INFERENCE SCHEMES FOR M BEST SOLUTIONS FOR SOFT CSPS Emma Rolln, - PowerPoint PPT Presentation

INFERENCE SCHEMES FOR M BEST SOLUTIONS FOR SOFT CSPS Emma Rollón, Natalia Flerova and Rina Dechter erollon@lsi.upc.edu, flerova@ics.uci.edu, dechter@ics.uci.edu Universitat Politècnica de Catalunya University of California, Irvine

Summary  Optimization problems:  Finding the best solution  Finding the m-best solutions  Applications of the m-best solutions:  Set of diverse solutions desired (e.g., haplotaping)  Constraints are hard to formalized (e.g., portfolio mgmt)  Sensitivity analysis (e.g., biological sequence alignment)

Summary  Previous works on the m-best tasks:  Compute the m-best solutions by successively computing the best solution, each time using a slightly different reformulation of the original problem.  Lawler, 1972; Nilsson, 1998; Yanover and Weiss, 2004;  Compute the m best solutions in a single pass of algorithm, using message passing/propagation  Seroussi and Golmard, 1994 ; Elliot, 2007;

Summary  Our contribution:  We provide a formalization of the m-best task within the unifying framework of c-semiring, making many known inference schemes immediately applicable.  In particular, we focus on Graphical Models and extend:  Bucket Elimination (exact algorithm)  Mini-Bucket Elimination (bounding algorithm)  We show how to tighten the bound on the best solution using a bound on the m-best solutions.

1-best vs. m-best optimization 1-best optimization m-best optimization Variables : X = {X 1 , X 2 , X 3 ,…, X n } Finite domain values: D = {D 1 , D 2 ,…, D n } n   Objective function: : F D A i  1 i A is a totally ordered set (<) of valuations   F ( t ) such that F ( 1 t ),..., F ( t ) such that m           t ' t : F ( t ) F ( t ' ) t ' t , t 1 i j m F ( t ) F ( t ) F ( t ' ) i j i j ฀ 

Graphical Model  X = {x 1 , …, x n }: a set of variables  D = {D 1 , …, D n }: a set of domain values  {f 1 , …, f e }: a set of local functions f j : D Y  A Y  X scope of f j  : combination operator over functions   Interaction graph: ฀  x x x x 2 3 n 1

Graphical Model e  f k F( X )   Global view (objective function): k  1  Reasoning task: X  F ( ) X  Particular instantiations: ฀    Task A    N U  min WCSP    max 0 ... 1 MPE ฀  ฀  ฀  ฀  ฀    ฀    P(e) 0 ... 1 ฀  ฀  ฀  ฀ 

Bucket Elimination   Select a var Combination Marginalization Output x 1 x 1   e  f k 5 4  ()   X   x 2 x 2 x 2 x 5 x 5 x 5 BE   k  1 ฀  x 4 x 4 x 3 ฀  x 4 x 3 x 3 ฀  Complete and correct : whenever the task can be defined over a semiring [Shafer et. Al, Srivanas et al, Kohlas et al.]

Bucket Elimination   Select a var Combination Marginalization Output x 1 x 1   e  f k 5 4  ()   X   x 2 x 2 x 2 x 5 x 5 x 5 BE   k  1 ฀  x 4 x 4 x 3 ฀  x 4 x 3 x 3     ฀  elim-opt min  () = c 1 = WCSP

Bucket Elimination   Select a var Combination Marginalization Output x 1 x 1   e  f k 5 4  ()   X   x 2 x 2 x 2 x 5 x 5 x 5 BE   k  1 ฀  x 4 x 4 x 3 ฀  x 4 x 3 x 3     ฀  elim-opt min  () = c 1 = WCSP   ??  ?? elim-m-opt  () = {c 1 ,...,c m } = m-best WCSP ฀  ฀ 

Bucket Elimination   Select a var Combination Marginalization Output x 1 x 1   e  f k 5 4  ()   X   x 2 x 2 x 2 x 5 x 5 x 5 BE   k  1 ฀  x 4 x 4 x 3 ฀  x 4 x 3 x 3     ฀  elim-opt min  () = c 1 Each tuple is the best = WCSP cost extension to x1   ??  ?? elim-m-opt  () = {c 1 ,...,c m } = m-best WCSP Each tuple has to be the m-best cost extensions to x1 ฀  ฀ 

From 1-best to m-best optimization m-best WCSP (m = 2) WCSP Functions   m f : l ( Y ) [ 0 ... 1 ] : ( ) [ 0 ... 1 ] f l Y   f ( t )  c 1 f ( t )  c 1 , c 2            ( ) 6 ; ( ) 5 f t g t f ( t ) 3 , 5 ; g ( t ) 1 , 6 ฀  ฀   ฀  min min             f ( x a ) 6 ; f ( x b ) 5 f ( x a ) 3 , 5 ; f ( x b ) 1 , 6 ฀ 

From 1-best to m-best optimization m-best WCSP (m = 2) WCSP Functions   m f : l ( Y ) [ 0 ... 1 ] : ( ) [ 0 ... 1 ] f l Y   f ( t )  c 1 f ( t )  c 1 , c 2            f ( t ) 6 ; g ( t ) 5 f ( t ) 3 , 5 ; g ( t ) 1 , 6 ฀  ฀           f ( t ) g ( t ) 6 5 11 f ( t ) g ( t ) 4 , 9 , 6 , 11 { 4 , 6 }  ฀  min min             f ( x a ) 3 , 5 ; f ( x b ) 1 , 6 f ( x a ) 6 ; f ( x b ) 5         min f min 5 , 6 5 min f 3 , 5 , 1 , 6 { 1 , 3 } x x ฀ 

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  2

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 1  4

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 3  1  2 4

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 3  1  2 nd best 2 4

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 3  1  2 nd best 2 4 1  5

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 3  1  2 nd best 2 4 1  3  5 4

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 3  1  2 nd best 3 rd best 2 4 1  3  5 4

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 3  1  2 nd best 3 rd best 2 4 1  3  5 4

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 3  1  2 nd best 3 rd best 2 4 1  3  6  5 4 2

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 3  1  2 nd best 3 rd best 2 4 1  3  6  4 th best 5 4 2

How to combine two ordered sets   { 1 , 3 , 6 } { 2 , 4 , 5 } S T 1  1 st best 2 3  1  2 nd best 3 rd best 2 4 1  3  6  4 th best 5 4 2 O(m 2 ) O(m * log(m+1))

Bucket Elimination   Select a var Combination Marginalization Output x 1 x 1   e  f k 5 4  ()   X   x 2 x 2 x 2 x 5 x 5 x 5 BE   k  1 ฀  x 4 x 4 x 3 ฀  x 4 x 3 x 3     ฀  elim-opt min  () = c 1 = WCSP     elim-m-opt min  () = {c 1 ,..,c m } = m-best WCSP

Bucket Elimination   Select a var Combination Marginalization Output x 1 x 1   e  f k 5 4  ()   X   x 2 x 2 x 2 x 5 x 5 x 5 BE   k  1 ฀  x 4 x 4 x 3 ฀  x 4 x 3 x 3     ฀  elim-opt min  () = c 1 = WCSP     elim-m-opt min  () = {c 1 ,..,c m } = m-best WCSP Correct and complete : the m-best problem can be formulated as a commutative semiring using the new operators

Semirings  A commutative semiring is a triplet (A, ⊗ , ⊕ ), where operators satisfy three axioms:  A1. The operation ⊕ is associative, commutative and idempotent, and there  is an additive identity element called 0 such that a ⊕ 0 = a for all a ∈ A.  A2. The operation ⊗ is also associative and commutative, and there is a  multiplicative identity element called 1 such that a ⊗ 1 = a for all a ∈ A  A3. ⊗ distributes over ⊕ , i.e., (a ⊗ b) ⊕ (a ⊗ c) = a ⊗ (b ⊕ c)  Example: MPE task is defined over semiring K = (R,×,max), a CSP is defined over semiring K = ({0, 1}, ∧ , ∨ ), and a Weighted CSP is defined over semiring K = (N ∪ {∞},+,min).  It was showed that the correctness of inference algorithms over a reasoning task P is ensured whenever P is defined over a semiring. [Shafer et. Al, Srivanas et al, Kohlas et al.]