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Exact Rule Learning Via Boolean Compressed Sensing

Anil Shanbhag
September 09, 2013

Exact Rule Learning Via Boolean Compressed Sensing

Present a novel approach to exact rule learning driven by an objective using techniques from compressed sensing and group testing

Anil Shanbhag

September 09, 2013

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  1. MoDvaDon   •  OrganizaDons  are  turning  to  predicDve   analysis

     to  support  decision  making   •  Output  given  to  people  with  limited  analyDcs,   data  and  modeling  literacy   •  Hence,  it  is  imperaDve  to  have  interpretable   machine  learning  methods  so  that  predicDons   are  beLer  adopted/trusted  by  decision   makers.    
  2. Brief  History   •  Old  methods  rely  on  heurisDcs,  bad

     in  worst  case   in  comparison  to  SVM   •  Renewed  interest  in  field  aRer  Ruckert  &   Kramer(2008)  –  first  to  introduce  rule-­‐learning   based  on  objecDve   •  Works  of  Gilbert(2008)  highlight  parallels   between  group  tesDng  and  sparse  signal  recovery  
  3. ContribuDons   The  paper  develops  a  new  approach  to  interpretable

      supervised  classificaDon  through  rule  learning  based  on   Boolean  compressed  sensing.       The  major  contribuDons  of  the  papers  are:  -­‐     •  Show  that  problem  of  learning  sparse  conjuncDve/   disjuncDve  clause  rules  from  training  samples  can  be   formulated  as  a  group  tesDng  problem   •  Reduce  the  NP  hard  problem  using  relaxaDon  to  resemble   the  basic  pursuit  algorithm  for  sparse  signal  recovery     •  Establish  condiDons  under  which  the  relaxaDon  recovers   exactly.    
  4. Parameters     •  y  I  is  the  value  of

     pool  i   •  Aij  is  0  or  1  based  on  whether  subject  j  is  part   of  ith  group  or  not       •  To  Find:  -­‐  wi  is  the  true  value  of  subject  i  which   should  be  recovered  
  5. Clause  Learning   •  Given  a  corpus  of  data  of

     training  samples       {Xi,  yi}  where  Xi  belongs  to  X  are  the  features   and  yi  =  0  or  1   •  We  would  like  to  learn  a  funcDon  to                           map  Xi  to  yi   •  Every  classifier  is  made  of  set  of  clauses,  each   clause  contains  a  set  of  boolean  terms  
  6. Clause  Learning  as  Boolean  TesDng   •  Consider    

    – Every  boolean  term  as  a  subject   – Every  yi  as  the  result  of  the  pool   – The  matrix  A  is  the  boolean  result  of  funcDon   applied  to  Xi  ie:  Aij  =  aj(xi)  
  7. AND  vs  OR   •  The  given  equaDons  learn  disjuncDve

     OR   clause.  To  learn  the  AND  clauses  (which  are   preferred)  we  just  need  to  make  a  small   modificaDon  :-­‐  
  8. Boolean  Compressed  Sensing   •  Compressed  sensing  is  a  signal

     processing   technique  for  effecDvely  measuring  and   reconstrucDng  signals   •  There  are  clear  similariDes,  however  we  restrict   boolean  algebra  instead  of  real  algebra     •  We  apply  similar  techniques  for  formulaDon  of   problem  and  use  LP  relaxaDon  from  compressed   sensing  
  9. Slack  to  accommodate  errors   •  There  may  be  no

     sparse  rules  to  approximate   labels  of  y  exactly  but  may  be  for   approximaDng  y  closely  
  10. Exact  Recovery   •  K-­‐Disjunct  :  A  measurement  matrix  A

     is  K-­‐ disjunct  if  the  union  of  any  K  columns  does   not  contain  any  other  column  of  A     •  If  there  exists  a  w*  with  K  non-­‐zero  entries   and  matrix  K-­‐disjunct,  then  the  LP  recovers  it   exactly    
  11. •  A  is  (e,  K)  disjunct  if  out  of  (n

     C  k)  K  subsets,   (1-­‐e)  fracDon  of  them  saDsfy  the  property  that   union  does  not  contain  any  other  column   •  If  matrix  A  is  (e,K)  disjunct,  then  LP  recover   the  correct  soluDon  with  probablity  1-­‐e  
  12. ConDnuous  Features   •  For  conDnuous  features  we  choose  thresholds

      suitably  spread  across  the  domain   •  T1  <=  T2  <=  T3  <=  T4  …  <=  Tn   •  Each  threshold  value  leads  to  two  indicator   funcDons  :  I(  xj  >=  T1  )  &  I(  xj  <  T1  )  
  13. Learning  Rule  Sets   •  We  use  the  Set  Covering

     Approach   •  This  is  a  common  technique   •  First  step  is  to  learn  an  AND  rule  (here  using  ideas   from  boolean  sensing)   •  Once  we  know  one  AND  rule,  remove  all  training   samples  which  are  idenDfied  by  the  rule   •  Now  repeat  the  learning  on  remaining  rules   •  This  leads  to  DNF  
  14. EvaluaDng  the  approach   •  I  have  tried  to  evaluate

     the  approach  used  in   the  paper  on  IRIS  dataset.   •  IRIS  dataset  is  a  set  of  150  tuples.  There  are   four  features  :  sepal  width,  sepal  length,  petal   length,  petal  width.  Each  tuple  is  indicaDve  of   an  iris  flower.  There  are  three  types  of   flowers  :-­‐  setosa,  versicolor,  virginica  
  15. Three  rules  learnt       •  Petal  length  <=

      5.4   •  Petal  Width  <=   1.7   •  Petal  Width  >  0.9  
  16. References     •  Malioutov,  Dmitry  M.,  and  Kush  R.

     Varshney,   Exact  Rule  Learning  via  Boolean  Compressed   Sensing.   •  Malioutov,  D.  and  Malyutov,  M.  Boolean   compressed  sensing:  LP  relaxaDon  for  group   tesDng.  In  Proc.  IEEE  Int.  Conf.  Acoust.  Speech   Signal  Process.,  pp.  3305–3308,  Kyoto,  Japan,   Mar.  2012.