Stability and Convergence

Stability refers to the equilibrium behaviour of the activation state of a neural network, whereas convergence refers to the adjustment behaviour of the weights during learning, which will eventually lead to minimization of error between the desired and actual outputs. Thus convergence is typically associated with supervised learning, although it is relevant in all cases of learning, both supervised and unsuper- vised. The objective of any learning law is that it should eventually lead to a set of weights which will capture the pattern information in the training set data.

In this section we will discuss the global behaviour of artificial neural whose activation dynamics is described by the following set of equations [Cohen and Grossberg, 19831:

Stability and Convergence 69 These equations represent a class of N-dimensional competitive systems. the previous activation models including the general shunting activation model form special cases of this system. In the activation state of the network starts from an initial state and follows a trajectory dictated by the dynamics of the equations. A network will be useful only if a trajectory leads eventually to an equilibrium state at which point there is no further change in the state. Such a state is also called a stable state, when a small perturbation of the state settles to the same state. Different initial states may follow different trajectories, all of which should terminate at some equilibrium states. There may be

trajectories that may at the same equilibrium state. The existence of such equilibrium states enables global pattern formation possible in a network. That is, an input pattern corresponding to a starting state will eventually lead to one of the global patterns, which can be interpreted as storage of the input pattern in long term memory. The global pattern thus formed will only change if there is a different external input. In some cases the network parameters such as weights may slowly change due to learning or self-organization. If the global pattern formation still occurs for choice of these parameters, then the resulting pattern is said to be absolutely stable or globally stable.

Under certain conditions, which will be discussed later, the set of equations (2.74) describing activation dynamics do exhibit stable states which are also called point equilibrium states. Such a network then can form global patterns at those states, and hence can be used for pattern storage. One of the conditions is that the weights should be symmetric = If the weights are not exactly symmetric, then the network may exhibit periodic oscillations of states in certain regions of the state space. These oscillatory regions are also stable, and hence can be used for pattern storage. Oscillatory stable states may also arise when there is some delay in the feedback of the outputs from other processing units to the current unit, even though the weights are exactly symmetric.

For some other conditions, the network may display chaotic changes of states in the regions of equilibrium, also called basins of attraction. Such a network is said to exhibit chaotic stability. Thus pattern storage is possible in any network that exhibits either fixed point stability or oscillatory stability or chaotic stability. However, it is to analyze and design a network suitable for the oscillatory and chaotic types of stabilities [Cohen and Grossberg, 1983; 19951.

A general network is more likely to exhibit random chaotic changes of states throughout due to nonlinearly coupled set of equations with delayed feedback. One has to carefully choose the parameters of the activation dynamics models for ensuring stable points. In general, it is difficult to know whether a network will have

70 Activation and S y m p t i c

stable points, and if so, how many. It is even more difficult to determine the behaviour of the network near the stable points to examine the nature of stability. However, in a few cases it is possible to predict the global pattern behaviour, if it is possible to show the existence of an energy function called Lyapunov function

19771. It is a scalar function of the parameters of the network, denoted by where x is the activation state vector of the network. is said to be a Lyapunov function if for all x. It is sufficient if we can find a Lyapunov function for a network in order to demonstrate the existence of stable equilibrium states. It is not a necessary condition, as the network may still have stable points, even though a Lyapunov function could not be found. The existence of Lyapunov function makes it easy to analyze the stability of the network.

If the Lyapunov is interpreted as an energy function, then the condition that means that any change in the energy due to change in the state of the network results in lowering the total energy. In other words, any change of the state of the network results in the trajectory of the state sliding along the energy surface in the state space towards lower energy. Eventually the leads to a state from where there is no further decrease in the energy due to changes in the state. Such a state corresponds to the energy minimum, at which = Normally there will be many states at which = All such states correspond to equilibrium points or stable states. All trajectories in the state space will eventually lead to one of these stable states.

In the following, three general theorems are given that describe the stability of a set of nonlinear systems. The first theorem, the Cohen-Grossberg theorem, is useful to show the stability of fixed weight autoassociative networks. The second theorem, the Cohen-Grossberg-Kosko theorem, is useful to show the stability of adaptive autoassociative networks. The third theorem, the adaptive bidirectional-associative memory theorem, is useful to show the stability of adaptive heteroassociative networks.

Cohen-Grossberg theorem: For a system of equations given by

Stability and Convergence Since

L

v

tonically nondecreasing function), are constant and do not change with time, and is symmetric. This last property was used to obtain the simplified expression for the derivative of the second of in Eq. (2.78).

Note that the function could be arbitrary, except that it should ensure the integrability of the in Thus is a global Lyapunov function, provided conditions are satisfied.

Cohen-Grossberg-Kosko theorem: For a system where both the activation state and the synaptic weights are changing simultaneously, the equations describing the dynamics may be expressed as follows [Kosko, 19881:

where is assumed to be a symmetric matrix. For such a system the following is a Lyapunov function.

Adaptive bldirectlonal memory theorem: The system of equations describing the activation and synaptic dynamics of a neural network consisting of two layers of processing units, a unit in each layer feeding its output to all the units in the other layer, is given as follows 19881:

Activation and Synaptic Dynamics

The following is a Lyapunov function for the above system: