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Distance-­‐Based	
  Bias	
  	
  
      in	
  Model-­‐Directed	
  Op3miza3on	
  	
  
of	
  Addi3vely	
  Decomposable	
  Problems	
  

   Mar3n	
  Pelikan	
  	
  and	
  	
  Mark	
  W.	
  Hauschild	
  
                              	
  
 Missouri	
  Es3ma3on	
  of	
  Distribu3on	
  Algorithms	
  Laboratory	
  
   Department	
  of	
  Mathema3cs	
  and	
  Computer	
  Science	
  
              University	
  of	
  Missouri,	
  St.	
  Louis,	
  MO	
  
                                      	
  
               E-­‐mail:	
  mar3n@mar3npelikan.net	
  
                WWW:	
  hKp://mar3npelikan.net/	
                            1
Background	
  
•  Model-­‐directed	
  op3mizers	
  (MDOs)	
  learn	
  and	
  use	
  
   models	
  in	
  op3miza3on	
  to	
  solve	
  difficult	
  
   op3miza3on	
  problems	
  scalably	
  and	
  reliably.	
  
•  MDOs	
  oPen	
  provide	
  more	
  than	
  the	
  solu3on;	
  
   they	
  provide	
  a	
  set	
  of	
  models	
  that	
  reveal	
  
   informa3on	
  about	
  the	
  problem.	
  
•  Learning	
  from	
  experience:	
  Use	
  models	
  from	
  prior	
  
   runs	
  of	
  MDOs	
  to	
  introduce	
  bias	
  when	
  solving	
  
   problems	
  of	
  similar	
  type	
  in	
  future.	
  
                                                                       2
Purpose	
  
•  Combine	
  prior	
  models	
  with	
  a	
  problem-­‐specific	
  
   distance	
  metric	
  to	
  solve	
  new	
  problem	
  instances	
  
   with	
  increased	
  speed,	
  accuracy,	
  reliability.	
  
•  Demonstrate	
  significant	
  speedups	
  across	
  a	
  broad	
  
   array	
  of	
  problem	
  domains.	
  
•  Focus	
  on	
  hBOA	
  algorithm	
  and	
  addi3vely	
  
   decomposable	
  func3ons,	
  although	
  the	
  approach	
  
   can	
  be	
  generalized	
  to	
  other	
  MDOs	
  and	
  other	
  
   problem	
  classes.	
  
                                                                      3
Outline	
  
1.  Hierarchical	
  BOA	
  (hBOA).	
  
2.  Distance	
  metric	
  for	
  ADFs.	
  
3.  Learning	
  from	
  experience	
  via	
  distance-­‐based	
  
    bias.	
  
4.  Experiments.	
  
5.  Summary	
  and	
  conclusions.	
  



                                                                    4
Hierarchical	
  Bayesian	
  Op3miza3on	
  
           Algorithm	
  (hBOA)	
  
 Current                                 Bayesian     New
population            Selection          network    population




[Pelikan, Goldberg, & Cantu-Paz, 2001]                           5
Decision	
  Trees	
  Represent	
  Dependencies	
  
                Dependency      X2	
  
                       X1	
     X3	
  
                                X4	
      Decision tree
  Probability table                      (more efficient)




                                                            6
Learning	
  from	
  Experience	
  	
  
              (Transfer	
  Learning)	
  
•  Mo3va3on	
  
   –  When	
  solving	
  a	
  problem,	
  hBOA	
  provides	
  the	
  user	
  
      with	
  a	
  set	
  of	
  probabilis3c	
  models.	
  
   –  Each	
  model	
  encodes	
  informa3on	
  about	
  the	
  problem,	
  
      such	
  as	
  dependencies	
  between	
  variables.	
  
   –  Why	
  not	
  use	
  this	
  informa3on	
  when	
  solving	
  new	
  
      problem	
  instances	
  of	
  similar	
  type?	
  
•  Example:	
  hBOA	
  solves	
  99	
  scheduling	
  problems;	
  
   why	
  not	
  use	
  the	
  knowledge	
  obtained	
  when	
  
   solving	
  the	
  100th	
  instance?	
  
                                                                                7
How	
  to	
  Make	
  it	
  Work?	
  
•  It	
  is	
  straighborward	
  to	
  keep	
  sta3s3cs	
  from	
  past	
  
   hBOA	
  runs,	
  for	
  example,	
  capturing	
  the	
  number	
  of	
  
   dependencies	
  between	
  any	
  pair	
  of	
  variables.	
  
•  In	
  hBOA,	
  this	
  can	
  be	
  done	
  by	
  looking	
  at	
  the	
  
   number	
  of	
  “splits”	
  on	
  variable	
  Xi	
  in	
  a	
  decision	
  tree	
  
   storing	
  dependencies	
  for	
  variable	
  Xj.	
  
•  But	
  it	
  is	
  important	
  to	
  ensure	
  that	
  the	
  sta3s3cs	
  are	
  
   meaningful	
  with	
  respect	
  to	
  the	
  problem	
  being	
  
   solved,	
  so	
  that	
  the	
  sta3s3cs	
  help	
  us	
  solve	
  future	
  
   problem	
  instances	
  faster	
  and	
  beKer.	
  
                                                                                         8
Learning	
  from	
  Experience	
  via	
  
       Distance-­‐Based	
  Bias:	
  Basic	
  Idea	
  
•  Learning	
  from	
  experience	
  using	
  distance-­‐based	
  bias	
  
    –  Define	
  distances	
  between	
  problem	
  variables.	
  
    –  Mine	
  probabilis3c	
  models	
  from	
  previous	
  runs	
  for	
  
       model	
  regulari3es	
  with	
  respect	
  to	
  distances.	
  
•  Mine	
  models	
  to	
  es3mate	
  how	
  strongly	
  variables	
  
   influence	
  each	
  other	
  depending	
  on	
  their	
  distance.	
  
    –  This	
  should	
  work	
  whenever	
  strength	
  of	
  dependencies	
  
       is	
  correlated	
  with	
  distance.	
  
•  Apply	
  idea	
  to	
  hBOA	
  and	
  addi3vely	
  decomposable	
  
   func3ons.	
  
                                                                               9
Addi3vely	
  Decomposable	
  Func3ons	
  
•  Addi3vely	
  decomposable	
  func3on	
  (ADF):	
  

	
  
	
  

       –  {Si}	
  are	
  subsets	
  of	
  variables.	
  
       –  {fi}	
  are	
  func3ons	
  defining	
  overall	
  solu3on	
  quality.	
  
•  Addi3vely	
  decomposable	
  func3ons	
  are	
  oPen	
  
   difficult	
  to	
  solve!	
  Many	
  NP-­‐complete	
  problems	
  
   are	
  ADFs	
  with	
  subproblems	
  of	
  2	
  or	
  3	
  variables.	
  
                                                                                     10
Define	
  Distance	
  Metric	
  for	
  ADFs	
  
         Using	
  Dependency	
  Graph	
  
•  Create	
  a	
  dependency	
  graph	
  where	
  variables	
  in	
  
   the	
  same	
  subset	
  Si	
  are	
  connected.	
  
•  Define	
  distance	
  between	
  variables	
  as	
  shortest	
  
   path	
  between	
  them	
  in	
  the	
  dependency	
  graph.	
  
•  If	
  there	
  exists	
  no	
  such	
  path,	
  set	
  distance	
  to	
  the	
  
   number	
  of	
  variables	
  (any	
  exis3ng	
  path	
  is	
  
   shorter).	
  


[Hauschild et al., 2008]                                                              11
Define	
  Distance	
  Metric	
  for	
  ADFs	
  
   Using	
  Dependency	
  Graph:	
  Example	
  




[Hauschild et al., 2008]                             12
Mo3va3ng	
  Example	
  
•  Propor3ons	
  of	
  splits	
  for	
  variables	
  at	
  various	
  distances	
  
   shows	
  evident	
  correla3on	
  between	
  the	
  two:	
  

               NK landscapes                                2D spin glass




                                                                                  13
Details	
  of	
  the	
  Approach	
  
•  Denote	
  by	
  M	
  the	
  set	
  of	
  models	
  from	
  prior	
  runs.	
  
•  Record	
  the	
  number	
  of	
  splits	
  on	
  any	
  variable	
  Xi	
  in	
  
   any	
  decision	
  tree	
  Xj	
  in	
  model	
  m	
  such	
  that	
  distance	
  of	
  
   Xi	
  and	
  Xj	
  is	
  d	
  	
  	
  
   	
  
•  Compute	
  probability	
  of	
  kth	
  split	
  on	
  variable	
  Xi	
  in	
  any	
  
   decision	
  tree	
  Xj	
  such	
  that	
  dist.	
  of	
  Xi	
  and	
  Xj	
  is	
  d	
  
   assuming	
  (k-­‐1)	
  such	
  splits:	
  


                                                                                        14
Details	
  of	
  the	
  Approach	
  
•  Set	
  prior	
  probability	
  of	
  network	
  structure	
  based	
  
   on	
  the	
  learned	
  probabili3es	
  (kappa	
  denotes	
  
   strength	
  of	
  bias)	
  




•  Evaluate	
  each	
  network	
  using	
  a	
  Bayesian	
  metric	
  


                                                                            15
Test	
  Problems	
  
•  Included	
  in	
  this	
  paper	
  
    –  NK	
  landscapes	
  with	
  nearest-­‐neighbor	
  interac3ons.	
  
    –  2D	
  spin	
  glass.	
  
•  Done	
  later	
  on	
  
    –  3D	
  spin	
  glass.	
  
    –  Minimum	
  vertex	
  cover	
  for	
  random	
  graphs.	
  
    –  MAXSAT	
  for	
  3-­‐CNF	
  formulas.	
  
•  Large	
  number	
  of	
  different	
  instances	
  for	
  each	
  
   problem	
  class	
  (100s	
  to	
  1000s	
  each).	
  
                                                                            16
Experimental	
  Methodology	
  
•  10-­‐fold	
  crossvalida3on	
  
    –  Divide	
  instances	
  into	
  10	
  sets.	
  
    –  Test	
  bias	
  from	
  models	
  on	
  9	
  sets	
  on	
  remaining	
  1	
  set,	
  
       repeat	
  for	
  every	
  set.	
  
    –  BoKom	
  line:	
  Any	
  problem	
  instance	
  is	
  never	
  used	
  for	
  
       both	
  crea3ng	
  the	
  bias	
  and	
  tes3ng	
  it.	
  
•  Bisec3on	
  for	
  gemng	
  popula3on	
  sizes,	
  10	
  runs	
  for	
  
   each	
  problem	
  instance.	
  
•  Focus	
  on	
  mul3plica3ve	
  speedups	
  
    –  How	
  many	
  3mes	
  faster	
  with	
  the	
  use	
  of	
  bias?	
                    17
Results	
  on	
  NK	
  Landscapes	
  




                                        18
Results	
  on	
  Minimum	
  Vertex	
  Cover	
  




                                                  19
Results	
  on	
  2D	
  Spin	
  Glass	
  




                                           20
Results	
  on	
  3D	
  Spin	
  Glass	
  




                                           21
Results	
  on	
  MAXSAT	
  




                              22
More	
  Results	
  to	
  be	
  Published	
  Soon	
  
•  Nearly	
  iden3cal	
  speedups	
  if	
  bias	
  is	
  based	
  on	
  
   problems	
  of	
  smaller	
  size.	
  
•  Significant	
  speedups	
  even	
  if	
  bias	
  is	
  based	
  on	
  
   another	
  class	
  of	
  ADFs	
  (e.g.	
  models	
  from	
  NK	
  
   landscapes	
  used	
  to	
  solve	
  MVC).	
  	
  
•  Nearly	
  mul3plica3ve	
  speedups	
  in	
  combina3on	
  
   with	
  other	
  efficiency	
  enhancements	
  (e.g.	
  sporadic	
  
   model	
  building).	
  
•  So	
  far	
  not	
  a	
  single	
  problem	
  class	
  for	
  which	
  the	
  bias	
  
   does	
  not	
  yield	
  significant	
  speedups.	
  	
                               23
Results	
  Applicable	
  in	
  Other	
  Contexts	
  
•  Approach	
  can	
  be	
  applied	
  to	
  other	
  model-­‐
   directed	
  op3mizers,	
  such	
  as	
  ECGA,	
  LTGA,	
  or	
  
   mGA.	
  
•  Approach	
  can	
  be	
  applied	
  to	
  other	
  problem	
  
   classes	
  for	
  which	
  a	
  distance	
  metric	
  can	
  be	
  
   defined,	
  such	
  as	
  QAP	
  or	
  scheduling	
  problems.	
  
•  This	
  work	
  demonstrates	
  the	
  poten3al,	
  but	
  more	
  
   work	
  to	
  be	
  done	
  in	
  future.	
  

                                                                         24
Summary	
  and	
  Conclusions	
  
•  Proposed	
  a	
  prac3cal	
  approach	
  to	
  using	
  models	
  
   from	
  prior	
  runs	
  of	
  model-­‐directed	
  op3mizers	
  to	
  
   bias	
  op3miza3on	
  of	
  future	
  problem	
  instances.	
  
•  Demonstrated	
  significant	
  speedups	
  across	
  a	
  
   number	
  of	
  problem	
  domains	
  and	
  semngs,	
  
   including	
  a	
  number	
  scenarios	
  that	
  are	
  not	
  possible	
  
   with	
  related	
  techniques	
  proposed	
  in	
  the	
  past.	
  
•  Approach	
  is	
  ready	
  to	
  be	
  applied	
  in	
  a	
  different	
  
   context.	
  
                                                                             25
Acknowledgments	
  
•  Support	
  was	
  provided	
  by	
  
   –  NSF	
  grants	
  ECS-­‐0547013	
  and	
  IIS-­‐1115352.	
  
   –  ITS	
  at	
  the	
  University	
  of	
  Missouri	
  in	
  St.	
  Louis.	
  
   –  University	
  of	
  Missouri	
  Bioinforma3cs	
  Consor3um.	
  


•  Get	
  the	
  papers	
  at	
  
   hKp://medal-­‐lab.org/files/2012001.pdf	
  
   hKp://medal-­‐lab.org/files/2012004.pdf	
  

                                                                                    26

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Distance-based bias in model-directed optimization of additively decomposable problems

  • 1. Distance-­‐Based  Bias     in  Model-­‐Directed  Op3miza3on     of  Addi3vely  Decomposable  Problems   Mar3n  Pelikan    and    Mark  W.  Hauschild     Missouri  Es3ma3on  of  Distribu3on  Algorithms  Laboratory   Department  of  Mathema3cs  and  Computer  Science   University  of  Missouri,  St.  Louis,  MO     E-­‐mail:  mar3n@mar3npelikan.net   WWW:  hKp://mar3npelikan.net/   1
  • 2. Background   •  Model-­‐directed  op3mizers  (MDOs)  learn  and  use   models  in  op3miza3on  to  solve  difficult   op3miza3on  problems  scalably  and  reliably.   •  MDOs  oPen  provide  more  than  the  solu3on;   they  provide  a  set  of  models  that  reveal   informa3on  about  the  problem.   •  Learning  from  experience:  Use  models  from  prior   runs  of  MDOs  to  introduce  bias  when  solving   problems  of  similar  type  in  future.   2
  • 3. Purpose   •  Combine  prior  models  with  a  problem-­‐specific   distance  metric  to  solve  new  problem  instances   with  increased  speed,  accuracy,  reliability.   •  Demonstrate  significant  speedups  across  a  broad   array  of  problem  domains.   •  Focus  on  hBOA  algorithm  and  addi3vely   decomposable  func3ons,  although  the  approach   can  be  generalized  to  other  MDOs  and  other   problem  classes.   3
  • 4. Outline   1.  Hierarchical  BOA  (hBOA).   2.  Distance  metric  for  ADFs.   3.  Learning  from  experience  via  distance-­‐based   bias.   4.  Experiments.   5.  Summary  and  conclusions.   4
  • 5. Hierarchical  Bayesian  Op3miza3on   Algorithm  (hBOA)   Current Bayesian New population Selection network population [Pelikan, Goldberg, & Cantu-Paz, 2001] 5
  • 6. Decision  Trees  Represent  Dependencies   Dependency X2   X1   X3   X4   Decision tree Probability table (more efficient) 6
  • 7. Learning  from  Experience     (Transfer  Learning)   •  Mo3va3on   –  When  solving  a  problem,  hBOA  provides  the  user   with  a  set  of  probabilis3c  models.   –  Each  model  encodes  informa3on  about  the  problem,   such  as  dependencies  between  variables.   –  Why  not  use  this  informa3on  when  solving  new   problem  instances  of  similar  type?   •  Example:  hBOA  solves  99  scheduling  problems;   why  not  use  the  knowledge  obtained  when   solving  the  100th  instance?   7
  • 8. How  to  Make  it  Work?   •  It  is  straighborward  to  keep  sta3s3cs  from  past   hBOA  runs,  for  example,  capturing  the  number  of   dependencies  between  any  pair  of  variables.   •  In  hBOA,  this  can  be  done  by  looking  at  the   number  of  “splits”  on  variable  Xi  in  a  decision  tree   storing  dependencies  for  variable  Xj.   •  But  it  is  important  to  ensure  that  the  sta3s3cs  are   meaningful  with  respect  to  the  problem  being   solved,  so  that  the  sta3s3cs  help  us  solve  future   problem  instances  faster  and  beKer.   8
  • 9. Learning  from  Experience  via   Distance-­‐Based  Bias:  Basic  Idea   •  Learning  from  experience  using  distance-­‐based  bias   –  Define  distances  between  problem  variables.   –  Mine  probabilis3c  models  from  previous  runs  for   model  regulari3es  with  respect  to  distances.   •  Mine  models  to  es3mate  how  strongly  variables   influence  each  other  depending  on  their  distance.   –  This  should  work  whenever  strength  of  dependencies   is  correlated  with  distance.   •  Apply  idea  to  hBOA  and  addi3vely  decomposable   func3ons.   9
  • 10. Addi3vely  Decomposable  Func3ons   •  Addi3vely  decomposable  func3on  (ADF):       –  {Si}  are  subsets  of  variables.   –  {fi}  are  func3ons  defining  overall  solu3on  quality.   •  Addi3vely  decomposable  func3ons  are  oPen   difficult  to  solve!  Many  NP-­‐complete  problems   are  ADFs  with  subproblems  of  2  or  3  variables.   10
  • 11. Define  Distance  Metric  for  ADFs   Using  Dependency  Graph   •  Create  a  dependency  graph  where  variables  in   the  same  subset  Si  are  connected.   •  Define  distance  between  variables  as  shortest   path  between  them  in  the  dependency  graph.   •  If  there  exists  no  such  path,  set  distance  to  the   number  of  variables  (any  exis3ng  path  is   shorter).   [Hauschild et al., 2008] 11
  • 12. Define  Distance  Metric  for  ADFs   Using  Dependency  Graph:  Example   [Hauschild et al., 2008] 12
  • 13. Mo3va3ng  Example   •  Propor3ons  of  splits  for  variables  at  various  distances   shows  evident  correla3on  between  the  two:   NK landscapes 2D spin glass 13
  • 14. Details  of  the  Approach   •  Denote  by  M  the  set  of  models  from  prior  runs.   •  Record  the  number  of  splits  on  any  variable  Xi  in   any  decision  tree  Xj  in  model  m  such  that  distance  of   Xi  and  Xj  is  d         •  Compute  probability  of  kth  split  on  variable  Xi  in  any   decision  tree  Xj  such  that  dist.  of  Xi  and  Xj  is  d   assuming  (k-­‐1)  such  splits:   14
  • 15. Details  of  the  Approach   •  Set  prior  probability  of  network  structure  based   on  the  learned  probabili3es  (kappa  denotes   strength  of  bias)   •  Evaluate  each  network  using  a  Bayesian  metric   15
  • 16. Test  Problems   •  Included  in  this  paper   –  NK  landscapes  with  nearest-­‐neighbor  interac3ons.   –  2D  spin  glass.   •  Done  later  on   –  3D  spin  glass.   –  Minimum  vertex  cover  for  random  graphs.   –  MAXSAT  for  3-­‐CNF  formulas.   •  Large  number  of  different  instances  for  each   problem  class  (100s  to  1000s  each).   16
  • 17. Experimental  Methodology   •  10-­‐fold  crossvalida3on   –  Divide  instances  into  10  sets.   –  Test  bias  from  models  on  9  sets  on  remaining  1  set,   repeat  for  every  set.   –  BoKom  line:  Any  problem  instance  is  never  used  for   both  crea3ng  the  bias  and  tes3ng  it.   •  Bisec3on  for  gemng  popula3on  sizes,  10  runs  for   each  problem  instance.   •  Focus  on  mul3plica3ve  speedups   –  How  many  3mes  faster  with  the  use  of  bias?   17
  • 18. Results  on  NK  Landscapes   18
  • 19. Results  on  Minimum  Vertex  Cover   19
  • 20. Results  on  2D  Spin  Glass   20
  • 21. Results  on  3D  Spin  Glass   21
  • 23. More  Results  to  be  Published  Soon   •  Nearly  iden3cal  speedups  if  bias  is  based  on   problems  of  smaller  size.   •  Significant  speedups  even  if  bias  is  based  on   another  class  of  ADFs  (e.g.  models  from  NK   landscapes  used  to  solve  MVC).     •  Nearly  mul3plica3ve  speedups  in  combina3on   with  other  efficiency  enhancements  (e.g.  sporadic   model  building).   •  So  far  not  a  single  problem  class  for  which  the  bias   does  not  yield  significant  speedups.     23
  • 24. Results  Applicable  in  Other  Contexts   •  Approach  can  be  applied  to  other  model-­‐ directed  op3mizers,  such  as  ECGA,  LTGA,  or   mGA.   •  Approach  can  be  applied  to  other  problem   classes  for  which  a  distance  metric  can  be   defined,  such  as  QAP  or  scheduling  problems.   •  This  work  demonstrates  the  poten3al,  but  more   work  to  be  done  in  future.   24
  • 25. Summary  and  Conclusions   •  Proposed  a  prac3cal  approach  to  using  models   from  prior  runs  of  model-­‐directed  op3mizers  to   bias  op3miza3on  of  future  problem  instances.   •  Demonstrated  significant  speedups  across  a   number  of  problem  domains  and  semngs,   including  a  number  scenarios  that  are  not  possible   with  related  techniques  proposed  in  the  past.   •  Approach  is  ready  to  be  applied  in  a  different   context.   25
  • 26. Acknowledgments   •  Support  was  provided  by   –  NSF  grants  ECS-­‐0547013  and  IIS-­‐1115352.   –  ITS  at  the  University  of  Missouri  in  St.  Louis.   –  University  of  Missouri  Bioinforma3cs  Consor3um.   •  Get  the  papers  at   hKp://medal-­‐lab.org/files/2012001.pdf   hKp://medal-­‐lab.org/files/2012004.pdf   26