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Marginal Inference in MRFs using Frank-Wolfe - CMAP, Polytechnique

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Dec 10, 2013 - Curvature + Convergence Rate. Cf = sup x,sâD;Î³â[0,1];y=x+Î³(sâx). 2 Î³2. (f (y) â f (x) â ãy

Marginal Inference in MRFs using Frank-Wolfe David Belanger, Daniel Sheldon, Andrew McCallum School of Computer Science University of Massachusetts, Amherst {belanger,sheldon,mccallum}@cs.umass.edu

December 10, 2013

Table of Contents

1

Markov Random Fields

2

Frank-Wolfe for Marginal Inference

3

Optimality Guarantees and Convergence Rate

4

Beyond MRFs

5

Fancier FW

December 10, 2013

2 / 26

Table of Contents

1

Markov Random Fields

2

Frank-Wolfe for Marginal Inference

3

Optimality Guarantees and Convergence Rate

4

Beyond MRFs

5

Fancier FW

December 10, 2013

3 / 26

Markov Random Fields

December 10, 2013

4 / 26

Markov Random Fields

Φθ (x) =

X

θc (xc )

c∈C

December 10, 2013

4 / 26

Markov Random Fields

Φθ (x) =

X

θc (xc )

c∈C

P(x) =

exp (Φθ (x)) log(Z )

December 10, 2013

4 / 26

Markov Random Fields

Φθ (x) =

X

θc (xc )

c∈C

P(x) =

x→µ

exp (Φθ (x)) log(Z )

December 10, 2013

4 / 26

Markov Random Fields

Φθ (x) =

X

θc (xc )

c∈C

P(x) =

exp (Φθ (x)) log(Z )

x→µ Φθ (x) → hθ, µi

December 10, 2013

4 / 26

Marginal Inference

µMARG = EPθ [µ]

December 10, 2013

5 / 26

Marginal Inference

µMARG = EPθ [µ] µMARG = arg max hµ, θi + HM (µ) µ∈M

December 10, 2013

5 / 26

Marginal Inference

µMARG = EPθ [µ] µMARG = arg max hµ, θi + HM (µ) µ∈M

µ ¯ approx = arg maxhµ, θi + HB (µ) µ∈L

December 10, 2013

5 / 26

Marginal Inference

µMARG = EPθ [µ] µMARG = arg max hµ, θi + HM (µ) µ∈M

µ ¯ approx = arg maxhµ, θi + HB (µ) µ∈L

HB (µ) =

X

Wc H(µc )

c∈C

December 10, 2013

5 / 26

MAP Inference

µMAP = arg max hµ, θi µ∈M

December 10, 2013

6 / 26

MAP Inference

µMAP = arg max hµ, θi µ∈M

✓

Black&Box&& MAP&Solver&

µMAP

December 10, 2013

6 / 26

MAP Inference

µMAP = arg max hµ, θi µ∈M

✓

✓

Black&Box&& MAP&Solver&

Gray&Box&& MAP&Solver&

µMAP

µMAP

December 10, 2013

6 / 26

Marginal → MAP Reductions

Hazan and Jaakkola [2012] Ermon et al. [2013]

December 10, 2013

7 / 26

Table of Contents

1

Markov Random Fields

2

Frank-Wolfe for Marginal Inference

3

Optimality Guarantees and Convergence Rate

4

Beyond MRFs

5

Fancier FW

December 10, 2013

8 / 26

Generic FW with Line Search

yt = arg minhx, −∇f (xt−1 )i x∈X

xt = min f ((1 − γ)xt + γyt ) γ∈[0,1]

December 10, 2013

9 / 26

Generic FW with Line Search

xt

Compute& &Gradient&

rf (xt

1)

Linear&& Minimiza