvedwormsinstances

CombiLP(VAC+VAC-integral+threshold+RASPS)CombiLP(VAC-in-preprocessing)

toulbar2(VAC+VAC-integral+threshold+RASPS)toulbar2(VAC-in-preprocessing)

18 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Numerical Results for CPD

0

5000

10000

15000

20000

25000

30000

35000

0 5 10 15 20

CPUtim

e(inse

conds)

NumberofsolvedCPDinstances

CombiLP(VAC+VAC-integral+threshold+RASPS)toulbar2(VAC+VAC-integral+threshold+RASPS)

CombiLP(VAC-in-preprocessing)toulbar2(VAC-in-preprocessing)

19 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Scatter Plots

20 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Future Work

Further exploring the connections between WCSP and ILP

Finding ways of making VAC more useful more often inWCSP (so it could potentially become the default option)

Testing these heuristics with Bayesian Network StructureLearning instances

21 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

References

Cooper, Martin and Schiex, Thomas (2004)

Arc consistency for soft constraints

Artificial Intelligence 154(1-2), 199 – 227.

Cooper, Martin C and De Givry, Simon and Sanchez, Martı andSchiex, Thomas and Zytnicki, Matthias and Werner, Tomas (2010)

Soft arc consistency revisited

Artificial Intelligence 174(7-8), 449 – 478.

Danna, Emilie and Rothberg, Edward and Le Pape, Claude (2005)

Exploring relaxation induced neighborhoods to improve MIP solutions

Mathematical Programming 102(1), 71 – 90.

Haller, Stefan and Swoboda, Paul and Savchynskyy, Bogdan (2018)

Exact MAP-Inference by Confining Combinatorial Search with LPRelaxation

AAAI.

22 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

References

Savchynskyy, Bogdan and Kappes, Jorg Hendrik and Swoboda, Pauland Schnorr, Christoph (2013)

Global MAP-optimality by shrinking the combinatorial search areawith convex relaxation

Advances in Neural Information Processing Systems 1950 – 1958.

23 / 24

Relaxation-Aware

Heuristics

Fulya Trosser

Introduction

Preliminaries

CSP

WCSP

Local Consistency

Contributions

VAC-integrality

Variable Ordering

UB Computation

Results

Future Work

References

Thank You

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