J. Cent. South Univ. Technol. (2007)05-0685-05

DOI: 10.1007/s11771-007-0131-z                                                                                  

Adaptive learning with guaranteed stability for

discrete-time recurrent neural networks

DENG Hua(邓  华)¹, WU Yi-hu(吴义虎)², DUAN Ji-an(段吉安)¹

(1. School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China;

2. School of Automobile and Mechanical Engineering, Changsha University of Science and Technology,

 Changsha 410076, China)

Abstract：To avoid unstable learning, a stable adaptive learning algorithm was proposed for discrete-time recurrent neural networks. Unlike the dynamic gradient methods, such as the backpropagation through time and the real time recurrent learning, the weights of the recurrent neural networks were updated online in terms of Lyapunov stability theory in the proposed learning algorithm, so the learning stability was guaranteed. With the inversion of the activation function of the recurrent neural networks, the proposed learning algorithm can be easily implemented for solving varying nonlinear adaptive learning problems and fast convergence of the adaptive learning process can be achieved. Simulation experiments in pattern recognition show that only 5 iterations are needed for the storage of a 15×15 binary image pattern and only 9 iterations are needed for the perfect realization of an analog vector by an equilibrium state with the proposed learning algorithm.

Key words: recurrent neural networks; adaptive learning; nonlinear discrete-time systems; pattern recognition                

1 Introduction

Recurrent neural networks (RNNs) are among the most useful neural networks and have played important roles for discrete-time nonlinear processes in pattern recognition, modeling, adaptive control and adaptive filtering^[1-8]. To train RNNs, two popular dynamic gradient methods, the backpropagation through time (BPTT)^[9] and the real time recurrent learning (RTRL)^[10], are frequently adopted. It is known that such on-line updating methods can be advantageous in problems that need only off-line training, e.g., for the standard backpropagation algorithm the on-line update is generally faster than the batch update^[11]. However, if the weights of RNNs are adjusted on-line with such dynamic gradient methods, special consideration must be given to the stability of the adaptation because it is not ensured^[12].

To ensure stable learning for RNNs, JIN and GUPTA^[13] proposed a stable dynamic learning algorithm for discrete-time RNNs. The dynamic learning algorithm was based on the gradient descent method and named as dynamic back propagation (DBP) by NARENDRA and PARTHASARATHY^[14]. By introducing explicit stability conditions into the weight updating algorithms, a checking phase was then added by JIN and GUPTA^[13] to the learning process to guarantee the asymptotic stability of the equilibrium points during such a DBP learning procedure. However, long convergence time of such a stable DBP is often needed. In Ref.[7], a normalized RTRL algorithm that outperforms the standard RTRL algorithm was proposed. The stability of the normalized RTRL algorithm was proven by removing the second- and higher-order partial derivatives of the Taylor series expansion of the RNN output error and assuming that the weight updating error of a RNN was statistically independent of all variables with all on-line updated parameters^[8]. Obviously, too many factors are neglected and the assumption is very strict.

In this paper, to solve varying nonlinear adaptive learning problems, a stable adaptive learning algorithm was proposed for discrete-time recurrent neural networks. Unlike the previous methods, the weights of the RNNs were updated online in terms of Lyapunov stability theory. The weight updating algorithm converged fast and was easily implemented. Two simulation examples in pattern recognition were also presented to validate the theoretical findings.

2 RNN and adaptive learning

2.1 RNN models

The discrete-time RNN model considered in this study has the following form:

,  i=1,2, …, n      (1)

where

,

  ,  ,

,

U₀(k-1)= [u(k-1), …, u(k-d₀)]^T,

U₁(k-1)= [, …, ]^T,

U_n(k-1)= [, …, ]^T,

0＜α＜1, 0＜|β_i|＜+∞,

 is the ith output and u(k) is an external input of the RNN, d denotes that the RNN has d inputs, α_i and β_i are undetermined constants, W_i and W_0i are weights and thresholds and σ(?) is the activation function of the RNN with σ(u)=tanh (u). The neural network model (1) is a combination of the RNN model in Ref.[13] and the recurrent NARMA model in Ref.[8]. Model (1) is more general and thus can be used to solve a wide class of nonlinear discrete-time problems.

The following properties are associated with the RNN model (1), which will be used in the convergence proof of learning algorithms.

Property 1:  in model (1) is bounded for all k.

Property 2: The region of the equilibrium points of model (1) is the n-dimensional hypercube^[13] as follows:

.

Property 3: -1＜＜1.                                                                                                                                                                                    

Property 1 can be verified by noticing that 0＜α_i＜1, 0＜|β_i|＜+∞ and -1＜σ(v_i(k))＜1. This property shows that there are no unstable equilibrium states in model (1) if conditions 0＜α_i＜1 and 0＜|β_i|＜+∞ are satisfied. Property 2 shows the region of the equilibrium points and the approximation capability of the RNN. The approximation performance of the RNN cannot be guaranteed when the states of a nonlinear discrete-time process are outside the region. Therefore, parameters α_i and β_i must be properly chosen so that the region is large enough to include all the states of the nonlinear process. Property 3 will be used in the convergence proof of the proposed adaptive learning algorithm only. To prove property 3, re-write model (1) as

.

Since -1＜σ(v_i(k))＜1,  the result is immediate.

2.3    Adaptive learning algorithm

Define the approximation error as

          (2)

where  y_i(k) is the ith target signal. The following learning algorithm is proposed to update the weights of the RNN model (1), which can guarantee the learning stability.

 where i=1,2,…, n,  ,

, 0＜η_i≤1 and 0≤μ_i＜1.

3 Stability and performance analysis

Stability and performance of the closed-loop system with RNN model (1) and weight updating law (3) are given in Theorem 1.

Theorem 1: For given RNN model (1) and input vector Z(k-1) of the RNN, if  then adaptive learning algorithm (3) guarantees that e_i(k)→0 as k→∞.

Proof  First, it will be proven that the approximation error (2) approaches zero.

Choose a Lyapnuov candidate . The first difference of V(k) is given as

  (4)

Using model (1), one has

       (5)

Substituting the RNN adaptive learning law (3) into Eqn.(5), one obtains

     -

(6)

Using Eqn.(6), (4) becomes

          (7)

As 0≤μ_i＜1, from Lyapunov stability theory, Eqn.(7) shows that when .

Second, it will be proven that the RNN weights in the closed-loop system are bounded.

Re-write Eqn.(3) as follows:

                    (8)

Combining Eqn.(8) yields

   (9)

where

          ,

         .

Choosing a non-singular matrix, one obtains      

As 0＜η_i＜1, all eigenvalues of matrix A₁ are in the interval (-1, 1) and thus A is stable. On the other hand, in terms of Eqn.(6),  in Eqn.(8) can be written as

           (10)

By property 3, -1＜＜1, then exists. Therefore,  0＜＜+∞ and the norm of B is bounded.  As A is stable and B is bounded, the boundedness of W_i(k) and W_0i(k) in Eqn.(9) is guaranteed, leading to the boundedness of W_i(k) and W_0i(k) in Eqn.(3).

Remark 1: In RNN model (1), α_i is the time-constant or linear feedback gain of the ith neuron^[13]. Usually, large time-constant result in that model (1) has slow dynamics. On the other hand, the capacity of the RNN model (1) decreases when α_i is too small. Therefore, typical value of the parameter α_i in adaptive learning algorithm (3) is between 0.1 and 0.4. After α_i is given, the parameter β_i is chosen to ensure that  and thus theorem 1 holds.

Remake 2: In adaptive learning algorithm (3), η_i is associated with the learning rate of RNN weights and μ_i is associated with the convergence rate of the RNN output error. Generally, during adaptive learning, the oscillation of the weights becomes larger as η_i increases and the oscillation of the RNN output error becomes larger as μ_i decreases. However, the learning stability is always guaranteed when 0＜η_i≤1 and 0＜μ_i≤1. For most learning problems, the learning rate and the oscillation of the learning process must be considered together to ensure a fast and stable learning process. Simulation experiments show that typical values of the parameters η_i and μ_i in adaptive learning algorithm (3) can be given between 0.1 and 0.5.

4　Simulation experiments

Example 1 (Binary image storage): A binary image pattern storage process^[13] is studied here to illustrate the applicability of the proposed learning algorithm. As shown in Fig.1(a), a 15×15 binary image is used as the target pattern. The recurrent neural network is described by the following equation:

i, j=1, 2, …, 15                 (11)

where  .

In RNN model (11), the double subscripts represent the two-dimension binary image.  is a state of the neuron(i, j), which corresponds to the image unit (i, j). is a weight between the neuron(i, j) and the neuron(h, l), and is a threshold of the neuron. In this example, the binary target pattern is desired to be stored directly at a stable equilibrium point of the network. For the initial stability of the network, all the iterative initial values of the weights and thresholds were chosen randomly in the interval [-0.001, 0.001] as in Ref.[13]. The parameters in Eqn.(11) and in the weight updating law (3) were chosen as ,,, . To observe the change of the binary image during the learning process, the pattern  was employed to represent the binary image at the iterative instant k, where  if ≥0.9, and y_i,j=-1 if ＜0.9.

Using the proposed adaptive learning algorithm, the dynamic learning process was completed at the iterative instant 5, and the 15×15 binary patterns were perfectly stored at a stable equilibrium point of the network. The proposed adaptive learning algorithm is very fast comparing with the learning algorithms in Ref.[13], in which 300 iterations are needed to complete the dynamic learning process. The recovered binary images at some iterative instant are depicted in Figs.1(b) and (c), and the error index and weight curves during the dynamic learning process are given in Fig.2. Both Figs.1 and 2 show that the proposed learning method has a very fast convergence speed and can be used effectively for large-scale pattern storage problems.

Fig.1 Target pattern and recovered binary images at different iterative instants

(a) Target pattern; (b) k = 2-4; (c) k = 5

Fig.2 Error index and weight curves for example 1 during adaptive learning process

To check the stable equilibrium point of the network, the target pattern in Fig.1(a) is distorted by randomly and independently reversing each pixel of the pattern from +1 to -1 and vice versa with a probability of 0.1 as shown in Fig.3(a). Then the corrupted pattern is used as a probe for the trained RNN. After 4 iterations, the RNN converged onto the exactly correct form of the target pattern. The recollection of the corrupted target pattern is shown in Fig.3. Furthermore, it is found that, the larger the probability of reversing each pixel of the pattern is, the more the iterations are needed to converge onto the target pattern. For instance, 6 iterations are needed with a probability of 0.25 and 10 iterations with a probability of 0.40.

Fig.3 Recollection of corrupted target pattern

(a) Corrupted target pattern; (b) k=2; (c) k=3; (d) k=4

Example 2 (Analog vector storage): A two- dimensional analog vector storage problem^[13] is considered in this example. The target pattern is t =   [0.5, 0.8]^T. A two-neuron model is employed, which is described as

                                                                            　　　　　　    (12)

where x₁ and x₂ are the states of the network. Two nonlinear equations are also introduced as follows, which are identical to that in Ref.[13].

      (13)

The objective of the simulation is to realize the target pattern t by a steady output vector y=[y₁, y₂]^T, where  and , and x=[x₁, x₂]^T is an equilibrium state of the network. In Ref.[13], the initial values of the weights and thresholds were chosen randomly at the interval [-0.5, 0.5]. The parameters in Eqn.(12) and in the weight updating law (3) were chosen as α_i=0.2，β_i=1，η_i=0.25，μ_i=0.1, i = 1, 2. Using the proposed adaptive learning method, the analog vector was perfectly realized by an equilibrium state with x₁= 0.707 0 and x₂ = 0.304 8, and the set of weights and thresholds were computed as

， .

The error index and weight curves during the dynamic learning process are given in Fig.4, and the trajectory of the equilibrium points at different iterative instants is depicted in Fig.5. Fig.4 shows that only 9 iterations are needed to complete the dynamic learning process. However, using the methods in Ref.[13], the dynamic learning process was completed at a learning time k = 160 for the same storage problem. This example also illustrates that the proposed adaptive learning algorithm is stable and its convergence is fast.

Fig.4 Error index and weight curves for example 2 during adaptive learning process

Fig. 5 Phase plane diagram of equilibrium point of network during adaptive learning (global stable equilibrium point is [0.7070, 0.3048])

5 Conclusions

1)    The learning stability of the proposed adaptive learning algorithm can be guaranteed as the weight updating law of the RNNs is derived based on Lyapunov stability theory.

2)    The proposed stable adaptive learning algorithm can be easily implemented with fast convergence for discrete-time recurrent neural networks.

3)    Simulation experiments show that only 5 iterations are needed for the storage of a 15?15 binary image pattern and only 9 iterations are needed for the perfect realization of an analog vector by using the proposed learning algorithm. These results also validate the theoretical findings.

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Foundation item: Project(50276005) supported by the National Natural Science Foundation of China; Projects (2006CB705400, 2003CB716206) supported by National Basic Research Program of China

Received date: 2007-03-20; Accepted date: 2007-04-23

Corresponding author: DENG Hua, Professor, PhD; Tel: +86-731-8836499; E-mail: hdeng@csu.edu.cn

(Edited by ZHAO Jun)