[PPT] - Unifying Local Consistency and MAX SAT Relaxations for Scalable PowerPoint Presentation

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Unifying Local Consistency and MAX SAT Relaxations for Scalable Inference with Rounding Guarantees

Stephen H. Bach Bert Huang Lise Getoor Maryland Virginia Tech UC Santa Cruz

AISTATS 2015

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This Talk

§ Markov random fields capture rich dependencies in structured data, but inference is NP-hard § Relaxed inference can help, but techniques have tradeoffs § Two approaches: Local Consistency Relaxation MAX SAT Relaxation

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Takeaways

§ We can combine their advantages: quality guarantees and highly scalable message-passing algorithms § New inference algorithm for broad class of structured, relational models

⊃

Local Consistency Relaxation MAX SAT Relaxation

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Modeling Relational Data with Markov Random Fields

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Markov Random Fields

§ Probabilistic model for high-dimensional data: § The random variables represent the data, such as whether a person has an attribute or whether a link exists § The potentials score different configurations of the data § The weights scale the influence of different potentials

x w φ

P(x) ∝ exp

w>φ(x)

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Markov Random Fields

§ Variables and potentials form graphical structure:

φ φ φ φ φ φ φ φ x x x x

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Modeling Relational Data

§ Many important problems have relational structure § Common to use logic to describe probabilistic dependencies § Relations in data map to logical predicates

crossing waiting queueing walking talking dancing jogging

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Logical Potentials

§ One way to compactly define MRFs is with first-order logic, e.g., Markov logic networks

[Richardson and Domingos, 2006]

§ Each first-order rule is a template for potentials

Ground out rule over relational data
The truth table of each ground rule is a potential
Each potential’s weight comes from the rule that templated it

5.0 : Friends(X, Y ) ∧ Smokes(X) = ⇒ Smokes(Y )

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Logical Potentials: Grounding

5.0 : Friends(X, Y ) ∧ Smokes(X) = ⇒ Smokes(Y )

Erin

Dave Claudia Bob Alexis

5.0 : Friends(Alexis, Bob) ∧ Smokes(Alexis) = ⇒ Smokes(Bob)

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Logical Potentials

§ Let be a set of rules, where each rule has the general form

Weights and sets and index variables

wj ≥ 0 I+

j

I−

j

R Rj

wj : B @ _

i∈I+

j

xi 1 C A _ B @ _

i∈I−

j

¬xi 1 C A

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§ MAP (maximum a posteriori) inference seeks a most- probable assignment to the unobserved variables § MAP inference is § This MAX SAT problem is combinatorial and NP-hard!

MAP Inference

arg max

x

P(x) ≡ arg max

x∈{0,1}n

X

Rj∈R

wj B @ B @ _

i∈I+ j

xi 1 C A _ B @ _

i∈I− j

¬xi 1 C A 1 C A

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Relaxed MAP Inference

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Approaches to Relaxed Inference

§ Local consistency relaxation

Developed in probabilistic graphical models community
ADVANTAGE: Many highly scalable algorithms available
DISADVANTAGE: No known quality guarantees for logical MRFs

§ MAX SAT relaxation

Developed in randomized algorithms community
ADVANTAGE: Provides strong quality guarantees
DISADVANTAGE: No algorithms designed for large-scale models

§ How can we combine these advantages?

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Local Consistency Relaxation

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Local Consistency Relaxation

§ LCR is a popular technique for approximating MAP in MRFs

Often simply called linear programming (LP) relaxation
Dual decomposition solves dual to LCR objective

§ Lots of work in PGM community, e.g.,

Globerson and Jaakkola, 2007
Wainwright and Jordan, 2008
Sontag et al. 2008, 2012

§ Idea: relax search over consistent marginals to simpler set

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Local Consistency Relaxation

φ φ φ φ φ φ φ φ x x x x

µ µ µ µ

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Local Consistency Relaxation

φ φ φ φ φ φ φ φ x x x x

θ θ

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Local Consistency Relaxation

θ µ : pseudomarginals over variable states

: pseudomarginals over joint potential states

x

φ(xj)

arg max

(θ,µ)∈L

X

Rj∈R

wj X

xj

θj(xj) φj(xj)

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MAX SAT Relaxation

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Approximate Inference

§ View MAP inference as optimizing rounding probabilities § Expected score of a clause is a weighted noisy-or function: § Then expected total score is § But, is highly non-convex!

arg maxp ˆ W

ˆ W = X

Rj∈R

wj B @1 − Y

i∈I+ j

(1 − pi) Y

i∈I− j

pi 1 C A wj B @1 − Y

i∈I+ j

(1 − pi) Y

i∈I− j

pi 1 C A

C

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Approximate Inference

§ It is the products in the objective that make it non-convex § The expected score can be lower bounded using the relationship between arithmetic and harmonic means: § This leads to the lower bound p1 + p2 + · · · + pk k ≥

k

√p1p2 · · · pk

Goemans and Williamson, 1994

X

Rj∈R

wj B @1 − Y

i∈I+ j

(1 − pi) Y

i∈I− j

pi 1 C A ≥ ✓ 1 − 1 e ◆ X

Rj∈R

wj min 8 > < > : X

i∈I+ j

pi + X

i∈I− j

(1 − pi), 1 9 > = > ;

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§ So, we solve the linear program § If we set , a greedy rounding method will find a

optimal discrete solution

§ If we set , it improves to ¾-optimal

arg max

y∈[0,1]n

X

Rj∈R

wj min 8 > < > : X

i∈I+

j

yi + X

i∈I−

j

(1 − yi), 1 9 > = > ;

Approximate Inference

pi = yi pi = 1 2yi + 1 4

✓ 1 − 1 e ◆

Goemans and Williamson, 1994

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Unifying the Relaxations

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Analysis

φ φ φ φ φ φ φ φ x x x x

j=1 j=2 j=3 and so on…

arg max

(θ,µ)∈L

X

Rj∈R

wj X

xj

θj(xj) φj(xj)

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Analysis

µ µ µ µ

arg max

µ∈[0,1]n

X

Rj∈R

ˆ φj(µ)

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Analysis

§ We can now analyze each potential’s parameterized subproblem in isolation: § Using the KKT conditions, we can find a simplified expression for each solution based on the parameters :

ˆ φj(µ) = max

θj|(θj,µ)∈L wj

X

xj

θj(xj) φj(xj)

µ

ˆ φj(µ) = wj min 8 > < > : X

i∈I+

j

µi + X

i∈I−

j

(1 − µi), 1 9 > = > ;

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arg max

µ∈[0,1]n

X

Rj∈R

ˆ φj(µ)

Analysis

µ µ µ µ

wj min 8 > < > : X i∈I+ j µi + X i∈I− j (1 − µi), 1 9 > = > ; wj min 8 > < > : X i∈I+ j µi + X i∈I− j (1 − µi), 1 9 > = > ; wj min 8 > < > : X i∈I+ j µi + X i∈I− j (1 − µi), 1 9 > = > ; wj min 8 > < > : X i∈I+ j µi + X i∈I− j (1 − µi), 1 9 > = > ; wj min 8 > < > : X i∈I+ j µi + X i∈I− j (1 − µi), 1 9 > = > ; wj min 8 > < > : X i∈I+ j µi + X i∈I− j (1 − µi), 1 9 > = > ; wj min 8 > < > : X i∈I+ j µi + X i∈I− j (1 − µi), 1 9 > = > ; wj min 8 > < > : X i∈I+ j µi + X i∈I− j (1 − µi), 1 9 > = > ;

Substitute back into

uter objective

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Analysis

§ Leads to simplified, projected LCR over :

arg max

µ∈[0,1]n m

X

j=1

wj min 8 > < > : X

i∈I+ j

µi + X

i∈I− j

(1 − µi), 1 9 > = > ;

µ

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arg max

µ∈[0,1]n

X

Rj∈R

wj min 8 > < > : X

i∈I+ j

µi + X

i∈I− j

(1 − µi), 1 9 > = > ;

Analysis

arg max

y∈[0,1]n

X

Rj∈R

wj min 8 > < > : X

i∈I+ j

yi + X

i∈I− j

(1 − yi), 1 9 > = > ;

Local Consistency Relaxation MAX SAT Relaxation

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Evaluation

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New Algorithm: Rounded LP

§ Three steps:

Solves relaxed MAP inference problem
Modifies pseudomarginals
Rounds to discrete solutions

§ We use the alternating direction method of multipliers (ADMM) to implement a message-passing approach

[Glowinski and Marrocco, 1975; Gabay and Mercier, 1976]

§ ADMM-based inference for MAX SAT form of problem was

riginally developed for hinge-loss MRFs

[Bach et al., 2015]

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Evaluation Setup

§ Compared with

MPLP
MPLP with cycle tightening

§ MPLP uses coordinate descent dual decomposition, so rounding not applicable § Solved MAP in social-network opinion models with super- and submodular features § Measured primal score, i.e., weighted sum of satisfied clauses

[Globerson and Jaakkola, 2007; Sontag et al. 2008, 2012]

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Results

§ Expected scores of Rounded LP are significantly better § Rounded LP’s final scores are even better § Cycle tightening has limited effect § Rounded LP does 20% better than MPLP, and only takes 1 minute for 1 million clauses

2 4 6 8 10 x 10 5 2 4 6 8 10 x 10 5 MPLP Primal Objective Primal Objective LP Upper Bound Rounded LP Rounded LP (Exp) MPLP w/ Cycles

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Conclusion

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Conclusion

§ Uniting local consistency and MAX SAT relaxation combines the benefits of both: scalability and accuracy § Rounding pseudomarginals can significantly improve quality over coordinate descent dual decomposition § Many applications to structured and relational data:

Social network analysis
Bioinformatics
Recommender systems
Text and video understanding

Thank You!

bach@cs.umd.edu @stevebach