Lyapunov Stability Analysis of Gradient Descent-Learning Algorithm in Network Training

Abstract

The Lyapunov stability theorem is applied to guarantee the convergence and stability of the learning algorithm for several networks. Gradient descent learning algorithm and its developed algorithms are one of the most useful learning algorithms in developing the networks. To guarantee the stability and convergence of the learning process, the upper bound of the learning rates should be investigated. Here, the Lyapunov stability theorem was developed and applied to several networks in order to guaranty the stability of the learning algorithm.

1. Introduction

Science has evolved from an attempt to understand and predict the behavior of the universe and the systems within it. Much of this owes to the development of suitable models, which agree with the observations. These models are either in a symbolic form which the humans use or in mathematical form that are found from physical laws. Most systems are causal, which can be categorized as either static, where the output depends on the current inputs, or dynamic, where the output depends on not only the current inputs but also past inputs and outputs. Many systems also possess unobservable inputs, which cannot be measured, but affect the system’s output, that is, time series systems. These inputs are known as disturbances and aggravate the modeling process.

To cope with the complexity of dynamic systems, there have been significant developments in the field of artificial neural network during last three decades which have been applied for identification and modeling [1–5]. One major issue that instigates for proposing these different types of networks is to predict the dynamic behavior of many complex systems existing in nature. ANN is a powerful method in approximating a nonlinear system and mapping between input and output data [1]. Recently, wavelet neural networks (WNNs) have been introduced [6–10]. Such types of networks employ wavelets as the activation function in a hidden layer. Because of the ability of the localized analysis of wavelets collectively in their frequency and time domains and the learning ability of ANN, the WNN prompts a superior system model for complex and seismic applications. The majority of the applications of wavelet function are limited to a small dimension [11] although WNN can handle large-dimension problems as well [6]. Due to the dynamic behavior of recurrent network, they are suitable in dealing with the modeling of dynamic systems as compared with the static behavior of feed-forward network [12–19]. It has already been shown that recurrent networks are less sensitive to noise with relatively smaller network size and simpler structure. Their long-term prediction property makes them more powerful in dealing with dynamic systems. Recurrent networks are less sensitive to noise because the recurrent network could recognize and generate periodic waves in spite of the existence of a large amount of noise. This means that the network is able to regenerate the original periodic waves in the process of learning the teachers' signals with noises [2]. For unknown dynamic systems, the recurrent network results in a smaller-sized network as compared with the feed-forward network [12, 20]. For the time-series modeling, it generates a simpler structure [15–23] and gives long-term predictions [22, 24]. The recurrent network for system modeling learns and memorizes information in terms of embedded weights [21].

Different methods have been introduced for learning the parameters onnetwork based of the gradient descent. All learning methods like backpropagation-through-time [16, 17] or real-time recurrent learning algorithm [18] can be applied in order to adjust parameters of the feed-forward or recurrent networks. In [19], the quasi-Newton method was applied to improve the rate of convergence. In [9, 23], using the Lyapunov stability theorem, a mathematical way was introduced for calculating the upper bound of the learning rate for recurrent and feed-forward wavelet neural network based on the network parameters. Here, the Lyapunov stability theorem is developed and applied to several networks, and the learning procedure of the proposed networks is considered.

2. Methodology

2.1. Gradient-Descent Algorithm

The Gradient-descent (GD) learning can be achieved by minimizing the performance index 𝐽 as follows:𝐽=12⋅𝑃⋅𝑦^2𝑟⋅^𝑃_𝑝=1𝑌(𝑝)−𝑌(𝑝)²,(2.1) where 𝑦_𝑟=(max^𝑃𝑝=1𝑌(𝑝)−min^𝑃𝑝=1𝑌(𝑝)), 𝑌 is the output of the known network, 𝑌 is the actual data, and 𝑃 is the number of dataset. The reason for using a normalized mean square error is that it provides a universal platform for modeling evaluation irrespective of the application and target value specification while selecting an input to the model.

In the batch-learning scheme employing the 𝑃-data set, achange in any parameter is covered by the following equation:Δ𝜐(𝑞)=^𝑃_𝑝=1Δ_𝑝𝜐(𝑞),(2.2) and the parametric update equation is𝜐(𝑞+1)=𝜐(𝑞)+𝜕𝐽𝜕𝜐.(2.3)

2.2. Lyapunov Method in Analysis of Stability

Consider a dynamic system, which satisfies𝑥̇=𝑓(𝑥,𝑡),𝑥𝑡₀=𝑥₀,𝑥∈𝑅.(2.4)

The equilibrium point 𝑥^∗=0 is stable (in the sense of Lyapunov) at 𝑡=𝑡₀ if for any 𝜀>0 there exists a 𝛿(𝑡₀,𝜀)>0 such that ‖‖𝑥𝑡₀‖‖<𝛿⟹‖𝑥(𝑡)‖<𝜀,∀𝑡≥𝑡₀.(2.5)

Lyapunov Stability Theorem
Let 𝑉(𝑥,𝑡) be a nonnegative function with the derivative 𝑉̇ along the trajectories of the system. Then(i)The origin of the system is locally stable (in the sense of Lyapunov) if 𝑉(𝑥,𝑡) is locally positive definite and −𝑉̇(𝑥,𝑡)≤0 is locally in 𝑥 and for all 𝑡;(ii)The origin of the system is globally uniformly asymptotically stable if 𝑉(𝑥,𝑡) is positive definite and excrescent and −𝑉̇(𝑥,𝑡) is positive definite.
To approve stability analysis of the networks based on GD learning algorithm, we can define discreet function as 𝑉(𝑘)=𝐸(𝑘)=12⋅[𝑒(𝑘)]².(2.6) Change of Lyapunov function is Δ𝑉(𝑘)=𝑉(𝑘+1)−𝑉(𝑘)=12⋅𝑒²(𝑘+1)−𝑒²(𝑘).(2.7)from𝑒(𝑘+1)=𝑒(𝑘)+Δ𝑒(𝑘)⟹𝑒²(𝑘+1)=𝑒²(𝑘)+Δ²𝑒(𝑘)+2⋅𝑒(𝑘)⋅Δ𝑒(𝑘).(2.8) Then Δ𝑉(𝑘)=Δ𝑒(𝑘)⋅𝑒(𝑘)+12⋅Δ𝑒(𝑘).(2.9) Difference of error is Δ𝑒(𝑘)=𝑒(𝑘+1)−𝑒(𝑘)≈𝜕𝑒(𝑘)𝜕𝜐^𝑇⋅Δ𝜐,(2.10) where 𝜐 is the learning parameter and 𝑒(𝑘)=𝑦(̂𝑘)−𝑦(𝑘) is error between output of plant and present output of network Δ𝜐=−𝜂⋅𝜕𝐽𝜕𝜐.(2.11)
By using (2.10) and (2.1) and putting them in (2.3), Δ𝑉(𝑘)=𝜕𝑒(𝑘)𝜕𝜐^𝑇⋅Δ𝜐⋅𝑒(𝑘)+12⋅𝜕𝑒𝜕(𝜐𝑘)^𝑇Δ𝑉(𝑘)=𝜕𝑒(𝑘)⋅Δ𝜐,𝜕𝜐^𝑇⋅−𝜂⋅𝜕𝐸(𝑘)𝜕𝜐⋅𝑒(𝑘)+21⋅𝜕𝑒𝜕(𝜐𝑘)^𝑇⋅−𝜂⋅𝜕𝐸(𝑘)Δ𝑉(𝑘)=𝜕𝑒(𝑘)𝜕𝜐,𝜕𝜐^𝑇⋅(−𝜂)⋅1𝑃⋅𝑦^2𝑟⋅𝑒(𝑘)⋅𝜕𝑦̂(𝑘)⋅𝑒(𝑘)+1𝜕𝜐2⋅𝜕𝑒𝜕(𝜐𝑘)^𝑇⋅(−𝜂)⋅1𝑃⋅𝑦^2𝑟⋅𝑒(𝑘)⋅𝜕𝑦̂(𝑘)Δ𝑉(𝑘)=𝑒₂𝜕𝜐,(𝑘)⋅−𝜕𝑦̂(𝑘)𝜕𝜐^𝑇⋅𝜂⋅1𝑃⋅𝑦^2𝑟⋅𝜕𝑦̂(𝑘)𝜕𝜐+12⋅𝜕𝑦𝜕̂(𝜐𝑘)^𝑇⋅𝜕𝑦̂(𝑘)𝜕𝜐^𝑇⋅𝜂²⋅1𝑃⋅𝑦^2𝑟²⋅𝜕𝑦̂(𝑘)Δ𝑉(𝑘)=−𝑒₂𝜕𝜐²(𝑘)⋅12⋅𝑃⋅𝜂𝑦^2𝑟⋅𝜕𝑦̂(𝑘)𝜕𝜐²⋅2−𝜂𝑃⋅𝑦^2𝑟⋅𝜕𝑦̂(𝑘)𝜕𝜐²,(2.12) where 𝑦_𝑟=(max^𝑃𝑝=1𝑦(𝑝)−min^𝑃𝑝=1𝑦(𝑝)).
Therefore Δ𝑉(𝑘)=−𝜆⋅𝑒²(𝑘),(2.13) where 𝜆=(1/2)⋅(𝜂/(𝑃⋅𝑦^2𝑟))⋅(𝜕𝑦(̂𝑘)/𝜕𝜐)²⋅{2−(𝜂/(𝑃⋅𝑦^2𝑟))⋅(𝜕𝑦(̂𝑘)/𝜕𝜐)²}.
From the Lyapunov stability theorem, the stability is guaranteed if 𝑉(𝑘) is positive and 𝑉(𝑘) is negative. From (2.6), 𝑉(𝑘) is already positive. The condition of stability depends on 𝑉(𝑘) being negative. Therefore, 𝜆>0 is considered for all models.
Because (1/2)⋅(𝜂/(𝑃⋅𝑦^2𝑟))⋅(𝜕𝑦(̂𝑘)/𝜕𝜐)²>0, then the convergence condition is limited to 2−𝜂𝑃⋅𝑦^2𝑟⋅𝜕𝑦̂(𝑘)𝜕𝜐²>0⟹𝜂𝑃⋅𝑦^2𝑟⋅𝜕𝑦̂(𝑘)𝜕𝜐²<2⟹𝜂<2⋅𝑃⋅𝑦^2𝑟(𝜕𝑦̂(𝑘)/𝜕𝜐)².(2.14) The maximum learning rate 𝜂 changes in a fixed range. Since 2⋅𝑃⋅𝑦^2𝑟 does not depend on the model, the value of 𝜂_Max guarantees that the convergence can be found by minimizing the term of |𝜕𝑦(̂𝑘)/𝜕𝜐_𝐼|. Therefore, 0<𝜂<𝜂_Max,(2.15) where 𝜂_Max=(2⋅𝑃⋅𝑦^2𝑟)/Max(𝜕𝑦(̂𝑘)/𝜕𝜐)².

3. Experimental Results

In this section, the proposed stability analysis is applied for some networks. The selected networks are neurofuzzy (ANFIA) [25, 26], Wavelet neurofuzzy, and recurrent wavelet network.

3.1. Example 1: Convergence Theorems of the TSK Neurofuzzy Model

TSKmodel has a linear or nonlinear relationship of inputs 𝑤^𝑚(𝑋) in the output space. The rules of TSK model are in the following way:𝑅^𝑚∶if𝐱is𝐀^mthen𝑦is𝑤^𝑚(𝑋).(3.1) A linear form of 𝑤^𝑚(𝑋) in (3.1) is as follows:𝑤^𝑚(𝑋)=𝑤^𝑚0+𝑤^𝑚1𝑥₁+⋯+𝑤^𝑚𝑛𝑥_𝑛.(3.2) By taking the Gaussian membership function and an equal number of fuzzy sets to the rules with respect to the inputs, the firing strength of rules (3.1) can be written as𝜇_𝐴^𝑚(𝐱)=^𝑛_𝑖=1exp−𝑥^𝑖−𝑥_𝑚𝑖𝜎_𝑚𝑖²,(3.3) where 𝑥_𝑚𝑖 and 𝜎_𝑚𝑖 are the center and standard deviation of the Gaussian membership functions, respectively. By applying the T-norm (product operator) of the membership functions of the premise parts of the rule and the weighted average gravity method for de-fuzzification, the output of the TSK model can be defined as𝑌=∑^𝑀𝑚=1𝜇_𝐴^𝑚(𝐱)⋅𝑤^𝑚(𝐱)∑^𝑀𝑚=1𝜇_𝐴^𝑚(𝐱).(3.4)

Theorem 3.1. The asymptotic learning convergence of TSK neurofuzzy is guaranteed if the learning rate for different learning parameters follows the upper bound as will be mentioned below: 0<𝜂_𝜎0<𝜂_𝑤<2⋅𝑃⋅y²_r,<2⋅𝑃⋅𝑦^2𝑟max_𝑚||𝑤(X)||²⋅2/𝜎^3min²0<𝜂_𝑥,<2⋅𝑃⋅𝑦^2𝑟max_𝑚||𝑤(X)||²⋅2/𝜎^2min².(3.5)

Proof. In equation (2.15) for neurofuzzy models can be written as 0<𝜂_𝜐<2⋅𝑃⋅𝑦^2𝑟||𝜕𝑌_NF/𝜕𝜐||^2max.(3.6) Because 𝛽_𝑚=𝜇_𝐴^𝑚(𝐗)/∑^𝑀𝑚=1𝜇_𝐴^𝑚(𝐗)≤1 for all 𝑚 and since local models have same variables, that is, 𝐗, therefore, from (3.7), (3.5) easily can be derived 𝜕𝑌_NF𝜕𝑤_𝑚0𝜕𝑌_NF=𝛽_𝑚,𝜕𝑤_𝑚𝑖𝜕𝑌_NF=𝑥_𝑖⋅𝛽_𝑚,𝜕𝑥_𝑚𝑖=𝑤_𝑚𝐗^⋅𝛽^𝑚𝜇_𝐴^𝑚⋅1−𝛽_𝑚⋅2⋅𝑥^𝑖−𝑥_𝑚𝑖𝜎^2𝑚𝑖𝜕𝑌_NF,𝜕𝜎_𝑚𝑖=𝑤_𝑚𝐗^⋅𝛽^𝑚𝜇_𝐴^𝑚⋅1−𝛽_𝑚⋅2⋅𝑥^𝑖−𝑥_𝑚𝑖²𝜎^3𝑚𝑖.(3.7)

3.2. Example 2: Convergence Theorems of Recurrent Wavelet Neuron Models

Each neuron model in the proposed recurrent neuron models is summation or multiplication of Sigmoid Activation Function (SAF) and Wavelet Activation Function (WAF) as shown in Figure 1. Morlet wavelet function is considered in the recurrent models. In the series of developing different recurrent networks and neuron models, the proposed neurons’ model is used in a one-hidden-layer feed-forward neural network as shown in Figure 2.

Figure 1 

Summation/product recurrent sigmoid-wavelet neuron model.

Figure 2 

Feed-forward neural network.

The output of feed-forward network is given in   the following equation:𝑌_WNN=^𝐿_𝑙=1𝑊_𝑙⋅𝑦_𝑙,(3.8) where 𝑦_𝑙 is the output of S-W neurons, 𝑊_𝑙 is the weights between hidden neuron and output neurons, and 𝐿 is the number of hidden neuron,𝑦_𝑗(𝑘)=𝑦^𝜃𝑗(𝑘)+𝑦^𝜓𝑗(𝑘).(3.9)

The functions 𝑦^𝜃𝑗 and 𝑦^𝜓𝑗 are output of SAF and WAF for 𝑗th S-W neuron, in the hidden layer, respectively. The functions 𝑦^𝜃𝑗 and 𝑦^𝜓𝑗 are expressed as follow.𝑦^𝜃𝑗(𝑘)=𝜃^𝑛_𝑖=1𝑦𝐶^𝑗𝑆_𝑖⋅𝑥_𝑖(𝑘)_𝜓𝑗,(𝑘)=𝜓^𝑛_𝑖=1𝐶^𝑗𝑊_𝑖⋅𝑥_𝑖(𝑘).(3.10)𝑥_𝑖 is 𝑖th input. 𝐶_𝑆 and 𝐶_𝑊 are weights to input signal for SAF and WAF, in each hidden neuron, respectively.

To prove convergence of the recurrent networks, these facts are needed:Fact 1: let 𝑔(𝑦)=𝑦𝑒^(−𝑦2). Then |𝑔(𝑦)|<1,forall𝑦∈ℜ.Fact 2: let 𝑓(𝑦)=𝑦²𝑒^(−𝑦2). Then |𝑓(𝑦)|<1,forall𝑦∈ℜ.Fact 3: let 𝜃(𝑦)=1/(1+𝑒^−𝑦)be a sigmoid function. Then |𝜃(𝑦)|<1,forall𝑦∈ℜFact 4: let 𝜓_𝑎,𝑏(𝑦)=𝑒^{−((𝑦−𝑏)/𝑎)2}cos(5((𝑦−𝑏)/𝑎)) be a Morlet wavelet function. Then |𝜓_𝑎,𝑏(𝑦)|<1,forall𝑦,𝑎,𝑏∈ℜ.

(a) Summation Sigmoid-Recurrent Wavelet
Suppose 𝑍=∑^𝑛𝑖=1𝐶^𝑗𝑆_𝑖⋅𝑥_𝑖(𝑘) and 𝑆=∑^𝑛𝑖=1𝐶^𝑗𝑊_𝑖⋅𝑥_𝑖(𝑘)+𝑄^𝑗𝑊⋅𝑦^𝑗𝜓(𝑘−1).
From the facts 3 and 4: For parameter 𝑊 in all models 𝜕𝑦̂𝜕𝑊_𝑗=𝑦_𝑗<||𝑦^𝑗𝜓+𝑦^𝑗𝜃||<1+1=2.(3.11) Therefore 0<𝜂_𝑊<(2⋅𝑃⋅𝑦^2𝑟)/2²=(𝑃⋅𝑦^2𝑟)/2.
Differential of output of the model for another learning parameter is 𝜕𝑦̂(𝑘)𝜕𝐶^𝑗𝑊_𝑖=𝑥_𝑖(𝑘)⋅𝑊^𝑗⋅𝜓^^𝑛_𝑖=1<1⋅1⋅|||−2𝐶^𝑗𝑊_𝑖⋅𝑥_𝑖(𝑘)+𝑄^𝑗𝑊⋅𝑦^𝑗𝜓(𝑘−1)𝑎⋅𝑆−𝑎𝑏⋅𝑒^{−((𝑆−𝑏)/𝑎)2}⋅cos5𝑆−𝑏𝑎−𝑒^{−((𝑆−𝑏)/𝑎)2}⋅5<2𝑎⋅sin5𝑆−𝑎𝑏|||𝑎_min⋅1⋅1+5𝑎_min⋅1<7.(3.12) Therefore, 0<𝜂_𝐶𝑊<(2⋅𝑃⋅𝑦^2𝑟)/7²=(2⋅𝑃⋅𝑦^2𝑟)/49𝜕𝑦̂(𝑘)𝜕𝐶^𝑗𝑆_𝑖=𝑥_𝑖(𝑘)⋅𝑊^𝑖⋅𝜃^^𝑛_𝑖=1<1⋅1⋅𝜃(𝑧)⋅(1−𝜃(𝑧𝐶))^𝑗𝑆_𝑖<⋅𝑥1_𝑖⋅(𝑘1)=1.(3.13) Therefore 0<𝜂_𝐶𝑆<(2⋅𝑃⋅𝑦^2𝑟)/1²=2⋅𝑃⋅𝑦^2𝑟𝜕𝑦̂(𝑘)𝜕𝑄^𝑗𝑊=𝑊^𝑗⋅𝑦^𝑗𝜓(𝑘−1)⋅𝜓^^𝑛_𝑖=1<1⋅1⋅|||−2𝐶^𝑗𝑊_𝑖⋅𝑥_𝑖(𝑘)+𝑄^𝑗𝑊⋅𝑦^𝑗𝜓(𝑘−1)𝑎⋅𝑆−𝑎𝑏⋅𝑒^{−((𝑆−𝑏)/𝑎)2}⋅cos5𝑆−𝑏𝑎−𝑒^{−((𝑆−𝑏)/𝑎)2}⋅5<2𝑎⋅sin5𝑆−𝑎𝑏|||𝑎_min⋅1⋅1+5𝑎_min⋅1<7.(3.14) Therefore, 0<𝜂_𝑄𝑊<(2⋅𝑃⋅𝑦^2𝑟)/7²=(2⋅𝑃⋅𝑦^2𝑟)/49.

(b) Multiplication Sigmoid-Recurrent Wavelet
From facts 3 and 4 suppose 𝑍=∑^𝑛𝑖=1𝐶^𝑗𝑆_𝑖⋅𝑥_𝑖(𝑘) and 𝑆=∑^𝑛𝑖=1𝐶^𝑗𝑊_𝑖⋅𝑥_𝑖(𝑘)+𝑄^𝑗𝑊⋅𝑦^𝑗𝜓(𝑘−1).
For parameter 𝑊 in all networks: 𝜕𝑦̂𝜕𝑊^𝑗=𝑦_𝑗=𝑦^𝑗𝜓⋅𝑦^𝑗𝜃<1⋅1<1.(3.15) Therefore, 0<𝜂_𝑊<(2⋅𝑃⋅𝑦^2𝑟)/1<2⋅𝑃⋅𝑦^2𝑟𝜕𝑦̂(𝑘)𝜕𝐶^𝑗𝑊_𝑖=𝑥_𝑖(𝑘)⋅𝑊^𝑗⋅𝜃^𝑛_𝑖=1𝐶^𝑗𝑆_𝑖⋅𝑥_𝑖(𝑘)⋅𝜓^^𝑛_𝑖=1<1⋅1⋅1⋅|||−2𝐶^𝑗𝑊_𝑖⋅𝑥_𝑖(𝑘)+𝑄^𝑗𝑊⋅𝑦^𝑗𝜓(𝑘−1)𝑎⋅𝑆−𝑎𝑏⋅𝑒^{−((𝑆−𝑏)/𝑎)2}⋅cos5𝑆−𝑏𝑎−𝑒^{−((𝑆−𝑏)/𝑎)2}⋅5<2𝑎⋅sin5𝑆−𝑎𝑏|||𝑎_min⋅1⋅1+5𝑎_min⋅1<7.(3.16) Therefore, 0<𝜂_𝐶𝑊<(2⋅𝑃⋅𝑦^2𝑟)/(7)²=(2⋅𝑃⋅𝑦^2𝑟)/49𝜕𝑦̂(𝑘)𝜕𝐶^𝑗𝑆_𝑖=𝑥_𝑖(𝑘)⋅𝑊^𝑗⋅𝜃^^𝑛_𝑖=1𝐶^𝑗𝑆_𝑖⋅𝑥_𝑖(𝑘)⋅𝜓^𝑛_𝑖=1<1⋅1⋅𝜃(𝑍)⋅(1−𝜃(𝑍))⋅1<1⋅1<1.𝐶^𝑗𝑊_𝑖⋅𝑥_𝑖(𝑘)+𝑄^𝑗𝑊⋅𝑦^𝑗𝜓(𝑘−1)(3.17) Therefore, 0<𝜂_𝐶𝑆<(2⋅𝑃⋅𝑦^2𝑟)/(1)²=2⋅𝑃⋅𝑦^2𝑟𝜕𝑦̂(𝑘)𝜕𝑄^𝑗𝑊=𝑊^𝑗⋅𝑦^𝑗𝜓(𝑘−1)⋅𝜃^𝑛_𝑖=1𝐶^𝑗𝑆_𝑖⋅𝑥_𝑖(𝑘)⋅𝜓^^𝑛_𝑖=1<1⋅1⋅1⋅|||−2𝐶^𝑗𝑊_𝑖⋅𝑥_𝑖(𝑘)+𝑄^𝑗𝑊⋅𝑦^𝑗𝜓(𝑘−1)𝑎⋅𝑆−𝑎𝑏⋅𝑒^{−((𝑆−𝑏)/𝑎)2}⋅cos5𝑆−𝑏𝑎−𝑒^{−((𝑆−𝑏)/𝑎)2}⋅5<2𝑎⋅sin5𝑆−𝑎𝑏|||𝑎_min⋅1⋅1+5𝑎_min⋅1<7.(3.18) Therefore, 0<𝜂_𝑄𝑊<(2⋅𝑃⋅𝑦^2𝑟)/(7)²=(2⋅𝑃⋅𝑦^2𝑟)/49

3.3. Example 3: Convergence Theorems of the Wavelet Nuro-Fuzzy (WNF) Model

The consequent part of each fuzzy rule corresponds to a sub-WNN consisting of wavelet with the specified dilation value, where, in the TSK fuzzy model, a linear function of inputs is used while 𝑤^𝑚(𝑋)=𝑌_WNN𝑚. Figure 1 shows the proposed WNN model which uses a combination of sigmoid and wavelet activation functions as a hidden neuron (Figure 2 without recurrent part) in the consequent part of each fuzzy rule.

Theorem 3.2. The asymptotic learning convergence is guaranteed if the learning rate for different learning parameters follows the upper bound as will be mentioned below: 0<𝜂_𝜎<2⋅𝑃⋅𝑦^2𝑟||𝑌_WNN||^2max⋅2/𝜎^3min²0<𝜂_𝑥,<2⋅𝑃⋅𝑦^2𝑟||𝑌_WNN||^2max⋅2/𝜎^2min²0<𝜂_𝑤,<2⋅𝑃⋅𝑦^2𝑟||𝜕𝑌_WNN/𝜕𝑤||^2max0<𝜂_𝐶𝑆,<2⋅𝑃⋅𝑦^2𝑟||𝜕𝑌_WNN/𝜕𝐶_𝑆||^2max0<𝜂_𝐶𝑊,<2⋅𝑃⋅𝑦^2𝑟||𝜕𝑌_WNN/𝜕𝐶_𝑊||^2max,(3.19)

where 𝜂_𝑤, 𝜂_𝐶𝑁, or 𝜂_𝐶𝑊 and 𝜂_𝜎 or 𝜂_𝑥 are the parameters’ learning rates of the consequent and the premise parts of the fuzzy rules. 𝐶_𝑆 and 𝐶_𝑊 are weights to inputs, signal for sigmoid and wavelet activation functions of local WNNs, in each hidden neuron, respectively. 𝑥_𝑚 and 𝜎_𝑚 are the center and standard deviation of the Gaussian membership functions of rule number m in WNF model, respectively.

Proof. In equation (2.15) for WNF models can be written as 0<𝜂_𝜐<2⋅𝑃⋅𝑦^2𝑟||𝜕𝑌_WNF/𝜕𝜐||^2max𝜕𝑌_WNF,𝜕𝑤=𝛽^𝑚⋅𝜕𝑌^WNN𝑚𝜕𝑌𝜕𝑤_WNF,𝜕𝐶_𝑁=𝛽_𝑚⋅𝜕𝑌^WNN𝑚𝜕𝐶_𝑁𝜕𝑌_WNF,𝜕𝐶_𝑊=𝛽_𝑚⋅𝜕𝑌^WNN𝑚𝜕𝐶_𝑊.(3.20) Because 𝛽_𝑚=𝜇_𝐴^𝑚(𝐗)/∑^𝑀𝑚=1𝜇_𝐴^𝑚(𝐗)≤1 for all 𝑚, therefore (3.13) to (3.15) are easily derived.
From (2.15) and (3.4) for parameters 𝜎 or 𝑥,there is 𝜕𝑌_WNF𝜕𝜎=𝑌^WNN𝑚⋅𝛽^𝑚𝜇_𝐴^𝑚⋅1−𝛽_𝑚⋅2⋅𝑥^𝑖−𝑥_𝑚𝑖²𝜎^3𝑚𝑖=𝑌_WNN𝑚⋅1−𝛽^𝑚∑^𝑀𝑚=1𝜇_𝐴^𝑚⋅2⋅𝑥^𝑖−𝑥_𝑚𝑖²𝜎^3𝑚𝑖𝜕𝑌_WNF,𝜕𝑥=𝑌^WNN𝑚⋅𝛽^𝑚𝜇_𝐴^𝑚⋅1−𝛽_𝑚⋅2⋅𝑥^𝑖−𝑥_𝑚𝑖𝜎^2𝑚𝑖=𝑌_WNN𝑚⋅1−𝛽^𝑚∑^𝑀𝑚=1𝜇_𝐴^𝑚⋅2⋅𝑥^𝑖−𝑥_𝑚𝑖𝜎^2𝑚𝑖(3.21) and therefore (3.19) arederived.

4. Conclusion

In this paper, a developed Lyapunov stability theorem was applied to guarantee the convergence of the gradient-descent learning algorithm in network training. The experimental examples showed that the upper bound of the learning parameter could be easily considered using this theorem. So, an adaptive learning algorithm can guaranty the fast and stable learning procedure.