Recurrent Networks: Second Order Properties and Pruning
Authors: Lars Kai Hansen Morten With Pedersen
Published in: Advances in Neural Information Processing Systems 7  (NIPS 1994) (Main Conference Track), 1994

Abstract: Second  order  properties  of cost  functions  for  recurrent  networks  are investigated.  We  analyze a layered fully recurrent  architecture,  the  virtue  of this  architecture  is  that  it features  the  conventional  feedforward architecture as a special case.  A detailed description of  recursive  computation of the full  Hessian of the network cost func(cid:173) tion is  provided.  We  discuss  the possibility of invoking simplifying  approximations of the Hessian and show how weight decays iron the  cost function and thereby greatly assist training. We present tenta(cid:173) tive  pruning results,  using  Hassibi  et  al.'s  Optimal Brain  Surgeon,  demonstrating  that  recurrent  networks  can  construct  an  efficient  internal memory. 
1  LEARNING IN RECURRENT NETWORKS 
Time  series  processing  is  an  important  application  area for  neural  networks  and  numerous architectures have been suggested, see e.g. (Weigend and Gershenfeld, 94).  The most general structure is a fully recurrent network and it may be adapted using  Real Time Recurrent Learning (RTRL) suggested by  (Williams and Zipser,  89).  By  invoking a  recurrent  network,  the  length of the network memory can  be  adapted to  the  given  time series,  while  it is  fixed  for  the  conventional  lag-space net  (Weigend  et  al.,  90).  In  forecasting,  however,  feedforward  architectures  remain  the  most  popular structures; only few  applications are reported based on the Williams&Zipser  approach.  The main difficulties experienced  using  RTRL are slow  convergence  and 
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Morten  With Pedersen,  Lars Kai Hansen 
lack  of generalization.  Analogous  problems  in  feedforward  nets  are  solved  using  second  order  methods  for  training  and  pruning  (LeCun  et  al.,  90;  Hassibi  et  al.,  92;  Svarer  et  al.,  93).  Also,  regularization  by  weight  decay  significantly  improves  training and generalization.  In this work we initiate the investigation of second order  properties for  RTRL; a detailed  calculation scheme for  the cost function  Hessian  is  presented,  the importance of weight decay is demonstrated, and preliminary pruning  results  using Hassibi et al.'s Optimal Brain Surgeon  (OBS)  are presented.  We find  that the recurrent  network  discards  the  available lag space  and constructs  its own  efficient  internal memory. 
1.1  REAL  TIME RECURRENT  LEARNING 
The fully  connected  feedback  nets studied  by  Williams&Zipser operate like a  state  machine, computing the outputs from the internal units according to a state vector  z(t)  containing previous external inputs and internal unit outputs.  Let  x(t) denote  a  vector  containing the external  inputs  to  the net  at time t,  and let y(t)  denote  a  vector  containing the outputs of the  units  in  the  net.  We now  arrange  the  indices  on  x  and y  so  that the elements of z(t)  can be defined  as 
,  k  E I  ,  k E U 
where I  denotes the set of indices for which  Zk  is  an input, and U denotes  the set of  indices for  which  Zk  is  the output of a unit in the net.  Thresholds are implemented  using  an  input  permanently  clamped  to  unity.  The  k'th  unit  in  the  net  is  now  updated according to 
where Wkj  denotes the weight to unit k from input/unit j  and "'0 is the activation  function  of the k'th unit.  When  used  for  time  series  prediction,  the  input  vector  (excluding  threshold)  is  usually defined  as  x(t) =  [x(t), . .. , x(t - L + 1)]  where  L denotes  the dimension of  the lag  space.  One of the units in the net is  designated to be the output unit Yo,  and  its activating function 10  is  often chosen  to be linear in order  to allow for  arbitrary  dynamical range.  The prediction of x(t + 1)  is x(t + 1)  = lo[so(t»).  Also,  if the first  prediction  is  at t =  1,  the first  example is  presented  at t = 0  and we 'set y(O)  =  O.  We  analyse here  a modification of the standard Williams&Zipser construction  that  is  appropriate for forecasting purposes.  The studied architecture is  layered.  Firstly,  we  remove  the external  inputs from  the linear  output unit in  order  to prevent  the  network from getting  trapped  in a linear mode.  The output then reads 
x(t + 1) = Yo(t + 1) = L WojYj(t)  + Wthres,o