On-line EM Algorithm for the Normalized Gaussian Network

A normalized gaussian network (NGnet) (Moody & Darken, 1989) is a network of local linear regression units. The model softly partitions the input space by normalized gaussian functions, and each local unit linearly approximates the output within the partition. In this article, we propose a new on-line EM algorithm for the NGnet, which is derived from the batch EM algorithm (Xu, Jordan, & Hinton 1995), by introducing a discount factor. We show that the on-line EM algorithm is equivalent to the batch EM algorithm if a specific scheduling of the discount factor is employed. In addition, we show that the on-line EM algorithm can be considered as a stochastic approximation method to find the maximum likelihood estimator. A new regularization method is proposed in order to deal with a singular input distribution. In order to manage dynamic environments, where the input-output distribution of data changes over time, unit manipulation mechanisms such as unit production, unit deletion, and unit division are also introduced based on probabilistic interpretation. Experimental results show that our approach is suitable for function approximation problems in dynamic environments. We also apply our on-line EM algorithm to robot dynamics problems and compare our algorithm with the mixtures-of-experts family.

[1]  Carlos S. Kubrusly,et al.  Stochastic approximation algorithms and applications , 1973, CDC 1973.

[2]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[3]  John Moody,et al.  Fast Learning in Networks of Locally-Tuned Processing Units , 1989, Neural Computation.

[4]  F. Girosi,et al.  Networks for approximation and learning , 1990, Proc. IEEE.

[5]  John C. Platt.  A Resource-Allocating Network for Function Interpolation , 1991, Neural Computation.

[6]  Steven J. Nowlan,et al.  Soft competitive adaptation: neural network learning algorithms based on fitting statistical mixtures , 1991 .

[7]  Geoffrey E. Hinton,et al.  Adaptive Mixtures of Local Experts , 1991, Neural Computation.

[8]  Robert A. Jacobs,et al.  Hierarchical Mixtures of Experts and the EM Algorithm , 1993, Neural Computation.

[9]  Geoffrey E. Hinton,et al.  An Alternative Model for Mixtures of Experts , 1994, NIPS.

[10]  Geoffrey E. Hinton,et al.  The delve manual , 1996 .

[11]  Peter I. Corke,et al.  A robotics toolbox for MATLAB , 1996, IEEE Robotics Autom. Mag..

[12]  Jürgen Schürmann,et al.  Pattern classification - a unified view of statistical and neural approaches , 2008 .

[13]  Steven R. Waterhouse,et al.  Classification and Regression using Mixtures of Experts , 1997 .

[14]  Shin Ishii,et al.  On-line EM algorithm and reconstruction of chaotic dynamics , 1998, Neural Networks for Signal Processing VIII. Proceedings of the 1998 IEEE Signal Processing Society Workshop (Cat. No.98TH8378).

[15]  Christopher G. Atkeson,et al.  Constructive Incremental Learning from Only Local Information , 1998, Neural Computation.

[16]  Radford M. Neal.  Assessing Relevance determination methods using DELVE , 1998 .

[17]  Geoffrey E. Hinton,et al.  A View of the Em Algorithm that Justifies Incremental, Sparse, and other Variants , 1998, Learning in Graphical Models.

[18]  G. McLachlan,et al.  Pattern Classification: A Unified View of Statistical and Neural Approaches. , 1998 .

[19]  Shin Ishii,et al.  Reinforcement Learning Based on On-Line EM Algorithm , 1998, NIPS.

[20]  Shin Ishii,et al.  Reconstruction of chaotic dynamics by on-line EM algorithm , 2001, Neural Networks.

[21]  Masa-aki Sato,et al.  Online Model Selection Based on the Variational Bayes , 2001, Neural Computation.