Neural Processing Letters 7: 81–89, 1998. 81 c 1998 Kluwer Academic Publishers. Printed in the Netherlands. On-Line Learning Fokker-Planck Machine J.A.K. SUYKENS, H. VERRELST and J. VANDEWALLE Katholieke Universiteit Leuven, Department of Electrical Engineering, ESAT-SISTA, Kardinaal Mercierlaan 94, B-3001 Leuven (Heverlee), Belgium E-mail: johan.suykens@esat.kuleuven.ac.be Key words: RBF networks, Gaussian mixture distribution, global optimization, Fokker-Planck equa- tion, constrained LMS, regularization Abstract. In this letter we present an on-line learning version of the Fokker-Planck machine. The method makes use of a regularized constrained normalized LMS algorithm in order to estimate the time-derivative of the parameter vector of a radial basis function network. The RBF network parametrizes a transition density which satisfies a Fokker-Planck equation, associated to continuous simulated annealing. On-line learning using the constrained normalized LMS method is necessary in order to make the Fokker-Planck machine applicable to large scale nonlinear optimization problems. 1. Introduction In (Suykens et al., 1996; Suykens & Vandewalle, 1995; Suykens & Vandewalle, 1996) the Fokker-Planck learning machine has been introduced as a new method for global optimization of differentiable cost functions. The method is derived from continuous simulated annealing (Gelfand & Mitter, 1991; Gelfand & Mitter, 1993; Kushner, 1987) (or recursive stochastic algorithms in a discrete time context) by considering the associated Fokker-Planck equation in the transition density. The step from the Fokker-Planck equation to the Fokker-Planck machine is made by parametrizing the density with a radial basis function network, corresponding to a Gaussian mixture distribution (Haykin, 1996; Amari, 1995; Streit & Luginbuhl, 1994) or Gaussian sum approximation (Alspach & Sorenson, 1972). By sampling the search space and evaluating the Fokker-Planck equation in these points, a set of equations is obtained in the time-derivative of the parameter vector of the RBF network. Hence the Fokker-Planck machine is a population based method like genetic algorithms (Goldberg, 1989). However it is not driven by cost function values (survival of the fittest) but by the local geometry at the sampling This research work was carried out at the ESAT laboratory and the Interdisciplinary Center of Neural Networks ICNN of the Katholieke Universiteit Leuven, in the following frameworks: the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture (IUAP P4–02 and IUAP P4–24), a Concerted Action Project MIPS (Modelbased Information Processing Systems) of the Flemish Community and the FWO (Fund for Scientific Research - Flanders) project G.0262.97 : Learning and Optimization: an Interdisciplinary Approach. The scientific responsibility rests with its authors.