{"title": "Computing Time Lower Bounds for Recurrent Sigmoidal Neural Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 503, "page_last": 510, "abstract": null, "full_text": "Computing Time  Lower Bounds for \n\nRecurrent  Sigmoidal  Neural  Networks \n\nMichael Schmitt \n\nLehrstuhl Mathematik und Informatik, Fakultat fUr  Mathematik \n\nRuhr-Universitat Bochum, D- 44780 Bochum,  Germany \n\nmschmitt@lmi.ruhr-uni-bochum.de \n\nAbstract \n\nRecurrent  neural networks of analog units are computers for  real(cid:173)\nvalued functions.  We  study the time complexity of real computa(cid:173)\ntion  in  general  recurrent  neural  networks.  These  have  sigmoidal, \nlinear,  and  product  units  of unlimited  order  as  nodes  and  no  re(cid:173)\nstrictions on the weights.  For networks operating in discrete time, \nwe  exhibit  a  family  of functions  with  arbitrarily high  complexity, \nand we derive almost tight bounds on the time required to compute \nthese functions.  Thus, evidence is  given of the computational lim(cid:173)\nitations  that  time-bounded  analog  recurrent  neural  networks  are \nsubject to. \n\n1 \n\nIntroduction \n\nAnalog recurrent neural networks are known to have computational capabilities that \nexceed those  of classical Turing machines  (see,  e.g.,  Siegelmann and Sontag,  1995; \nKilian  and  Siegelmann,  1996;  Siegelmann,  1999).  Very  little,  however,  is  known \nabout  their  limitations.  Among  the  rare  results  in  this  direction,  for  instance, \nis  the  one  of  Sima  and  Orponen  (2001)  showing  that  continuous-time  Hopfield \nnetworks  may  require  exponential time  before  converging to  a  stable  state.  This \nbound,  however,  is  expressed  in terms  of the  size  of the  network and,  hence,  does \nnot  apply  to  fixed-size  networks  with  a  given  number  of  nodes.  Other  bounds \non the computational power of analog recurrent networks have been established by \nMaass and Orponen (1998)  and Maass and Sontag (1999).  They show that discrete(cid:173)\ntime recurrent  neural networks  recognize only a  subset of the regular languages in \nthe presence  of noise.  This model of computation in recurrent  networks,  however, \nreceives  its  inputs  as  sequences.  Therefore,  computing time  is  not  an  issue  since \nthe  network  halts  when  the  input  sequence  terminates.  Analog  recurrent  neural \nnetworks,  however, can also be run as  \"real\"  computers that get  as  input a  vector \nof  real  numbers  and,  after  computing  for  a  while,  yield  a  real  output  value.  No \nresults  are  available  thus  far  regarding  the  time  complexity  of  analog  recurrent \nneural networks with given size. \n\nWe  investigate here the time complexity of discrete-time recurrent neural networks \nthat compute functions over the reals.  As  network nodes we  allow sigmoidal units, \nlinear  units,  and  product  units-\nthat  is,  monomials  where  the  exponents  are  ad-\n\n\fjustable weights  (Durbin  and  Rumelhart,  1989) .  We  study  the  complexity of real \ncomputation in the sense of Blum et aI.  (1998).  That means, we  consider real num(cid:173)\nbers as entities that are represented exactly and processed without restricting their \nprecision.  Moreover, we  do not assume that the information content of the network \nweights is bounded (as done, e.g., in the works of Balcazar et aI. , 1997; Gavalda and \nSiegelmann, 1999).  With such a  general type of network, the question arises which \nfunctions  can be computed with a  given number of nodes  and a  limited amount of \ntime.  In  the following, we  exhibit a  family of real-valued functions  ft, l  2:  1,  in one \nvariable that is computed by some fixed  size network in time O(l).  Our main result \nis,  then,  showing  that  every  recurrent  neural network  computing the  functions  ft \nrequires  at least  time  nW /4).  Thus,  we  obtain almost  tight  time  bounds  for  real \ncomputation in recurrent neural networks. \n\n2  Analog  Computation in Recurrent  Neural  Networks \n\nWe study a  very comprehensive type of discrete-time recurrent neural network that \nwe  call  general  recurrent  neural network (see  Figure 1).  For every k, n  E N there is \na  recurrent  neural architecture consisting of k  computation  nodes  YI ,  . . . , Yk  and n \ninput nodes Xl ,  ... , x n .  The size of a network is defined to be the number ofits com(cid:173)\nputation nodes.  The computation nodes form a  fully  connected recurrent network. \nEvery computation node also receives connections from every input node.  The input \nnodes play the role of the input variables of the system.  All connections are param(cid:173)\neterized  by  real-valued  adjustable  weights.  There  are  three  types  of computation \nnodes:  product units,  sigmoidal units,  and linear units.  Assume that computation \nnode i  has  connections from  computation nodes weighted by Wil, ... ,Wi k  and from \ninput nodes weighted by ViI, .. .  ,Vi n.  Let YI (t) , . . . ,Yk (t)  and Xl (t), ... ,Xn (t)  be the \nvalues of the computation nodes  and input nodes  at time t,  respectively.  If node i \nis  a  product  unit,  it computes at time t + 1 the value \n\n(1) \n\nthat  is,  after  weighting  them  exponentially,  the  incoming  values  are  multiplied. \nSigmoidal and linear units have an additional parameter associated with them, the \nthreshold or bias ()i .  A  sigmoidal  unit computes the value \n\nwhere  (J  is  the  standard sigmoid (J( z )  =  1/ (1  + e- Z ).  If node  i  is  a  linear  unit,  it \nsimply outputs the weighted sum \n\nWe allow the networks to be heterogeneous, that is, they may contain all three types \nof computation nodes simultaneously.  Thus, this model encompasses a wide class of \nnetwork types  considered in research and  applications.  For  instance, architectures \nhave  been proposed that include a  second layer of linear computation nodes which \nhave no recurrent connections to computation nodes but serve as output nodes  (see, \ne.g. , Koiran and Sontag, 1998;  Haykin,  1999;  Siegelmann, 1999).  It is  clear that in \nthe definition given here, the linear units can function  as  these output nodes if the \nweights  of the  outgoing  connections  are  set  to  O.  Also  very  common  is  the  use \nof  sigmoidal  units  with  higher-order  as  computation  nodes  in  recurrent  networks \n(see,  e.g.,  Omlin and  Giles,  1996;  Gavalda and  Siegelmann,  1999;  Carrasco et  aI., \n2000).  Obviously, the model here includes these higher-order networks as a  special \ncase  since  the  computation of a  higher-order  sigmoidal unit  can  be  simulated  by \nfirst  computing the higher-order terms using  product units  and then  passing their \n\n\fcomputation \n\nnodes \n\nI \nsigmoidal,  product, and  linear units \n\nI \n\n. \n\nYl \n\n. \n\nYk \n\nt \n\ninput nodes \n\nXl \n\nXn \n\nI \n\nFigure 1:  A general recurrent neural network of size k.  Any computation node may \nserve as output node. \n\noutputs to a  sigmoidal unit.  Product units, however,  are even more powerful than \nhigher-order terms  since  they  allow  to  perform  division operations  using  negative \nweights.  Moreover,  if a  negative  input  value  is  weighted  by  a  non-integer  weight, \nthe output of a  product unit may be a  complex number.  We  shall ensure here that \nall  computations are  real-valued.  Since  we  are  mainly interested in  lower  bounds, \nhowever,  these  bounds obviously remain valid if the computations of the networks \nare extended to the complex domain. \nWe now define what it means that a recurrent neural network N  computes a function \nf  : ~n  --+  llt  Assume  that N  has  n  input  nodes  and  let  x  E  ~n.  Given  tE N, \nwe  say  that  N  computes  f(x)  in  t  steps  if  after  initializing  at  time  0  the  input \nnodes  with x  and the computation nodes  with some fixed  values,  and performing t \ncomputation steps as defined in Equations (1) ,  (2) , and (3) , one of the computation \nnodes  yields  the  value  f(x).  We  assume  that  the  input  nodes  remain  unchanged \nduring the  computation.  We  further  say  that N  computes f  in  time t  if for  every \nx  E  ~n ,  network  N  computes  f  in  at  most  t  steps.  Note  that  t  may  depend \non  f  but  must  be  independent  of  the  input  vector.  We  emphasize  that  this  is \na  very  general  definition  of analog  computation  in  recurrent  neural  networks.  In \nparticular, we  do not specify any definite output node but allow the output to occur \nat  any  node.  Moreover,  it  is  not  even  required  that  the  network  reaches  a  stable \nstate, as with attractor or Hopfield  networks.  It is  sufficient that the output value \nappears  at  some  point  of the  trajectory the  network  performs.  A  similar view  of \ncomputation in recurrent networks is captured in a  model proposed by Maass et al. \n(2001).  Clearly,  the  lower  bounds  remain  valid  for  more  restrictive  definitions  of \nanalog  computation  that  require  output  nodes  or  stable  states.  Moreover,  they \nhold  for  architectures  that  have  no  input  nodes  but  receive  their  inputs  as  initial \nvalues of the computation nodes.  Thus, the bounds serve  as lower bounds  also  for \nthe transition times between real-valued states of discrete-time dynamical systems \ncomprising the networks considered here. \n\nOur  main  tool  of  investigation  is  the  Vapnik-Chervonenkis  dimension  of  neural \nnetworks.  It is defined as follows (see also Anthony and Bartlett, 1999):  A dichotomy \nof  a  set  S  ~ ~n  is  a  partition  of  S  into  two  disjoint  subsets  (So , Sd  satisfying \nSo  U S1  =  S.  A  class  :F  of functions  mapping  ~n  to  {O, I} is  said  to  shatter S  if \nfor  every  dichotomy  (So , Sd  of S  there  is  some  f  E  :F  that  satisfies  f(So)  ~ {O} \nand  f(S1)  ~ {I}.  The  Vapnik-Chervonenkis  (VC)  dimension  of :F  is  defined  as \n\n\f4\"'+4\",IL \n\n' I  -1---Y-2----Y-5~1 \n\nS~ \n\nY5 \n\noutput \n\nY4 \n\nFigure 2:  A  recurrent neural network computing the functions  fl  in time 2l + 1. \n\nthe  largest  number  m  such  that  there  is  a  set  of m  elements  shattered  by  F.  A \nneural  network  given  in  terms  of  an  architecture  represents  a  class  of  functions \nobtained by assigning real numbers to all its adjustable parameters, that is, weights \nand  thresholds  or  a  subset  thereof.  The  output  of the  network is  assumed  to  be \nthresholded at some fixed  constant  so  that the output  values  are  binary.  The  VC \ndimension of a  neural network is  then  defined  as  the VC  dimension of the class  of \nfunctions  computed by this network. \nIn  deriving lower  bounds  in the  next  section,  we  make  use  of the  following  result \non networks with product and sigmoidal units that has been previously established \n(Schmitt,  2002).  We  emphasize that the  only constraint on the  parameters of the \nproduct units is  that they yield real-valued, that is,  not complex-valued, functions. \nThis means further that the statement holds for networks of arbitrary order, that is, \nit does  not impose any restrictions on the magnitude of the weights of the product \nunits. \nProposition 1.  (Schmitt,  2002,  Theorem  2)  Suppose  N  is  a  feedforward  neural \nnetwork  consisting  of sigmoidal,  product,  and linear  units.  Let k  be  its  size  and W \nthe  number of adjustable  weights.  The  VC dimension  of N  restricted to  real-valued \nfunctions  is  at most 4(Wk)2 + 20Wk log(36Wk). \n\n3  Bounds on  Computing  Time \n\nWe establish bounds on the time required by recurrent neural networks for  comput(cid:173)\ning a  family of functions  fl  : JR  -+  JR,  l  2::  1,  where l  can be considered as  a  measure \nof the complexity of fl.  Specifically,  fl  is  defined in terms of a  dynamical system as \nthe lth iterate of the logistic map \u00a2>(x)  =  4x(1  - x),  that is, \n\nfl(X) \n\n{ \n\n\u00a2>(x) \n\u00a2>(fl- l (x)) \n\nl  = 1, \nl >  2. \n\nWe  observe that there is  a  single recurrent  network capable of computing every  fl \nin time O(l). \nLemma 2.  There  is  a  general  recurrent  neural  network  that  computes  fl  in  time \n2l + 1 for  every l. \n\nProof.  The  network  is  shown  in  Figure  2.  It consists  of  linear  and  second-order \nunits.  All  computation nodes  are initialized with 0, except  Yl,  which starts with 1 \nand  outputs  0  during  all following  steps.  The  purpose  of Yl  is  to let  the  input  x \n\n\foutput \n\nFigure 3:  Network Nt. \n\nenter  node  Y2  at  time  1  and  keep  it  away  at  later  times.  Clearly,  the  value  fl (x) \nresults at node Y5  after 2l + 1 steps. \nD \nThe network used for  computing fl  requires only linear and second-order units.  The \nfollowing  result  shows  that  the  established  upper  bound  is  asymptotically almost \ntight,  with a  gap only of order four .  Moreover, the lower bound holds for  networks \nof unrestricted order and with sigmoidal units. \nTheorem 3.  Every general recurrent neural network of size k  requires  at least time \ncl l / 4 j k  to  compute  function  fl'  where  c> 0  is  some  constant. \nProof.  The  idea is  to  construct  higher-order  networks Nt  of small  size  that  have \ncomparatively large VC dimension.  Such a network will consist of linear and product \nunits and hypothetical units that compute functions  fJ  for  certain values of j.  We \nshall derive a  lower bound on the VC  dimension of these networks.  Assuming that \nthe hypothetical units can be replaced by time-bounded general recurrent networks, \nwe  determine  an  upper  bound  on  the  VC  dimension  of the  resulting  networks  in \nterms of size and computing time using an idea from Koiran and Sontag (1998)  and \nProposition 1.  The comparison of the lower  and upper VC  dimension bounds will \ngive  an estimate of the time required for  computing k \nNetwork Nt, shown in Figure 3, is a feedforward network composed of three networks \n(/1) \n\u2022 r(1) \nJVI \nand  2l  + 2  computation  nodes  yb/1),  ... , Y~r~l  (see  Figure  4).  There  is  only  one \nadjustable parameter in Nt, denoted w, all other weights are fixed.  The computation \nnodes  are defined as follows  (omitting time parameter t): \n\nlnput  no  es  Xl' .. . , x I \n\n1  2  3  h \n,  ,  ,  as \n\n.r(3)  E  h \n\nac  networ \n\nk  \u2022 r(/1) \n\nJVI \n\n,J.L  = \n\n\u2022 r(2) \n\n, JVI \n\n, JVI \n\nl\u00b7 \n\n. \n\nd \n\n(/1) \n\nfor  J.L  = 3, \nfor  J.L  =  1,2, \n\nfll'--1 (Y~~)l) for  i  =  1, ... ,l and J.L  =  1,2,3, \n\ny~/1) . x~/1),  for  i  =  1, .. . ,l and J.L  =  1,2,3, \n\n(/1) \nYIH  + ... + Y21 \n\n(/1)  c \n\nlor  J.L  -\n\n- 1  2  3 \n,  \u2022 \n\n, \n\ny~/1) \n\ny}~{ \n\n(/1) \n\nY21+l \n\nThe  nodes  Yb/1)  can be  considered as  additional input  nodes  for  N//1),  where N;(3) \ngets  this  input  from  w,  and  N;(/1)  from  N;(/1+l)  for  J.L  =  1,2.  Node  Y~r~l  is  the \noutput node of N;(/1),  and node Y~~~l is also the output node of Nt.  Thus, the entire \nnetwork has 3l + 6 nodes that are linear or product units and 3l nodes that compute \nfunctions  h, fl'  or  f12. \n\n\foutput \n\n8 \n\nr - - - - - - - - - - - - '  ..... L - - - - - - - - - - - ,  \n\nI \n\nB \nt \nI x~p)1 \n\nI \n\nB \nt \n~ \n\n-----\n\nt \n\ninput:  w  or \noutput of N;(P+1) \n\nFigure 4:  Network N;(p). \n\nWe show that Ni shatters some set of cardinality [3,  in particular, the set S  =  ({ ei  : \n\ni  =  1, . .. , [})3,  where  ei  E  {O, 1}1  is  the  unit  vector  with  a  1  in  position  i  and \u00b0 \n\nelsewhere.  Every dichotomy of S  can be programmed into the network parameter \nw  using  the  following  fact  about  the  logistic  function  \u00a2  (see  Koiran  and  Sontag, \n1998,  Lemma 2):  For  every  binary vector  b E  {O, l}m, b =  b1  .\u2022. bm ,  there  is  some \nreal number w  E  [0,1]  such that for  i  =  1, ... , m \n\nE \n\n{ \n\n[0,1 /2) \n\n(1/2,1] \n\nif bi  =  0, \nif bi  =  1. \n\nHence,  for  every dichotomy (So, Sd of S  the parameter w  can be chosen such that \nevery  (ei1' ei2 , ei3)  E  S  satisfies \n\n1/2 \n\n1/2 \n\nif (eillei2,eis)  E  So, \nif (eillei2,eiJ  E  S1. \n\nSince h +i2 H i 3 .12 (w)  =  \u00a2i1 (\u00a2i2'1 (\u00a2i3 .12 (w))),  this  is  the  value  computed  by Ni  on \ninput  (eill ei2' ei3),  where ei\"  is  the input given to network N;(p).  (Input ei\"  selects \nthe function  li\"'I,,-1  in N;(p).)  Hence,  S  is  shattered by Ni,  implying that Ni  has \nVC  dimension at least [3. \n\n\fAssume now that Ii can be computed by a  general recurrent neural network of size \nat most  kj  in time tj.  Using  an  idea of Koiran and  Sontag  (1998),  we  unfold  the \nnetwork to obtain a feedforward network of size at most kjtj computing fj.  Thus we \ncan replace the nodes computing ft, ft, fl2  in Nz  by networks of size k1t1, kltl, k12t12, \nrespectively,  such  that  we  have  a  feedforward  network '!J  consisting of sigmoidal, \nproduct,  and  linear  units.  Since  there  are  3l  units  in  Nl  computing  ft, ft,  or  fl2 \nand  at  most  3l  + 6  product  and  linear  units,  the  size  of Nt  is  at  most  c1lkl2tl2 \nfor  some  constant  C1  > O.  Using that Nt  has  one  adjustable  weight,  we  get  from \nProposition 1 that its VC  dimension is  at most c2l2kr2tr2  for  some constant C2  > o. \nOn the other hand,  since Nz  and Nt  both shatter S,  the VC  dimension of Nt  is  at \nleast  l3.  Hence,  l3  ~ C2l2 kr2 tr2 holds,  which  implies  that  tl2  2:  cl 1/ 2 / kl2 for  some \nc > 0,  and hence  tl  2:  cl1/4 / kl. \nD \nLemma  2  shows  that  a  single  recurrent  network  is  capable  of  computing  every \nfunction  fl  in time O(l).  The following consequence of Theorem 3 establishes that \nthis  bound cannot be much improved. \n\nCorollary  4.  Every general  recurrent neural network requires  at least  time 0(ll /4 ) \nto  compute  the  functions  fl. \n\n4  Conclusions  and  Perspectives \n\nWe  have  established  bounds  on  the  computing  time  of  analog  recurrent  neural \nnetworks.  The result shows that for  every network of given size there are functions \nof  arbitrarily  high  time  complexity.  This  fact  does  not  rely  on  a  bound  on  the \nmagnitude  of weights.  We  have  derived  upper  and  lower  bounds  that  are  rather \ntight- with  a  polynomial  gap  of  order  four- and  hold  for  the  computation  of  a \nspecific  family  of  real-valued  functions  in  one  variable.  Interestingly,  the  upper \nbound  is  shown  using second-order networks  without sigmoidal units,  whereas the \nlower bound is  valid even for  networks with sigmoidal units and arbitrary product \nunits.  This  indicates that  adding these  units  might  decrease  the  computing time \nonly marginally.  The derivation made use of an upper bound on the VC  dimension \nof higher-order sigmoidal networks.  This  bound is  not  known to  be optimal.  Any \nfuture  improvement will  therefore  lead to  a  better  lower  bound  on the  computing \ntime. \n\nWe have focussed  on product and sigmoidal units as nonlinear computing elements. \nHowever,  the  construction presented here  is  generic.  Thus,  it is  possible to  derive \nsimilar results for  radial basis function  units,  models of spiking neurons,  and other \nunit  types  that  are  known  to  yield  networks  with  bounded  VC  dimension.  The \nquestions whether such results can be obtained for continuous-time networks and for \nnetworks operating in the  domain of complex numbers,  are challenging.  A  further \nassumption made  here  is  that  the  networks  compute the  functions  exactly.  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