ECE 6504: Advanced Topics in Machine Learning

Probabilistic Graphical Models and Large-Scale Learning

Dhruv Batra Virginia Tech

Topics –  Markov Random Fields: MAP Inference

–  Integer Programming, LP formulation –  Dual Decomposition

Readings: KF 13.1-5, Barber 5.1,28.9

Administrativia •  HW1

–  Solutions and grades released

•  HW2 –  Solutions released –  Grades next week

•  Project Presentations –  When: April 22, 24 –  Where: in class –  5 min talk

•  Main results •  Semester completion 2 weeks out from that point so nearly finished

results expected •  Slides due: April 21 11:55pm

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Recap of Last Time

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MAP Inference

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MAP

Inference

Most Likely Assignment

y1

y2

…

yn

Person

Table Plate

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S(y) =�

i∈Vθi(yi) +

�

(i,j)∈E

θij(yi, yj)

P (y) =1

Z eS(y)

kx1 1 1 10 0

kxk

10

10 10

10

0

0

Node Scores / Local Rewards

Edge Scores / Distributed Prior

P (y)

y

MAP Inference •  Why is MAP difficult?

•  What if we independently maximize the terms?

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MAP in Pairwise MRFs

•  Over-Complete Representation

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G = (V, E)

X1

X2

…

Xn Xi kxk

… …

θij(1, 1) θij(1, k)

θij(k, 1) θij(k, k)

θ1(1)

θ1(k)

…

kx1

x1 (x1, x2) (xn−1, xn)

…

θn(1)

θn(k)

… …

θij(1, 1) θij(1, k)

θij(k, 1) θij(k, k)

… xn…

kx1

µi(s)1

1

0

0

0

0

0

0

θ = θ1(1) . . . θ1(k) θn(1) . . . θn(k) θn−1,n(1, 1) . . . θn−1,n(k, k)θ12(1, 1) . . . θ12(k, k)

1

0

0

0

xi = 1

0

1

0

0

xi = 2

µ1(1) . . . µ1(k) µn(1) . . . µn(k)

xi

MAP in Pairwise MRFs

•  Over-Complete Representation

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G = (V, E)

x1

x2

…

xn Xi

x1 (x1, x2) (xn−1, xn)… xn…

θ = θ1(1) . . . θ1(k) θn(1) . . . θn(k) θn−1,n(1, 1) . . . θn−1,n(k, k)θ12(1, 1) . . . θ12(k, k)

k2x1

µij(s, t)

xi

xj

1 0 0 0 0 0 0 0 0 0 0 0

xi = 1xj = 1

0 1 0 0 0 0 0 0 0 0 0 0

xj = 2xi = 1

µ1(1) . . . µ1(k) µn(1) . . . µn(k) µ12(1, 1) . . . µ12(k, k) µn−1,n(1, 1) . . . µn−1,n(k, k)µx =

S(x) = θ · µx

MAP in Pairwise MRFs •  Integer Program

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�

s

µij(s, t) = µj(t)

�

s,t

µi,j(s, t) = 1

�

s

µi(s) = 1

µij(s, t) ∈ {0, 1}

µi(s) ∈ {0, 1}Indicator Variables

Unique Label

Consistent Assignments

maxµ

θTµ

MAP in Pairwise MRFs •  MAP Integer Program

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µx5

µx4µx3

µx2

µx1

maxµ

θTµ

s.t. Aµ = b

µ(·) ∈ {0, 1}

MAP in Pairwise MRFs •  MAP Linear Program

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µx5

µx4µx3

µx2

µx1

A = O(|E|)

O(|E|)

Off-the-shelf solvers CPLEX Mosek

etc

maxµ

θTµ

s.t. Aµ = b

µ(·) ∈ [0, 1]

Plan for today •  MRF Inference

–  (Specialized) MAP Inference •  Integer Programming Formulation •  Linear Programming Relaxation

–  Understanding the LP better –  When is it tight? –  When is it not?

•  Dual Decomposition –  Algorithm for solving this LP

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MAP in Pairwise MRFs •  MAP Integer Program

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µx5

µx4µx3

µx2

µx1

maxµ

θTµ

s.t. Aµ = b

µ(·) ∈ {0, 1}

Marginal Polytope

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•  a

Figure Credit: David Sontag

MAP in Pairwise MRFs •  MAP Linear Program

•  Properties –  If LP-opt is integral, MAP is found –  LP always integral for trees –  Efficient message-passing schemes for solving LP

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µx5

µx4µx3

µx2

µx1

maxµ

θTµ

s.t. Aµ = b

µ(·) ∈ [0, 1]

LP Relaxation •  Block Co-ordinate / Sub-gradient Descent on Dual

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A = O(|E|)

O(|E|)

λij→j

λji→i

LP Relaxation •  Block Co-ordinate / Sub-gradient Descent on Dual

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A = O(|E|)

O(|E|)

λ(t)ji→i

λ(t)ij→j

LP Relaxation •  Block Co-ordinate / Sub-gradient Descent on Dual

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A = O(|E|)

O(|E|)

λ(t+1)ij→j

λ(t+1)ji→i

Distributed Message-Passing

Still inefficient!

Linear Programming Duality

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Dual Decomposition •  For MAP Inference

–  On board

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MAP in Pairwise MRFs •  MAP Integer Program

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maxµ

θTµ

s.t. Aµ = b

µ(·) ∈ {0, 1}

µx5

µx4µx3

µx2

µx1

MAP LP •  Lagrangian Relaxation

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Dual

Convex (Non-smooth)

Upper-Bound on MAP

f(λ) =

Subgradient Descent

∇f(λ(0))

λ(0)

∇f(λ(1))

λ(1)

maxµ∈C

�

i

θi · µi +�

(i,j)

θij · µij

s.t. µi(·),µij(·) ∈ {0, 1}

minλ≥0

f(λ)

λ

MAP score

f(λ)

−λ · (Aµ− b)