We show in a very general framework the a.s. convergence of the one-dimensional Kohonen algorithm–after self-organization–to its unique equilibrium when the learning rate decreases to 0 in a suitable way. The main requirement is a log-concavity assumption on the stimuli distribution which includes all the usual (truncated) probability distributions (uniform, exponential, gamma distribution with parameter ≥ 1, etc.). For the constant step algorithm, the weak convergence of the invariant distributions towards equilibrium as the step goes to 0 is established too. The main ingredients of the proof are the Poincaré-Hopf Theorem and a result of Hirsch on the convergence of cooperative dynamical systems.

This paper investigates the dynamical properties of a class of urn processesand recursive stochastic algorithms with constant gain which arise frequentlyin control, pattern recognition, learning theory, and elsewhere.

It is shown that, under suitable conditions, invariant measures of theprocess tend to concentrate on the Birkhoff center of irreducible (i.e.chain transitive) attractors of some vector field $F: {\Bbb R}^d \rightarrow{\Bbb R}^d$ obtained by averaging. Applications are given to simplesituations including the cases where $F$ is Axiom A or Morse–Smale, $F$isgradient-like, $F$ is a planar vector field, $F$ has finitely many alpha andomega limit sets.

We consider stochastic processes {xn}n≥0 of the form

where F: ℝm → ℝm is C2, {λi}i≥1 is a sequence of positive numbers decreasing to 0 and {Ui}i≥1 is a sequence of uniformly bounded ℝm-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Γ, the probability of sample paths approaching Γ is positive. This gives the generic behavior of three-color urn models.