Generalization and Scaling in Reinforcement Learning
Authors: David H. Ackley, Michael L. Littman
Published in: Advances in Neural Information Processing Systems 2  (NIPS 1989) (Main Conference Track), 1989

Abstract: In  associative reinforcement learning,  an environment generates input  vectors, a  learning system generates possible output vectors, and a  re(cid:173) inforcement function computes feedback signals from the input-output  pairs.  The  task is  to discover  and  remember  input-output  pairs  that  generate  rewards.  Especially  difficult  cases  occur  when  rewards  are  rare, since the expected time for any algorithm can grow exponentially  with the size  of the problem.  Nonetheless, if a  reinforcement function  possesses regularities, and a learning algorithm exploits them, learning  time  can be reduced  below that  of non-generalizing  algorithms.  This  paper  describes  a  neural  network algorithm called  complementary  re(cid:173) inforcement  back-propagation (CRBP),  and  reports simulation results  on problems designed to offer differing opportunities for generalization. 
1  REINFORCEMENT LEARNING REQUIRES  SEARCH  Reinforcement learning (Sutton, 1984; Barto &  Anandan, 1985; Ackley,  1988; Allen,  1989)  requires more from  a learner than does the more familiar  supervised learning  paradigm.  Supervised learning supplies the correct answers to the learner,  whereas  reinforcement  learning  requires  the  learner  to  discover  the  correct  outputs  before  they  can  be  stored.  The  reinforcement  paradigm  divides  neatly  into  search  and  learning  aspects:  When  rewarded  the  system makes internal adjustments to learn  the discovered input-output pair;  when punished the system makes internal adjust(cid:173) ments to search elsewhere. 
Generalization and Scaling in Reinforcement Learning 
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1.1  MAKING REINFORCEMENT INTO  ERROR 
Following work by Anderson (1986)  and Williams  (1988),  we  extend the backprop(cid:173) agation  algorithm  to associative  reinforcement  learning.  Start with  a  "garden  va(cid:173) riety"  backpropagation  network:  A  vector  i  of n  binary  input  units  propagates  through  zero  or  more  layers  of hidden  units,  ultimately  reaching  a  vector  8  of m  sigmoid  units,  each  taking  continuous  values in  the range  (0,1).  Interpret each  8j  as  the  probability that  an  associated  random  bit  OJ  takes on  value  1.  Let  us  call  the  continuous,  deterministic  vector  8  the  search  vector to distinguish  it from  the  stochastic binary output  vector o. 
Given  an  input  vector,  we  forward  propagate  to  produce  a  search  vector  8,  and  then  perform  m  independent  Bernoulli trials  to  produce  an  output  vector  o.  The  i  - 0  pair  is  evaluated  by  the  reinforcement  function  and  reward  or  punishment  ensues.  Suppose  reward occurs.  We  therefore  want  to make  0  more  likely given  i.  Backpropagation will  do just that if we  take  0  as  the desired  target to produce  an  error vector (0 - 8)  and adjust weights normally.  Now suppose punishment occurs, indicating 0  does not correspond with i.  By choice  of error vector, backpropagation allows us to push the search vector in any direction;  which way should we go?  In absence of problem-specific information, we cannot pick  an appropriate  direction  with certainty.  Any decision  will  involve assumptions.  A  very minimal "don't be like 0"  assumption-employed in Anderson (1986), Williams  (1988),  and Ackley  (1989)-pushes  s directly  away from 0 by taking (8 - 0)  as the  error vector.  A slightly stronger  "be like not-o"  assumption-employed in Barto &  Anandan  (1985)  and Ackley  (1987)-pushes s  directly  toward the  complement of 0  by taking  ((1  - 0)  - 8)  as  the  error  vector.  Although  the  two  approaches always  agree  on  the  signs  of the  error  terms,  they  differ  in  magnitudes.  In  this  work,  we  explore  the second  possibility,  embodied in  an algorithm called  complementary  reinforcement  back-propagation ( CRBP). 
Figure 1 summarizes the CRBP algorithm.  The algorithm in the figure reflects three  modifications to the basic approach just sketched.  First, in step 2,  instead of using  the  8j'S  directly as probabilities,  we  found  it advantageous to  "stretch"  the values  using  a  parameter  v.  When  v  < 1,  it is  not  necessary for  the  8i'S  to reach zero or  one  to  produce  a  deterministic  output.  Second,  in  step  6,  we  found  it  important  to use  a smaller learning rate for  punishment compared to reward.  Third, consider  step 7:  Another  forward  propagation is  performed,  another  stochastic binary out(cid:173) put  vector  0*  is  generated  (using  the procedure  from  step  2),  and  0*  is  compared  to  o.  If they  are  identical  and  punishment  occurred,  or  if they  are  different  and  reward occurred, then another error vector is generated and another weight update  is performed.  This loop continues until a  different  output is generated  (in the case  of failure)  or  until the original  output is  regenerated  (in  the case  of success).  This  modification improved performance significantly, and added only a small percentage  to the total number of weight updates performed. 
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O.  Build  a  back  propagation  network  with  input  dimensionality  n  and  output 
dimensionality m.  Let t  =  0 and te  =  O. 

Pick random i  E 2n  and forward  propagate to produce  a/s.  2.  Generate a binary output vector o.  Given a uniform random variable ~ E  [0,1] 

and parameter 0 < v  < 1, 
OJ  = {1,  if(sj - !)/v+! ~ ~j