Using Curvature Information for Fast Stochastic Search
Authors: Todd K. Leen Genevieve B. Orr
Published in: Advances in Neural Information Processing Systems 9  (NIPS 1996) (Main Conference Track), 1996

Abstract: We  present an algorithm for  fast  stochastic gradient descent  that  uses  a  nonlinear adaptive momentum scheme to optimize the late  time convergence rate.  The algorithm makes  effective  use  of cur(cid:173) vature information,  requires  only  O(n)  storage and computation,  and  delivers  convergence  rates  close  to  the theoretical  optimum.  We demonstrate the technique on linear and large nonlinear back(cid:173) prop networks. 
Improving Stochastic  Search 
Learning algorithms that perform gradient  descent  on a  cost function can  be for(cid:173) mulated in either stochastic (on-line)  or batch form.  The stochastic version  takes  the form 
Wt+l  =  Wt  +  J1.t  G( Wt, Xt  ) 
(1)  where  Wt  is  the  current  weight  estimate,  J1.t  is  the learning  rate,  G  is  minus  the  instantaneous gradient  estimate, and  Xt  is  the input at time  t i .  One obtains the  corresponding batch mode learning rule by taking J1.  constant and averaging Gover  all  x. 
Stochastic  learning  provides  several  advantages  over  batch  learning.  For  large  datasets the batch average is  expensive to compute.  Stochastic learning eliminates  the averaging.  The stochastic update can  be regarded as  a  noisy  estimate of the  batch update, and this intrinsic noise can reduce the likelihood of becoming trapped  in poor local optima [1,  2J. 
1 We  assume  that  the  inputs  are i.i.d.  This is  achieved  by random  sampling with  re(cid:173)
placement from  the training data. 
Using Curvature Informationfor Fast Stochastic Search 
607 
The noise must be reduced late in the training to allow  weights to converge.  After  settling within the basin of a local optimum W.,  learning rate annealing allows con(cid:173) vergence of the weight error v ==  W  - w •.  It is well-known that the expected squared  weight  error, E[lv12]  decays at its maximal rate ex:  l/t with the annealing schedule  flo/to  FUrthermore to achieve this rate one must have flo  > flcnt  =  1/(2Am in) where  Amin  is the smallest eigenvalue of the Hessian at w.  [3,  4,  5, and references therein].  Finally the optimal flo, which gives the lowest possible value of E[lv12] is flo  = 1/ A.  In multiple dimensions the optimal learning  rate  matrix is  fl(t) = (l/t) 1-£-1 ,where  1-£  is  the Hessian at the local optimum.  Incorporating this curvature information into stochastic learning is  difficult for  two  reasons.  First, the Hessian is  not available since the point of stochastic learning is  not to  perform averages  over  the training data.  Second,  even if the Hessian  were  available, optimal learning requires its inverse - which is  prohibitively expensive to  compute 2. 
The primary result of this paper is that one can achieve an algorithm that behaves  optimally,  i.e.  as if one  had incorporated the inverse  of the full  Hessian,  without  the storage or  computational  burden.  The  algorithm,  which  requires  only  V(n)  storage and computation (n = number of weights in the network), uses an adaptive  momentum parameter, extending our earlier work  [7]  to fully  non-linear problems.  We demonstrate the performance on several large back-prop networks trained with  large datasets. 
Implementations of stochastic learning typically use a constant learning rate during  the early part of training (what Darken and Moody [4]  call the search phase) to ob(cid:173) tain exponential convergence towards a local optimum, and then switch to annealed  learning (called the converge phase).  We use Darken and Moody's adaptive search  then converge (ASTC)  algorithm to determine the point at which to switch to  l/t  annealing.  ASTC was  originally conceived as a  means to insure flo  > flcnt  during  the annealed phase, and we  compare its performance with adaptive momentum as  well.  We  also provide a  comparison with conjugate gradient optimization. 
1  Momentum in Stochastic Gradient Descent 
The adaptive momentum algorithm we  propose was  suggested by  earlier  work  on  convergence rates for  annealed learning with constant momentum.  In this  section  we  summarize the relevant results of that work. 
Extending (1)  to include momentum leaves  the learning rule 
wt+ 1  = Wt  + flt  G ( Wt, x t)  + f3  ( Wt  - Wt -1 ) 
(2)  where f3  is  the momentum parameter constrained so  that  0  < f3  <  1.  Analysis  of  the dynamics of the expected squared weight  error E[ Ivl2  ]  with flt = flo/t learning  rate annealing [7, 8]  shows that at late times, learning proceeds as for the algorithm  without momentum, but with a  scaled or  effective  learning rate 
flo  fleff  =  1 _  f3