Discrete Optimization Lecture 1 M. Pawan Kumar Slides available online

Outline Problem Formulation –Energy Function –Energy Minimization –Computing min-marginals Reparameterization Energy Minimization for Trees Loopy Belief Propagation

Energy Function VaVa VbVb VcVc VdVd Label l 0 Label l 1 Random Variables V = {V a, V b, ….} Labels L = {l 0, l 1, ….} Labelling f: {a, b, …. }  {0,1, …}

Energy Function VaVa VbVb VcVc VdVd Q(f) = ∑ a  a;f(a) Unary Potential Label l 0 Label l 1 Easy to minimize Neighbourhood

Energy Function VaVa VbVb VcVc VdVd E : (a,b)  E iff V a and V b are neighbours E = { (a,b), (b,c), (c,d) } Label l 0 Label l 1

Energy Function VaVa VbVb VcVc VdVd +∑ (a,b)  ab;f(a)f(b) Pairwise Potential Label l 0 Label l 1 Q(f) = ∑ a  a;f(a)

Energy Function VaVa VbVb VcVc VdVd Parameter Label l 0 Label l 1 +∑ (a,b)  ab;f(a)f(b) Q(f;  )= ∑ a  a;f(a)

Outline Problem Formulation –Energy Function –Energy Minimization –Computing min-marginals Reparameterization Energy Minimization for Trees Loopy Belief Propagation

Energy Minimization VaVa VbVb VcVc VdVd Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) Label l 0 Label l 1

Energy Minimization VaVa VbVb VcVc VdVd Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) = 13 Label l 0 Label l 1

Energy Minimization VaVa VbVb VcVc VdVd Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) Label l 0 Label l 1

Energy Minimization VaVa VbVb VcVc VdVd Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) = 27 Label l 0 Label l 1

Energy Minimization VaVa VbVb VcVc VdVd Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) f* = arg min Q(f;  ) q* = min Q(f;  ) = Q(f*;  ) Label l 0 Label l 1

Energy Minimization f(a)f(b)f(c)f(d) Q(f;  ) possible labellings f(a)f(b)f(c)f(d) Q(f;  ) f* = {1, 0, 0, 1} q* = 13

Outline Problem Formulation –Energy Function –Energy Minimization –Computing min-marginals Reparameterization Energy Minimization for Trees Loopy Belief Propagation

Min-Marginals VaVa VbVb VcVc VdVd f* = arg min Q(f;  ) such that f(a) = i Min-marginal q a;i Label l 0 Label l 1

Min-Marginals 16 possible labellings q a;0 = 15 f(a)f(b)f(c)f(d) Q(f;  ) f(a)f(b)f(c)f(d) Q(f;  )

Min-Marginals 16 possible labellings q a;1 = 13 f(a)f(b)f(c)f(d) Q(f;  ) f(a)f(b)f(c)f(d) Q(f;  )

Min-Marginals and MAP Minimum min-marginal of any variable = energy of MAP labelling min f Q(f;  ) such that f(a) = i q a;i min i min i ( ) V a has to take one label min f Q(f;  )

Summary Energy Minimization f* = arg min Q(f;  ) Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) Min-marginals q a;i = min Q(f;  ) s.t. f(a) = i Energy Function

Outline Problem Formulation Reparameterization Energy Minimization for Trees Loopy Belief Propagation

Reparameterization VaVa VbVb f(a)f(b) Q(f;  ) Add a constant to all  a;i Subtract that constant from all  b;k

Reparameterization f(a)f(b) Q(f;  ) Add a constant to all  a;i Subtract that constant from all  b;k Q(f;  ’) = Q(f;  ) VaVa VbVb

Reparameterization VaVa VbVb f(a)f(b) Q(f;  ) Add a constant to one  b;k Subtract that constant from  ab;ik for all ‘i’ - 3

Reparameterization VaVa VbVb f(a)f(b) Q(f;  ) Q(f;  ’) = Q(f;  ) Add a constant to one  b;k Subtract that constant from  ab;ik for all ‘i’

Reparameterization VaVa VbVb VaVa VbVb VaVa VbVb  ’ a;i =  a;i  ’ b;k =  b;k  ’ ab;ik =  ab;ik + M ab;k - M ab;k + M ba;i - M ba;i Q(f;  ’) = Q(f;  )

Reparameterization Q(f;  ’) = Q(f;  ), for all f  ’ is a reparameterization of , iff  ’    ’ b;k =  b;k  ’ a;i =  a;i  ’ ab;ik =  ab;ik + M ab;k - M ab;k + M ba;i - M ba;i Equivalently Kolmogorov, PAMI, 2006 VaVa VbVb

Recap Energy Minimization f* = arg min Q(f;  ) Q(f;  ) = ∑ a  a;f(a) + ∑ (a,b)  ab;f(a)f(b) Min-marginals q a;i = min Q(f;  ) s.t. f(a) = i Q(f;  ’) = Q(f;  ), for all f  ’   Reparameterization

Outline Problem Formulation Reparameterization Energy Minimization for Trees Loopy Belief Propagation Pearl, 1988

Belief Propagation Belief Propagation is exact for chains Some problems are easy Exact MAP for trees Clever Reparameterization

Two Variables VaVa VbVb VaVa VbVb Choose the right constant  ’ b;k = q b;k Add a constant to one  b;k Subtract that constant from  ab;ik for all ‘i’

VaVa VbVb VaVa VbVb Choose the right constant  ’ b;k = q b;k  a;0 +  ab;00 =  a;1 +  ab;10 = min M ab;0 = Two Variables

VaVa VbVb VaVa VbVb Choose the right constant  ’ b;k = q b;k Two Variables

VaVa VbVb VaVa VbVb Choose the right constant  ’ b;k = q b;k f(a) = 1  ’ b;0 = q b;0 Two Variables Potentials along the red path add up to 0

VaVa VbVb VaVa VbVb Choose the right constant  ’ b;k = q b;k  a;0 +  ab;01 =  a;1 +  ab;11 = min M ab;1 = Two Variables

VaVa VbVb VaVa VbVb Choose the right constant  ’ b;k = q b;k f(a) = 1  ’ b;0 = q b;0 f(a) = 1  ’ b;1 = q b;1 Minimum of min-marginals = MAP estimate Two Variables

VaVa VbVb VaVa VbVb Choose the right constant  ’ b;k = q b;k f(a) = 1  ’ b;0 = q b;0 f(a) = 1  ’ b;1 = q b;1 f*(b) = 0f*(a) = 1 Two Variables

VaVa VbVb VaVa VbVb Choose the right constant  ’ b;k = q b;k f(a) = 1  ’ b;0 = q b;0 f(a) = 1  ’ b;1 = q b;1 We get all the min-marginals of V b Two Variables

Recap We only need to know two sets of equations General form of Reparameterization  ’ a;i =  a;i  ’ ab;ik =  ab;ik + M ab;k - M ab;k + M ba;i - M ba;i  ’ b;k =  b;k Reparameterization of (a,b) in Belief Propagation M ab;k = min i {  a;i +  ab;ik } M ba;i = 0

Three Variables VaVa VbVb VcVc Reparameterize the edge (a,b) as before l0l0 l1l1

VaVa VbVb VcVc Reparameterize the edge (a,b) as before f(a) = Three Variables l0l0 l1l1

VaVa VbVb VcVc Reparameterize the edge (a,b) as before f(a) = 1 Potentials along the red path add up to Three Variables l0l0 l1l1

VaVa VbVb VcVc Reparameterize the edge (b,c) as before f(a) = 1 Potentials along the red path add up to Three Variables l0l0 l1l1

VaVa VbVb VcVc Reparameterize the edge (b,c) as before f(a) = 1 Potentials along the red path add up to 0 f(b) = 1 f(b) = Three Variables l0l0 l1l1

VaVa VbVb VcVc Reparameterize the edge (b,c) as before f(a) = 1 Potentials along the red path add up to 0 f(b) = 1 f(b) = 0 q c;0 q c; Three Variables l0l0 l1l1

VaVa VbVb VcVc f(a) = 1 f(b) = 1 f(b) = 0 q c;0 q c;1 f*(c) = 0 f*(b) = 0f*(a) = 1 Generalizes to any length chain Three Variables l0l0 l1l1

VaVa VbVb VcVc f(a) = 1 f(b) = 1 f(b) = 0 q c;0 q c;1 f*(c) = 0 f*(b) = 0f*(a) = 1 Dynamic Programming Three Variables l0l0 l1l1

Dynamic Programming 3 variables  2 variables + book-keeping n variables  (n-1) variables + book-keeping Start from left, go to right Reparameterize current edge (a,b) M ab;k = min i {  a;i +  ab;ik }  ’ ab;ik =  ab;ik + M ab;k - M ab;k  ’ b;k =  b;k Repeat

Dynamic Programming Start from left, go to right Reparameterize current edge (a,b) M ab;k = min i {  a;i +  ab;ik } Repeat MessagesMessage Passing Why stop at dynamic programming?  ’ ab;ik =  ab;ik + M ab;k - M ab;k  ’ b;k =  b;k

VaVa VbVb VcVc Reparameterize the edge (c,b) as before Three Variables l0l0 l1l1

VaVa VbVb VcVc Reparameterize the edge (c,b) as before  ’ b;i = q b;i Three Variables l0l0 l1l1

VaVa VbVb VcVc Reparameterize the edge (b,a) as before Three Variables l0l0 l1l1

VaVa VbVb VcVc Reparameterize the edge (b,a) as before  ’ a;i = q a;i Three Variables l0l0 l1l1

VaVa VbVb VcVc Forward Pass   Backward Pass All min-marginals are computed Three Variables l0l0 l1l1

Belief Propagation on Chains Start from left, go to right Reparameterize current edge (a,b) M ab;k = min i {  a;i +  ab;ik }  ’ ab;ik =  ab;ik + M ab;k - M ab;k  ’ b;k =  b;k Repeat till the end of the chain Start from right, go to left Repeat till the end of the chain

Belief Propagation on Chains A way of computing reparam constants Generalizes to chains of any length Forward Pass - Start to End MAP estimate Min-marginals of final variable Backward Pass - End to start All other min-marginals

Computational Complexity Each constant takes O(|L|) Number of constants - O(|E||L|) O(|E||L| 2 ) Memory required ? O(|E||L|)

Belief Propagation on Trees VbVb VaVa Forward Pass: Leaf  Root All min-marginals are computed Backward Pass: Root  Leaf VcVc VdVd VeVe VgVg VhVh

Outline Problem Formulation Reparameterization Energy Minimization for Trees Loopy Belief Propagation Pearl, 1988; Murphy et al., 1999

Belief Propagation on Cycles VaVa VbVb VdVd VcVc Where do we start? Arbitrarily  a;0  a;1  b;0  b;1  d;0  d;1  c;0  c;1 Reparameterize (a,b)

Belief Propagation on Cycles VaVa VbVb VdVd VcVc  a;0  a;1  ’ b;0  ’ b;1  d;0  d;1  c;0  c;1 Potentials along the red path add up to 0

Belief Propagation on Cycles VaVa VbVb VdVd VcVc  a;0  a;1  ’ b;0  ’ b;1  d;0  d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0

Belief Propagation on Cycles VaVa VbVb VdVd VcVc  a;0  a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0

Belief Propagation on Cycles VaVa VbVb VdVd VcVc  ’ a;0  ’ a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0

Belief Propagation on Cycles VaVa VbVb VdVd VcVc  ’ a;0  ’ a;1  ’ b;0  ’ b;1  ’ d;0  ’ d;1  ’ c;0  ’ c;1 Potentials along the red path add up to 0 -  a;0 -  a;1

Belief Propagation on Cycles VaVa VbVb VdVd VcVc  a;0  a;1  b;0  b;1  d;0  d;1  c;0  c;1 Any suggestions?Fix V a to label l 0

Belief Propagation on Cycles VaVa VbVb VdVd VcVc Any suggestions?Fix V a to label l 0  a;0  b;0  b;1  d;0  d;1  c;0  c;1 Equivalent to a tree-structured problem

Belief Propagation on Cycles VaVa VbVb VdVd VcVc  a;1  b;0  b;1  d;0  d;1  c;0  c;1 Any suggestions?Fix V a to label l 1 Equivalent to a tree-structured problem

Belief Propagation on Cycles Choose the minimum energy solution VaVa VbVb VdVd VcVc  a;0  a;1  b;0  b;1  d;0  d;1  c;0  c;1 This approach quickly becomes infeasible

Loopy Belief Propagation V1V1 V2V2 V3V3 V4V4 V5V5 V6V6 V7V7 V8V8 V9V9 Keep reparameterizing edges in some order Hope for convergence and a good solution

Belief Propagation Generalizes to any arbitrary random field Complexity per iteration ? O(|E||L| 2 ) Memory required ? O(|E||L|)

Graph Cuts as Reparameterization?