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Kernel, RKHS, and Gaussian Processes

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Sungjoon Choi
Sungjoon Choi

Basic definitions and theorems for constructing RKHS Random processes to Gaussian processes Leveraged Gaussian processes and Leverage optimization

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Kernel, RKHS, and Gaussian Processes Caution! No proof will be given. Sungjoon Choi, SNU
Leveraged Gaussian  Process  Regression
Leveraged  Gaussian  Processes The  original  Gaussian  process  regression   anchors  positive training  data. The  proposed  leveraged  Gaussian  process  regression  anchors  positive data   while  avoiding  negative data. Moreover,  it  is  possible  to  vary  the  leverage   of  each  training  data  from  -­‐1  to  +1.
Positive and  Negative Motion   Control  Data  for  Autonomous   Navigation
Training  Phase
Execution  Phase
Real  World  Experiments
Leverage  Optimization
Leverage  Optimization • The  key  intuition  behind  the  leverage  optimization  is  that  we   cast  the  leverage  optimization  problem  into  a  model  selection   problem in  Gaussian  process  regression.   • However,  the  number  of  leverage  parameters  is  equivalent  to   the  number  of  training  data.   • To  handle  this  issue,  we  propose  a  sparse  constrained  leverage   optimization where  we  assume  that  the  majority  of  leverage   parameters  are  +1.  
Leverage  Optimization  (v1) Using  proximal  linearized  minimization  [1],  the  update  rule  for  solving  above   optimization  is   [1]  J.  Bolte,  S.  Sabach,  and  M.  Teboulle,  “Proximal  alternating  linearized  minimization  for  nonconvex  and  nonsmooth  problems,”   Mathematical  Programming,  vol.  146,  no.  1-­2,  pp.  459–494,  2014. where  the  proximal  mapping  becomes  soft-­‐thresholding: where   𝛾̅ = 𝛾 − 1.  
Leverage  Optimization  (v2) We  propose  a  new  leverage  optimization  method  by  doubling the  leverage   parameters  to  positive  and  negative  parts.   Furthermore,  we  assume  multiple  demonstrations  are  collected  from  one   demonstrator.   By  doing  so,  we  have  two  major  benefits: 1. As  the  L1-­‐norm  regularizer is  only  on  the  positive  parts  of  the  leverages,   negative  parts  of  the  leverages  can  be  optimized  more  accurately.   2. By  doubling  the  variables,  proximal  mapping  is  no  longer  needed.  
Sensory  Field  Reconstruction Sensory  observations  from   correlated  sensory  fields Leverage  Optimization   Collect  observations
Sensory  observations  from   correlated  sensory  fields Proposed  Method Previous  Method Without  Optimization Sensory  Field  Reconstruction
Autonomous  Driving  Experiments Driving  demonstrations  with   mixed  qualities Leverage  Optimization   Collect  observations
Proposed  Method Previous  Method Autonomous  Driving  Experiments
Proposed  MethodWithout  Optimization Autonomous  Driving  Experiments
Kernel, RKHS, and Gaussian Processes

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Kernel, RKHS, and Gaussian Processes

  • 1. Kernel, RKHS, and Gaussian Processes Caution! No proof will be given. Sungjoon Choi, SNU
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  • 32. Leveraged  Gaussian  Processes The  original  Gaussian  process  regression   anchors  positive training  data. The  proposed  leveraged  Gaussian  process  regression  anchors  positive data   while  avoiding  negative data. Moreover,  it  is  possible  to  vary  the  leverage   of  each  training  data  from  -­‐1  to  +1.
  • 33. Positive and  Negative Motion   Control  Data  for  Autonomous   Navigation
  • 38. Leverage  Optimization • The  key  intuition  behind  the  leverage  optimization  is  that  we   cast  the  leverage  optimization  problem  into  a  model  selection   problem in  Gaussian  process  regression.   • However,  the  number  of  leverage  parameters  is  equivalent  to   the  number  of  training  data.   • To  handle  this  issue,  we  propose  a  sparse  constrained  leverage   optimization where  we  assume  that  the  majority  of  leverage   parameters  are  +1.  
  • 39. Leverage  Optimization  (v1) Using  proximal  linearized  minimization  [1],  the  update  rule  for  solving  above   optimization  is   [1]  J.  Bolte,  S.  Sabach,  and  M.  Teboulle,  “Proximal  alternating  linearized  minimization  for  nonconvex  and  nonsmooth  problems,”   Mathematical  Programming,  vol.  146,  no.  1-­2,  pp.  459–494,  2014. where  the  proximal  mapping  becomes  soft-­‐thresholding: where   𝛾̅ = 𝛾 − 1.  
  • 40. Leverage  Optimization  (v2) We  propose  a  new  leverage  optimization  method  by  doubling the  leverage   parameters  to  positive  and  negative  parts.   Furthermore,  we  assume  multiple  demonstrations  are  collected  from  one   demonstrator.   By  doing  so,  we  have  two  major  benefits: 1. As  the  L1-­‐norm  regularizer is  only  on  the  positive  parts  of  the  leverages,   negative  parts  of  the  leverages  can  be  optimized  more  accurately.   2. By  doubling  the  variables,  proximal  mapping  is  no  longer  needed.  
  • 41. Sensory  Field  Reconstruction Sensory  observations  from   correlated  sensory  fields Leverage  Optimization   Collect  observations
  • 42. Sensory  observations  from   correlated  sensory  fields Proposed  Method Previous  Method Without  Optimization Sensory  Field  Reconstruction
  • 43. Autonomous  Driving  Experiments Driving  demonstrations  with   mixed  qualities Leverage  Optimization   Collect  observations