MAP Inference in Discrete Models

M. Pawan Kumar, Stanford University

Recent Advances in Convex

Relaxations

Outline

• Revisiting the LP relaxation

• Rounding Schemes and Move Making

• Beyond the LP relaxation

Linear Programming Relaxation

min Ty

ya;i [0,1]

∑i ya;i = 1

∑k yab;ik = ya;i

No reason why we can’t solve this*

*memory requirements, time complexity

Linear Programming Relaxation

Primal formulation is useful

Easier to analyze

LP better than a large class of relaxations

- QP (Ravikumar, Lafferty 2006)

- SOCP (Muramatsu, Suzuki 2003)

Kumar, Kolmogorov and Torr, NIPS 2007

Linear Programming Relaxation

Primal fractional solution is useful

Multiplicative Bounds

Type of Problem Bound

Potts 2

Truncated Linear 2 + √2

Truncated Quadratic O(√M)

General Metric O(log |L|)

Outline

• Revisiting the LP relaxation

• Rounding Schemes and Move Making

• Beyond the LP relaxation

Randomized Rounding

0 y’a;0 y’a;i y’a;k y’a;h = 1

y’a;i = ya;0 + ya;1 + … + ya;i

Choose an interval of length L’

Randomized Rounding

0 y’a;0 y’a;i y’a;k y’a;h = 1

y’a;i = ya;0 + ya;1 + … + ya;i

Generate a random number r (0,1]

r

Randomized Rounding

0 y’a;0 y’a;i y’a;k y’a;h = 1

y’a;i = ya;0 + ya;1 + … + ya;i

Assign label next to r (if within the interval)

r

Move Making

Va Vb

• Initialize the labeling

• Choose interval I of L’ labels

• Each variable can

• Retain old label

• Choose a label from I

• Choose best labeling

Iterate over intervals

Truncated Convex Models

Two Problems

Va Vb

• Choose interval I of L’ labels

• Each variable can

• Retain old label

• Choose a label from I

• Choose best labeling

Large L’ => Non-submodular

Non-submodular

First Problem

Va Vb Submodular problem

Ishikawa, 2003; Veksler, 2007

First Problem

Va Vb Non-submodular

Problem

First Problem

Va Vb Submodular problem

Veksler, 2007

First Problem

Va Vb

am+1

am+2

an

t

am+2

bm+1

bm+2

bn

bm+2

First Problem

Va Vb

am+1

am+2

an

t

am+2

bm+1

bm+2

bn

bm+2

First Problem

Va Vb

am+1

am+2

an

t

am+2

bm+1

bm+2

bn

bm+2

First Problem

Va Vb

am+1

am+2

an

t

am+2

bm+1

bm+2

bn

bm+2

First Problem

Va Vb

Model unary potentials exactly

am+1

am+2

an

t

am+2

bm+1

bm+2

bn

bm+2

First Problem

Va Vb

Similarly for Vb

am+1

am+2

an

t

am+2

bm+1

bm+2

bn

bm+2

First Problem

Va Vb

Model convex pairwise costs

am+1

am+2

an

t

am+2

bm+1

bm+2

bn

bm+2

First Problem

Va Vb

Overestimated pairwise potentials

Wanted to model

ab;ik = wab min{ d(i-k), M }

For all li, lk I

Have modelled

ab;ik = wab d(i-k)

For all li, lk I

Second Problem

Va Vb

• Choose interval I of L’ labels

• Each variable can

• Retain old label

• Choose a label from I

• Choose best labeling

Non-submodular problem !!

Second Problem

Va Vb

am+1

am+2

an

t

bm+1

bm+2

bn

Previous labels may not lie in interval

Second Problem

Va Vb

am+1

am+2

an

t

bm+1

bm+2

bn

s

ua ub

ua and ub : unary potentials for previous labels

Second Problem

Va Vb

am+1

am+2

an

t

bm+1

bm+2

bn

s

ua ub

Pab : pairwise potential for previous labels

ab

Pab

MM

Second Problem

Va Vb

am+1

am+2

an

t

bm+1

bm+2

bn

wab d(i-k)

s

ua ub

ab

Pab

MM

Second Problem

Va Vb

am+1

am+2

an

t

bm+1

bm+2

bn

wab ( d(i-m-1) + M )

s

ua ub

ab

Pab

MM

Second Problem

Va Vb

am+1

am+2

an

t

bm+1

bm+2

bn

Pab

s

ua ub

ab

Pab

MM

Graph Construction

Va Vb

Find st-MINCUT.

Retain old labeling

if energy increases.

am+1

am+2

an

bm+1

bm+2

bn

t

ITERATE

Move Making

LP Bounds

Kumar and Torr, NIPS 08

In General?

Type of Problem Bound

Potts 2

Truncated Linear 2 + √2

Truncated Quadratic O(√M)

General Metric O(log |L|)

Kumar and Koller, UAI 09

Outline

• Revisiting the LP relaxation

• Rounding Schemes and Move Making

• Beyond the LP relaxation

LP over a Frustrated Cycle

Va Vb

1

0 0

1

0

0

0

0l0

l1

Vb Vc

1

0 0

1

0

0

0

0

Vc Va

1

0 0

1

0

0

0

0

Optimal labeling has energy = 1

One takes label l0, two take label l1

One takes label l1, two take label l0

LP optimal solution

Va Vb

0

0.5 0.5

0

0.5

0.5

0.5

0.5l0

l1

Vb Vc

0

0.5 0.5

0

0.5

0.5

0.5

0.5

Vc Va

0

0.5 0.5

0

0.5

0.5

0.5

0.5

Optimal fractional labeling has energy = 0

Need tighter relaxations

Cycle Inequalities

Vb

Va Vc

At least two variables take same label

Cycle Inequalities

Vb

Va Vc

Va and Vc take label 0, yac;00 = 1

Cycle Inequalities

Vb

Va Vc

Or Va and Vc take label 1, yac;11 = 1

Cycle Inequalities

Vb

Va Vc

∑ yab;00 + yab;11 ≥ 1

LP optimal solution

Va Vb

0

0.5 0.5

0

0.5

0.5

0.5

0.5l0

l1

Vb Vc

0

0.5 0.5

0

0.5

0.5

0.5

0.5

Vc Va

0

0.5 0.5

0

0.5

0.5

0.5

0.5

Does not satisfy cycle inequality

Cycle Inequalities

Generalizes to cycles of arbitrary length

Barahona and Mahjoub, 1986

Generalizes to arbitrary label sets

Chopra and Rao, 1991

Sontag and Jaakkola, 2007

Modifies the primal

But weren’t we solving the dual?

Modifying the Dual

Do operations on trees and cycles

Which algorithm? Which cycles?

Kumar and Torr, 2008

TRW-S All cycles of length 3 and 4

Komodakis and Paragios, 2008

Dual Decomposition All frustrated cycles

Sontag et al, 2008

MPLP Iteratively add cycles

Maximum increase in the dual

Questions?