Some Features of Neural Networks as Nonlinearly Parameterized Models of Unknown Systems Using an Online Learning Algorithm

This paper deals with deriving the properties of updated neural network model that is exploited to identify an unknown nonlinear system via the standard gradient learning algorithm. The convergence of this algorithm for online training the three-layer neural networks in stochastic environment is studied. A special case where an unknown nonlinearity can exactly be approximated by some neural network with a nonlinear activation function for its output layer is considered. To analyze the asymptotic behavior of the learning processes, the so-called Lyapunov-like approach is utilized. As the Lyapunov function, the expected value of the square of approximation error depending on network parameters is chosen. Within this approach, sufficient conditions guaranteeing the convergence of learning algorithm with probability 1 are derived. Simulation results are presented to support the theoretical analysis.


Introduction
Design of mathematical models for technical, economic, social and other systems with uncertainties is the important problem from both theoretical and practical points of view.This problem attracts close attention of many researches.
The significant progress in this scientific area has been achieved last time.Within this area, new methods and modern intelligent algorithms dealing with uncertain systems have recently been proposed in [1] [2] [3] [4].They include some new optimization approaches advanced, in particular, in the papers [2] [4].
Over the past decades, interest has been increasing toward the use multilayer neural networks applied among other as models for the adaptive identification of nonlinearly parameterized dynamic systems [5] [6] [7] [8].This has been motivated by the theoretical works of several researches [9] [10] who proved that, even with one hidden layer, neural network can uniformly approximate any continuous mapping over a compact domain, provided that the network has sufficient number of neurons with corresponding weights.The theoretical background on neural network modeling may be found in the book [11].
Different learning methods for updating the weights of neural networks have been reported in literature.Most of these methods rely on the gradient concept [8].One of these methods is based on utilizing the Lyapunov stability theory [6] [12].
The convergence of the online gradient training procedure dealing with input signals that have deterministic (non-stochastic) nature was studied by many authors [13]- [23].Several of these authors assumed that training set must be finite whereas in online identification schemes, this set is theoretically infinite.Nevertheless, recently we observed a non-stochastic learning process when this procedure did not converge for certain infinite sequence of training examples [24].
The probabilistic asymptotic analysis on convergence of the online gradient training algorithms has been conducted in [25]- [33].Several of their results make it possible to employ a constant learning rate [28] [30].To the best of author's knowledge, there are no general results in literature concerning the global convergence properties of training procedures with a fixed learning rate applicable to the case of infinite learning set.
A popular approach to analyze the asymptotic behavior of online gradient algorithms in stochastic case is based on Martingale convergence theory [34].This approach has been exploited by the authors in [24] to derive some local convergence in stochastic framework for standard online gradient algorithms with the constant learning rate.
The difficulties associated with convergence properties of online gradient learning algorithms are how to guarantee the boundedness of the network weights biases assuming the learning process to be theoretically infinite.To overcome these difficulties, the penalty term to an error function has been introduced in [33].Recently we however established in [35] that the global convergence of these algorithms with probability 1 can be achieved without any additional term, at least, in the case when the activation function of the network output layer is linear.
This work has been motivated by the fact that the standard gradient algorithm is widely exploited for online updating the neural network weights in accordance with the gradient-descent principle whereas the following important question related to its ultimate properties remained in part open as yet: when does the sequential procedure based on this algorithm converge if the learning rate is constant?As pointed out in [23], the answer to the question on convergence properties of this standard algorithm which should shed some light on asymptotic features of multilayer neural networks using the gradient-like training technique is the first step toward a full understanding of other more generic training algorithms based on regularization, conjugate gradient, and Newton optimization methods, etc.
Novelty of this paper which extends the basic ideas of [35] to the case where the activation function of the output layer is nonlinear, consists in establishing sufficient conditions under which the gradient algorithm for learning neural networks will globally converge in the sense almost sure for the case when the learning rate can be constant.The proposed approach to deriving these convergence results is based on utilizing the Lyapunov methodology [36].They make it possible to reveal some new features of the multilayer neural networks with nonlinear activation function in output layer which use the online gradient-type training algorithms having a constant learning rate.

Description of Learning Neural Network System: Problem Formulation
Consider the typical three-layer feedforward neural network containing a hidden layer and p inputs, q hidden neurons, and one output neuron.Denote by the weight matrix connecting the input and hidden layers, and define the so-called bias vector 0 w as T 0 01 0 , , , which is the threshold in the hidden-layer output.Further, let T 1 , , , be the weight vector between the hidden and output layers, and 0 ω be the bias in the output layer.As in [33], the activation functions used in the hidden neurons are all the same denoted by : g → R R, and the activation function for the output layer is by adding the column 0 w to W and the extended vector R , and also the function Then the for an input vector T 1 , , , the output vector of hidden layer can be written as is used, and the final output NN y ∈ R of the neural network can be expressed as follows: ( ) ( ) Let ( ) with : p ϕ → R R be an unknown and bounded nonlinearity given over the bounded either finite or infinite sets p X ⊂ R which are depicted in Figure 1 for the case 2 p = .This function needs to be approximated by the neural network (1) via suitable choice of ω  and W  .By virtue of (2) the approximation error depends on x for any fixed ( ) Now, suppose that some complex system to be identified is described at each nth time instant by the equation , ; , , , ; , , Introducing the notation of the extended weight and bias vector ( ) , and considering the Equations ( 5) and ( 6) in conjunction, rewrite the online gradient learning algorithm for updating n θ in a general form (as in [33]) ; , , where ( ) 2 ; , e The problem formulated in this paper consists in analyzing asymptotic properties of the learning neural network system presented above.More certainly, it is required to derive conditions under which the learning procedure will be convergent meaning the existence of a limit in some sense [24].

Preliminaries
Suppose that there is a multilayer neural network described by where θ is some fixed parameter vector.According to [9] [10], the require- ment evaluating an accuracy of the approximation of ( ) , x ψ θ can be satisfied for any 0 ε > via suitable choice of θ and the number of the neurons in its layers.On the other hand, the performance index of the neural network model with a fixed number of these neurons defining its approximation capability might naturally be expressed as follows: ( In fact, the desired (optimal) vector * 0 θ θ = will then be specified from ( 9) as the variable θ minimizing ( ) Nevertheless, all researches which employ online learning procedures in stochastic environment "silently" replace ( ) θ given by (10) as n → ∞ .Now, consider a special case when the unknown function (2) can exactly be approximated by the neural network ( ) In this case called in ( [8], p. 304) by the ideal case, we have ( ) any x from X and, consequently, ( ) If the condition given in identity ( 11) is satisfied, then the learning rate n η in (7) may be constant: see ( [37], sect.3.13).
Note that the property (11) may take place, in particular, when of training examples provided that their number does not exceed the dimension of θ .To understand this fact, according to (11) write the set of K equations with respect to the unknown θ .They are compatible if ( ) to (2) together with the definition of * θ it can be concluded that their solution is just

Some Feature of Multilayer Neural Network
It turns out that if the activation functions g of the hidden layer are nonlinear, then for an arbitrary fixed vector θ ′ there is, at least, one vector θ ′′ such that the network outputs for these different vectors are the same even though the output activation function f is linear, i.e. if ( ) The feature (12) gives that in the presence of nonlinear g there exist, at least, two different * s θ .For example, let 1, 1 p q = = and ( ) ( ) ( ) will also satisfy (12); see [35].Therefore, the set of * s θ will be not one-point if g is nonlinear.

An Observation
To study some asymptotic properties of sequence { }  , where { } n x was chosen as a non-stochastic sequence.It can be observed that in this example, the variable shown in Figure 3(b) has no limit implying that the learning algorithm (7) may not be convergent: in this case, the limit (8) does not exist, see Figure 3(c).Journal of Applied Mathematics and Physics

Sufficient Conditions for the Probabilistic Convergence of Learning Procedure
The following basic assumption concerning { } 0 which is bounded stochastic sequence (since X is bounded) is made: arise randomly in accordance with a probability distribution ( ) P x if X is finite, and with probability density ( ) Within assumption (A1), the expected value (mean) of ( ; , , e y x y x ; , if is finite set, ; , ; , d if is infinite set.
x X x X e y x P x X E e y x e y x p x x X To derive the main theoretical result we need Assumption (A1) and the following additional assumptions: (A2) the identity (11) holds; (A3) the activation functions used in the hidden neurons and output neuron are the same ( ) ( ) ( ) Then the algorithm (7) yields lim 0 n n V →∞ = a.s., where ( ) Related results followed from the Theorem 3' of [36] is satisfied. Next, we are able to present the convergence result summarized in the theorem below.
Theorem.Suppose Assumption (A2) holds.Then the gradient algorithm (7) with a constant learning rate, n η η ≡ , will converge with probability 1 (in the sense that 0 for any initial 0 θ chosen randomly so that the conditions (20) with Θ and τ determined by Θ ≡ Θ and n τ τ ≡ with Θ and τ determined by ( 22) and ( 23), respectively, we can conclude that the conditions 1), 2) of the Key Technical Lemma are satisfied.Applying its Corollary it proves that lim 0 Due to the fact that ( ) ( ) { }

Simulations and a Discussion
To demonstrate theoretical result given in Subsection 4.3, several simulations were conducted.First, we dealt with the same neural network and the same training samples as in ( [33], p. 1052).Namely, they were chosen as follows: Contrary to [33] the learning rate was chosen as 0.01 η = in order to implement the algorithms ( 5), (6) with no penalty term.
Results of two simulation experiments whose durations were 10000 iteration steps are presented in Figure 4

Conclusion
In this paper, some important features of multilayer neural networks which are utilized as nonlinearly parameterized models of unknown nonlinear systems to be identified have been derived.A special case where the nonlinearity can exactly be approximated by a three-layer neural network has been studied.Contrary to previous author's papers we dealt with the neural network having a nonlinear activation function for its output layer.It was shown that if the activation function of the hidden layer is nonlinear, then, for any input variables, there are, at least, two different network parameter vectors under which the network outputs will be the same even though the output activation function is linear.This feature gives that the standard gradient online training algorithm with a constant learning rate may not be convergent if the training sequence is non-stochastic.Nevertheless, provided that this sequence is stochastic, it has theoretically been established that, under certain conditions, such algorithm will converge with probability one.However, ultimate values of network parameters may be different.These facts were confirmed by simulation experiments.
in which nx X ∈ and n y ∈ R are its input and output signals, respectively available for measurement.Based on the infinite sequence of the training examples { } 0 , by (4), the outline learning algorithm for updating the weight and biases in (1) is defined as the standard gradient-descent iteration procedure ( )1 2

Figure 1 .
Figure 1.Training sets: (a) X is an infinite set of xs; (b) X is a finite set of xs.
θ calculated at the nth time instant.Thus, the Equation (7) together with the expression ( ) neural network system necessary to identify the nonlinearity (2).For better understanding the performance of this system, its structure is depicted in Figure2, where the notation

Figure 2 .
Figure 2. Configuration of learning neural network system.
in the non-stochastic case, simulation experiments with the scalar nonlinear system (2) having the nonlinearity

Figure 3
Figure 3 illustrates the results of the one simulation experiment with 0.01 η =

Figure 3 .
Figure 3. Behaviour of learning algorithm (7) in non-stochastic case: (a) inputs n e ; (b) the variable Assumption (A2), result(21) follows. and Figure5in which the components of n θ simulation experiments were also conducted.In contrast with previous experiments, they dealt with an infinite training sets X Namely, the two simulations with the same nonlinear function as in Subsection 4.2 were first conducted, provided that X is the infinite bounded set given by as the stochastic sequence.Namely, it was generated as a pseudorandom i.i.d.sequence.Two numerical examples were considered.In Example 3, the initial values of neural network weights and biases were taken as follows: 0 1 0.529 w = ,