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Fundamentals Adaptive systems are at the very core of modern digital signal processing. There are many reasons for this, foremost amongst these is that adaptive ﬁltering, prediction or identiﬁcation do not require explicit a priori statistical knowledge of the input data. Adaptive systems are employed in numerous areas such as biomedicine, communications, control, radar, sonar and video processing (Haykin 1996a).

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  1. Recurrent Neural Networks for Prediction Authored by Danilo P. Mandic, Jonathon A. Chambers Copyright c 2001 John Wiley & Sons Ltd ISBNs: 0-471-49517-4 (Hardback); 0-470-84535-X (Electronic) 2 Fundamentals 2.1 Perspective Adaptive systems are at the very core of modern digital signal processing. There are many reasons for this, foremost amongst these is that adaptive ﬁltering, prediction or identiﬁcation do not require explicit a priori statistical knowledge of the input data. Adaptive systems are employed in numerous areas such as biomedicine, communica- tions, control, radar, sonar and video processing (Haykin 1996a). 2.1.1 Chapter Summary In this chapter the fundamentals of adaptive systems are introduced. Emphasis is ﬁrst placed upon the various structures available for adaptive signal processing, and includes the predictor structure which is the focus of this book. Basic learning algo- rithms and concepts are next detailed in the context of linear and nonlinear structure ﬁlters and networks. Finally, the issue of modularity is discussed. 2.2 Adaptive Systems Adaptability, in essence, is the ability to react in sympathy with disturbances to the environment. A system that exhibits adaptability is said to be adaptive. Biological systems are adaptive systems; animals, for example, can adapt to changes in their environment through a learning process (Haykin 1999a). A generic adaptive system employed in engineering is shown in Figure 2.1. It consists of • a set of adjustable parameters (weights) within some ﬁlter structure; • an error calculation block (the diﬀerence between the desired response and the output of the ﬁlter structure); • a control (learning) algorithm for the adaptation of the weights. The type of learning represented in Figure 2.1 is so-called supervised learning, since the learning is directed by the desired response of the system. Here, the goal
  2. 10 ADAPTIVE SYSTEMS Comparator _ Filter + Structure Σ Desired Input Signal Response Control Algorithm Error Figure 2.1 Block diagram of an adaptive system is to adjust iteratively the free parameters (weights) of the adaptive system so as to minimise a prescribed cost function in some predetermined sense. 1 The ﬁlter structure within the adaptive system may be linear, such as a ﬁnite impulse response (FIR) or inﬁnite impulse response (IIR) ﬁlter, or nonlinear, such as a Volterra ﬁlter or a neural network. 2.2.1 Conﬁgurations of Adaptive Systems Used in Signal Processing Four typical conﬁgurations of adaptive systems used in engineering are shown in Figure 2.2 (Jenkins et al. 1996). The notions of an adaptive ﬁlter and adaptive system are used here interchangeably. For the system identiﬁcation conﬁguration shown in Figure 2.2(a), both the adap- tive ﬁlter and the unknown system are fed with the same input signal x(k). The error signal e(k) is formed at the output as e(k) = d(k) − y(k), and the parameters of the adaptive system are adjusted using this error information. An attractive point of this conﬁguration is that the desired response signal d(k), also known as a teaching or training signal, is readily available from the unknown system (plant). Applications of this scheme are in acoustic and electrical echo cancellation, control and regulation of real-time industrial and other processes (plants). The knowledge about the system is stored in the set of converged weights of the adaptive system. If the dynamics of the plant are not time-varying, it is possible to identify the parameters (weights) of the plant to an arbitrary accuracy. If we desire to form a system which inter-relates noise components in the input and desired response signals, the noise cancelling conﬁguration can be implemented (Figure 2.2(b)). The only requirement is that the noise in the primary input and the reference noise are correlated. This conﬁguration subtracts an estimate of the noise from the received signal. Applications of this conﬁguration include noise cancellation 1 The aim is to minimise some function of the error e. If E[e2 ] is minimised, we consider minimum mean squared error (MSE) adaptation, the statistical expectation operator, E[ · ], is due to the random nature of the inputs to the adaptive system.
  3. FUNDAMENTALS 11 y(k) x(k) Adaptive y(k) Adaptive Filter Filter _ _ e(k) e(k) Σ Σ + + d(k) x(k) Unknown d(k) N1(k) s(k) + No(k) Input System Output Reference input Primary input (a) System identiﬁcation conﬁguration (b) Noise cancelling conﬁguration Unknown Adaptive y(k) System Filter x(k) Adaptive y(k) _ Delay e(k) Filter Σ _ + e(k) Σ d(k) + Delay d(k) x(k) (c) Prediction conﬁguration (d) Inverse system conﬁguration Figure 2.2 Conﬁgurations for applications of adaptive systems in acoustic environments and estimation of the foetal ECG from the mixture of the maternal and foetal ECG (Widrow and Stearns 1985). In the adaptive prediction conﬁguration, the desired signal is the input signal advanced relative to the input of the adaptive ﬁlter, as shown in Figure 2.2(c). This conﬁguration has numerous applications in various areas of engineering, science and technology and most of the material in this book is dedicated to prediction. In fact, prediction may be considered as a basis for any adaptation process, since the adaptive ﬁlter is trying to predict the desired response. The inverse system conﬁguration, shown in Figure 2.2(d), has an adaptive system cascaded with the unknown system. A typical application is adaptive channel equal- isation in telecommunications, whereby an adaptive system tries to compensate for the possibly time-varying communication channel, so that the transfer function from the input to the output of Figure 2.2(d) approximates a pure delay. In most adaptive signal processing applications, parametric methods are applied which require a priori knowledge (or postulation) of a speciﬁc model in the form of diﬀerential or diﬀerence equations. Thus, it is necessary to determine the appropriate model order for successful operation, which will underpin data length requirements. On the other hand, nonparametric methods employ general model forms of integral
  4. 12 GRADIENT-BASED LEARNING ALGORITHMS s(k) x(k) Adaptive y(k) Zero d(k) Channel Memory Equaliser Nonlinearity _ + Σ Figure 2.3 Block diagram of a blind equalisation structure equations or functional expansions valid for a broad class of dynamic nonlinearities. The most widely used nonparametric methods are referred to as the Volterra–Wiener approach and are based on functional expansions. 2.2.2 Blind Adaptive Techniques The presence of an explicit desired response signal, d(k), in all the structures shown in Figure 2.2 implies that conventional, supervised, adaptive signal processing tech- niques may be applied for the purpose of learning. When no such signal is available, it may still be possible to perform learning by exploiting so-called blind, or unsuper- vised, methods. These methods exploit certain a priori statistical knowledge of the input data. For a single signal, this knowledge may be in the form of its constant mod- ulus property, or, for multiple signals, their mutual statistical independence (Haykin 2000). In Figure 2.3 the structure of a blind equaliser is shown, notice the desired response is generated from the output of a zero-memory nonlinearity. This nonlinear- ity is implicitly being used to test the higher-order (i.e. greater than second-order) statistical properties of the output of the adaptive equaliser. When ideal convergence of the adaptive ﬁlter is achieved, the zero-memory nonlinearity has no eﬀect upon the signal y(k) and therefore y(k) has identical statistical properties to that of the channel input s(k). 2.3 Gradient-Based Learning Algorithms We provide a brief introduction to the notion of gradient-based learning. The aim is to update iteratively the weight vector w of an adaptive system so that a nonnegative error measure J ( · ) is reduced at each time step k, J (w + ∆w) < J (w), (2.1) where ∆w represents the change in w from one iteration to the next. This will gener- ally ensure that after training, an adaptive system has captured the relevant properties of the unknown system that we are trying to model. Using a Taylor series expansion
  5. FUNDAMENTALS 13 x1 w1 =10 x2 w2 =1 Σ y x3 w3 =0.1 w4 =0.01 x4 Figure 2.4 Example of a ﬁlter with widely diﬀering weights to approximate the error measure, we obtain 2 ∂J (w) J (w) + ∆w + O(w2 ) < J (w). (2.2) ∂w This way, with the assumption that the higher-order terms in the left-hand side of (2.2) can be neglected, (2.1) can be rewritten as ∂J (w) ∆w < 0. (2.3) ∂w From (2.3), an algorithm that would continuously reduce the error measure on the run, should change the weights in the opposite direction of the gradient ∂J (w)/∂w, i.e. ∂J ∆w = −η , (2.4) ∂w where η is a small positive scalar called the learning rate, step size or adaptation parameter. Examining (2.4), if the gradient of the error measure J (w) is steep, large changes will be made to the weights, and conversely, if the gradient of the error measure J (w) is small, namely a ﬂat error surface, a larger step size η may be used. Gradient descent algorithms cannot, however, provide a sense of importance or hierarchy to the weights (Agarwal and Mammone 1994). For example, the value of weight w1 in Figure 2.4 is 10 times greater than w2 and 1000 times greater than w4 . Hence, the component of the output of the ﬁlter within the adaptive system due to w1 will, on the average, be larger than that due to the other weights. For a conventional gradient algorithm, however, the change in w1 will not depend upon the relative sizes of the coeﬃcients, but the relative sizes of the input data. This deﬁciency provides the motivation for certain partial update gradient-based algorithms (Douglas 1997). It is important to notice that gradient-descent-based algorithms inherently forget old data, which leads to a problem called vanishing gradient and has particular importance for learning in ﬁlters with recursive structures. This issue is considered in more detail in Chapter 6. 2 The explanation of the O notation can be found in Appendix A.
  6. 14 A GENERAL CLASS OF LEARNING ALGORITHMS 2.4 A General Class of Learning Algorithms To introduce a general class of learning algorithms and explain in very crude terms relationships between them, we follow the approach from Guo and Ljung (1995). Let us start from the linear regression equation, y(k) = xT (k)w(k) + ν(k), (2.5) where y(k) is the output signal, x(k) is a vector comprising the input signals, ν(k) is a disturbance or noise sequence, and w(k) is an unknown time-varying vector of weights (parameters) of the adaptive system. Variation of the weights at time k is denoted by n(k), and the weight change equation becomes w(k) = w(k − 1) + n(k). (2.6) Adaptive algorithms can track the weights only approximately, hence for the following ˆ analysis we use the symbol w. A general expression for weight update in an adaptive algorithm is w(k + 1) = w(k) + ηΓ (k)(y(k) − xT (k)w(k)), ˆ ˆ ˆ (2.7) where Γ (k) is the adaptation gain vector, and η is the step size. To assess how far an adaptive algorithm is from the optimal solution we introduce the weight error vector, ˘ w(k), and a sample input matrix Σ(k) as w(k) = w(k) − w(k), ˘ ˆ Σ(k) = Γ (k)xT (k). (2.8) Equations (2.5)–(2.8) yield the following weight error equation: w(k + 1) = (I − ηΣ(k))w(k) − ηΓ (k)ν(k) + n(k + 1). ˘ ˘ (2.9) For diﬀerent gains Γ (k), the following three well-known algorithms can be obtained from (2.7). 3 1. The least mean square (LMS) algorithm: Γ (k) = x(k). (2.10) 2. Recursive least-squares (RLS) algorithm: Γ (k) = P (k)x(k), (2.11) 1 P (k − 1)x(k)x (k)P (k − 1) T P (k) = P (k − 1) − η . (2.12) 1−η 1 − η + ηxT (k)P (k − 1)x(k) 3. Kalman ﬁlter (KF) algorithm (Guo and Ljung 1995; Kay 1993): P (k − 1)x(k) Γ (k) = , (2.13) R + ηxT (k)P (k − 1)x(k) ηP (k − 1)x(k)xT (k)P (k − 1) P (k) = P (k − 1) − + ηQ. (2.14) R + ηxT (k)P (k − 1)x(k) 3 Notice that the role of η in the RLS and KF algorithm is diﬀerent to that in the LMS algorithm. For RLS and KF we may put η = 1 and introduce a forgetting factor instead.
  7. FUNDAMENTALS 15 The KF algorithm is the optimal algorithm in this setting if the elements of n(k) and ν(k) in (2.5) and (2.6) are Gaussian noises with a covariance matrix Q > 0 and a scalar value R > 0, respectively (Kay 1993). All of these adaptive algorithms can be referred to as sequential estimators, since they reﬁne their estimate as each new sample arrives. On the other hand, block-based estimators require all the measurements to be acquired before the estimate is formed. Although the most important measure of quality of an adaptive algorithm is gen- erally the covariance matrix of the weight tracking error E[w(k)wT (k)], due to the ˘ ˘ statistical dependence between x(k), ν(k) and n(k), precise expressions for this covari- ance matrix are extremely diﬃcult to obtain. To undertake statistical analysis of an adaptive learning algorithm, the classical approach is to assume that x(k), ν(k) and n(k) are statistically independent. Another assumption is that the homogeneous part of (2.9) w(k + 1) = (I − ηΣ(k))w(k) ˘ ˘ (2.15) and its averaged version E[w(k + 1)] = (I − ηE[Σ(k)])E[w(k)] ˘ ˘ (2.16) are exponentially stable in stochastic and deterministic senses (Guo and Ljung 1995). 2.4.1 Quasi-Newton Learning Algorithm The quasi-Newton learning algorithm utilises the second-order derivative of the objec- tive function 4 to adapt the weights. If the change in the objective function between iterations in a learning algorithm is modelled with a Taylor series expansion, we have ∆E(w) = E(w + ∆w) − E(w) ≈ (∇w E(w))T ∆w + 1 ∆wT H∆w. 2 (2.17) After setting the diﬀerential with respect to ∆w to zero, the weight update equation becomes ∆w = −H −1 ∇w E(w). (2.18) The Hessian H in this equation determines not only the direction but also the step size of the gradient descent. To conclude: adaptive algorithms mainly diﬀer in their form of adaptation gains. The gains can be roughly divided into two classes: gradient-based gains (e.g. LMS, quasi-Newton) and Riccati equation-based gains (e.g. KF and RLS). 2.5 A Step-by-Step Derivation of the Least Mean Square (LMS) Algorithm Consider a set of input–output pairs of data described by a mapping function f : d(k) = f (x(k)), k = 1, 2, . . . , N. (2.19) 4 The term objective function will be discussed in more detail later in this chapter.
  8. 16 A STEP-BY-STEP DERIVATION OF THE LMS ALGORITHM x(k) x(k-1) x(k-2) x(k-N+1) z -1 z -1 z -1 z -1 w1(k) w2(k) w3(k) wN (k) y(k) Figure 2.5 Structure of a ﬁnite impulse response ﬁlter The function f ( · ) is assumed to be unknown. Using the concept of adaptive systems explained above, the aim is to approximate the unknown function f ( · ) by a function F ( · , w) with adjustable parameters w, in some prescribed sense. The function F is deﬁned on a system with a known architecture or structure. It is convenient to deﬁne an instantaneous performance index, J(w(k)) = [d(k) − F (x(k), w(k))]2 , (2.20) which represents an energy measure. In that case, function F is most often just the inner product F = xT (k)w(k) and corresponds to the operation of a linear FIR ﬁlter structure. As before, the goal is to ﬁnd an optimisation algorithm that minimises the cost function J(w). The common choice of the algorithm is motivated by the method of steepest descent, and generates a sequence of weight vectors w(1), w(2), . . . , as w(k + 1) = w(k) − ηg(k), k = 0, 1, 2, . . . , (2.21) where g(k) is the gradient vector of the cost function J(w) at the point w(k) ∂J(w) g(k) = . (2.22) ∂w w=w(k) The parameter η in (2.21) determines the behaviour of the algorithm: • for η small, algorithm (2.21) converges towards the global minimum of the error performance surface; • if the value of η approaches some critical value ηc , the trajectory of convergence on the error performance surface is either oscillatory or overdamped; • if the value of η is greater than ηc , the system is unstable and does not converge. These observations can only be visualised in two dimensions, i.e. for only two param- eter values w1 (k) and w2 (k), and can be found in Widrow and Stearns (1985). If the approximation function F in the gradient descent algorithm (2.21) is linear we call such an adaptive system a linear adaptive system. Otherwise, we describe it as a nonlinear adaptive system. Neural networks belong to this latter class. 2.5.1 The Wiener Filter Suppose the system shown in Figure 2.1 is modelled as a linear FIR ﬁlter (shown in Figure 2.5), we have F (x, w) = xT w, dropping the k index for convenience. Con- sequently, the instantaneous cost function J(w(k)) is a quadratic function of the
  9. FUNDAMENTALS 17 weight vector. The Wiener ﬁlter is based upon minimising the ensemble average of this instantaneous cost function, i.e. JWiener (w(k)) = E[[d(k) − xT (k)w(k)]2 ] (2.23) and assuming d(k) and x(k) are zero mean and jointly wide sense stationary. To ﬁnd the minimum of the cost function, we diﬀerentiate with respect to w and obtain ∂JWiener = −2E[e(k)x(k)], (2.24) ∂w where e(k) = d(k) − xT (k)w(k). At the Wiener solution, this gradient equals the null vector 0. Solving (2.24) for this condition yields the Wiener solution, w = R−1 r x,d , x,x (2.25) where Rx,x = E[x(k)xT (k)] is the autocorrelation matrix of the zero mean input data x(k) and r x,d = E[x(k)d(k)] is the crosscorrelation between the input vector and the desired signal d(k). The Wiener formula has the same general form as the block least-squares (LS) solution, when the exact statistics are replaced by temporal averages. The RLS algorithm, as in (2.12), with the assumption that the input and desired response signals are jointly ergodic, approximates the Wiener solution and asymptot- ically matches the Wiener solution. More details about the derivation of the Wiener ﬁlter can be found in Haykin (1996a, 1999a). 2.5.2 Further Perspective on the Least Mean Square (LMS) Algorithm To reduce the computational complexity of the Wiener solution, which is a block solution, we can use the method of steepest descent for a recursive, or sequential, computation of the weight vector w. Let us derive the LMS algorithm for an adaptive FIR ﬁlter, the structure of which is shown in Figure 2.5. In view of a general adaptive system, this FIR ﬁlter becomes the ﬁlter structure within Figure 2.1. The output of this ﬁlter is y(k) = xT (k)w(k). (2.26) Widrow and Hoﬀ (1960) utilised this structure for adaptive processing and proposed instantaneous values of the autocorrelation and crosscorrelation matrices to calcu- late the gradient term within the steepest descent algorithm. The cost function they proposed was J(k) = 1 e2 (k), 2 (2.27) which is again based upon the instantaneous output error e(k) = d(k) − y(k). In order to derive the weight update equation we start from the instantaneous gradient ∂J(k) ∂e(k) = e(k) . (2.28) ∂w(k) ∂w(k)
  10. 18 ON GRADIENT DESCENT FOR NONLINEAR STRUCTURES x(k) x(k-1) x(k-2) x(k-N+1) z -1 z -1 z -1 z -1 w1(k) w2(k) w3(k) wN(k) Φ y(k) Figure 2.6 The structure of a nonlinear adaptive ﬁlter Following the same procedure as for the general gradient descent algorithm, we obtain ∂e(k) = −x(k) (2.29) ∂w(k) and ﬁnally ∂J(k) = −e(k)x(k). (2.30) ∂w(k) The set of equations that describes the LMS algorithm is given by  N   y(k) = T  xi (k)wi (k) = x (k)w(k),  i=1 (2.31) e(k) = d(k) − y(k),      w(k + 1) = w(k) + ηe(k)x(k). The LMS algorithm is a very simple yet extremely popular algorithm for adaptive ﬁltering. It is also optimal in the H ∞ sense which justiﬁes its practical utility (Hassibi et al. 1996). 2.6 On Gradient Descent for Nonlinear Structures Adaptive ﬁlters and neural networks are formally equivalent, in fact, the structures of neural networks are generalisations of linear ﬁlters (Maass and Sontag 2000; Nerrand et al. 1991). Depending on the architecture of a neural network and whether it is used online or oﬄine, two broad classes of learning algorithms are available: • techniques that use a direct computation of the gradient, which is typical for linear and nonlinear adaptive ﬁlters; • techniques that involve backpropagation, which is commonplace for most oﬄine applications of neural networks. Backpropagation is a computational procedure to obtain gradients necessary for adaptation of the weights of a neural network contained within its hidden layers and is not radically diﬀerent from a general gradient algorithm. As we are interested in neural networks for real-time signal processing, we will analyse online algorithms that involve direct gradient computation. In this section we introduce a learning algorithm for a nonlinear FIR ﬁlter, whereas learning algorithms for online training of recurrent neural networks will be introduced later. Let us start from a simple nonlinear FIR ﬁlter, which consists of the standard FIR ﬁlter cascaded
  11. FUNDAMENTALS 19 with a memoryless nonlinearity Φ as shown in Figure 2.6. This structure can be seen as a single neuron with a dynamical FIR synapse. This FIR synapse provides memory to the neuron. The output of this ﬁlter is given by y(k) = Φ(xT (k)w(k)). (2.32) The nonlinearity Φ( · ) after the tap-delay line is typically a sigmoid. Using the ideas from the LMS algorithm, if the cost function is given by J(k) = 1 e2 (k) 2 (2.33) we have e(k) = d(k) − Φ(xT (k)w(k)), (2.34) w(k + 1) = w(k) − η∇w J(k), (2.35) where e(k) is the instantaneous error at the output neuron, d(k) is some teach- ing (desired) signal, w(k) = [w1 (k), . . . , wN (k)]T is the weight vector and x(k) = [x1 (k), . . . , xN (k)]T is the input vector. The gradient ∇w J(k) can be calculated as ∂J(k) ∂e(k) = e(k) = −e(k)Φ (xT (k)w(k))x(k), (2.36) ∂w(k) ∂w(k) where Φ ( · ) represents the ﬁrst derivative of the nonlinearity Φ(·) and the weight update Equation (2.35) can be rewritten as w(k + 1) = w(k) + ηΦ (xT (k)w(k))e(k)x(k). (2.37) This is the weight update equation for a direct gradient algorithm for a nonlinear FIR ﬁlter. 2.6.1 Extension to a General Neural Network When deriving a direct gradient algorithm for a general neural network, the network architecture should be taken into account. For large networks for oﬄine processing, classical backpropagation is the most convenient algorithm. However, for online learn- ing, extensions of the previous algorithm should be considered. 2.7 On Some Important Notions From Learning Theory In this section we discuss in more detail the inter-relations between the error, error function and objective function in learning theory. 2.7.1 Relationship Between the Error and the Error Function The error at the output of an adaptive system is deﬁned as the diﬀerence between the output value of the network and the target (desired output) value. For instance,
  12. 20 ON SOME IMPORTANT NOTIONS FROM LEARNING THEORY the instantaneous error e(k) is deﬁned as e(k) = d(k) − y(k). (2.38) The instantaneous error can be positive, negative or zero, and is therefore not a good candidate for the criterion function for training adaptive systems. Hence we look for another function, called the error function, that is a function of the instantaneous error, but is suitable as a criterion function for learning. Error functions are also called loss functions. They are deﬁned so that an increase in the error function corresponds to a reduction in the quality of learning, and they are nonnegative. An error function can be deﬁned as N E(N ) = e2 (i) (2.39) i=0 or as an average value N ¯ 1 E(N ) = e2 (i). (2.40) N +1 i=0 2.7.2 The Objective Function The objective function is a function that we want to minimise during training. It can be equal to an error function, but often it may include other terms to introduce con- straints. For instance in generalisation, too large a network might lead to overﬁtting. Hence the objective function can consist of two parts, one for the error minimisa- tion and the other which is either a penalty for a large network or a penalty term for excessive increase in the weights of the adaptive system or some other chosen function (Tikhonov et al. 1998). An example of such an objective function for online learning is N 1 J(k) = (e2 (k − i + 1) + G( w(k − i + 1) 2 )), 2 (2.41) N i=1 where G is some linear or nonlinear function. We often use symbols E and J inter- changeably to denote the cost function. 2.7.3 Types of Learning with Respect to the Training Set and Objective Function Batch learning Batch learning is also known as epochwise, or oﬄine learning, and is a common strategy for oﬄine training. The idea is to adapt the weights once the whole training set has been presented to an adaptive system. It can be described by the following steps. 1. Initialise the weights 2. Repeat • Pass all the training data through the network
  13. FUNDAMENTALS 21 • Sum the errors after each particular pattern • Update the weights based upon the total error • Stop if some prescribed error performance is reached The counterpart of batch learning is so-called incremental learning, online, or pat- tern learning. The procedure for this type of learning is as follows. 1. Initialise the weights 2. Repeat • Pass one pattern through the network • Update the weights based upon the instantaneous error • Stop if some prescribed error performance is reached The choice of the type of learning is very much dependent upon application. Quite often, for networks that need initialisation, we perform one type of learning in the initialisation procedure, which is by its nature an oﬄine procedure, and then use some other learning strategy while the network is running. Such is the case with recurrent neural networks for online signal processing (Mandic and Chambers 1999f). 2.7.4 Deterministic, Stochastic and Adaptive Learning Deterministic learning is an optimisation technique based on an objective function which always produces the same result, no matter how many times we recompute it. Deterministic learning is always oﬄine. Stochastic learning is useful when the objective function is aﬀected by noise and local minima. It can be employed within the context of a gradient descent learning algorithm. The idea is that the learning rate gradually decreases during training and hence the steps on the error performance surface in the beginning of training are large which speeds up training when far from the optimal solution. The learning rate is small when approaching the optimal solution, hence reducing misadjustment. This gradual reduction of the learning rate can be achieved by e.g. annealing (Kirkpatrick et al. 1983; Rose 1998; Szu and Hartley 1987). The idea behind the concept of adaptive learning is to forget the past when it is no longer relevant and adapt to the changes in the environment. The terms ‘adaptive learning’ or ‘gear-shifting’ are sometimes used for gradient methods in which the learning rate is changed during training. 2.7.5 Constructive Learning Constructive learning deals with the change of architecture or interconnections in the network during training. Neural networks for which topology can change over time are called ontogenic neural networks (Fiesler and Beale 1997). Two basic classes of constructive learning are network growing and network pruning. In the network growing approach, learning begins with a network with no hidden units, and if the
  14. 22 ON SOME IMPORTANT NOTIONS FROM LEARNING THEORY error is too big, new hidden units are added to the network, training resumes, and so on. The most used algorithm based upon network growing is the so-called cascade- correlation algorithm (Hoehfeld and Fahlman 1992). Network pruning starts from a large network and if the error in learning is smaller than allowed, the network size is reduced until the desired ratio between accuracy and network size is reached (Reed 1993; Sum et al. 1999). 2.7.6 Transformation of Input Data, Learning and Dimensionality A natural question is whether to linearly/nonlinearly transform the data before feed- ing them to an adaptive processor. This is particularly important for neural networks, which are nonlinear processors. If we consider each neuron as a basic component of a neural network, then we can refer to a general neural network as a system with compo- nentwise nonlinearities. To express this formally, consider a scalar function σ : R → R and systems of the form, y(k) = σ(Ax(k)), (2.42) where the matrix A is an N × N matrix and σ is applied componentwise σ(x1 (k), . . . , xN (k)) = (σ(x1 (k)), . . . , σ(xN (k))). (2.43) Systems of this type arise in a wide variety of situations. For a linear σ, we have a linear system. If the range of σ is ﬁnite, the state vector of (2.42) takes values from a ﬁnite set, and dynamical properties can be analysed in time which is polynomial in the number of possible states. Throughout this book we are interested in functions, σ, and combination matrices, A, which would guarantee a ﬁxed point of this mapping. Neural networks are commonly of the form (2.42). In such a context we call σ the activation function. Results of Siegelmann and Sontag (1995) show that saturated linear systems (piecewise linear) can represent Turing machines, which is achieved by encoding the transition rules of the Turing machine in the matrix A. The curse of dimensionality The curse of dimensionality (Bellman 1961) refers to the exponential growth of com- putation needed for a speciﬁc task as a function of the dimensionality of the input space. In neural networks, a network quite often has to deal with many irrelevant inputs which, in turn, increase the dimensionality of the input space. In such a case, the network uses much of its resources to represent and compute irrelevant informa- tion, which hampers processing of the desired information. A remedy for this prob- lem is preprocessing of input data, such as feature extraction, and to introduce some importance function to the input samples. The curse of dimensionality is particularly prominent in unsupervised learning algorithms. Radial basis functions are also prone to this problem. Selection of a neural network model must therefore be suited for a particular task. Some a priori information about the data and scaling of the inputs can help to reduce the severity of the problem.
  15. FUNDAMENTALS 23 Transformations on the input data Activation functions used in neural networks are centred around a certain value in their output space. For instance, the mean of the logistic function is 0.5, whereas the tanh function is centred around zero. Therefore, in order to perform eﬃcient prediction, we should match the range of the input data, their mean and variance, with the range of the chosen activation function. There are several operations that we could perform on the input data, such as the following. 1. Normalisation, which in this context means dividing each element of the input vector x(k) by its squared norm, i.e. xi (k) ∈ x(k) → xi (k)/ x(k) 2 . 2 2. Rescaling, which means transforming the input data in the manner that we multiply/divide them by a constant and also add/subtract a constant from the data. 5 3. Standardisation, which is borrowed from statistics, where, for instance, a ran- dom Gaussian vector is standardised if its mean is subtracted from it, and the vector is then divided by its standard deviation. The resulting random vari- able is called a ‘standard normal’ random variable with zero mean and unity standard deviation. Some examples of data standardisation are • Standardisation to zero mean and unity standard deviation can be per- formed as iXi i (Xi− mean)2 mean = , std = . N N −1 The standardised quantity becomes Si = (Xi − mean)/std. • Standardise X to midrange 0 and range 2. This can be achieved by midrange = 1 (max Xi + min Xi ), 2 range = max Xi − min Xi , i i i i Xi − midrange Si = . range/2 4. Principal component analysis (PCA) represents the data by a set of unit norm vectors called normalised eigenvectors. The eigenvectors are positioned along the directions of greatest data variance. The eigenvectors are found from the covariance matrix R of the input dataset. An eigenvalue λi , i = 1, . . . , N , is associated with each eigenvector. Every input data vector is then represented by a linear combination of eigenvectors. As pointed out earlier, standardising input variables has an eﬀect on training, since steepest descent algorithms are sensitive to scaling due to the change in the weights being proportional to the value of the gradient and the input data. 5 In real life a typical rescaling is transforming the temperature from Celsius into Fahrenheit scale.
  16. 24 LEARNING STRATEGIES Nonlinear transformations of the data This method to transform the data can help when the dynamic range of the data is too high. In that case, for instance, we typically apply the log function to the input data. The log function is often applied in the error and objective functions for the same purposes. 2.8 Learning Strategies To construct an optimal neural approximating model we have to determine an appro- priate training set containing all the relevant information of the process and deﬁne a suitable topology that matches the complexity and performance requirements. The training set construction issue requires four entities to be considered (Alippi and Piuri 1996; Bengio 1995; Haykin and Li 1995; Shadafan and Niranjan 1993): • the number of training data samples ND ; • the number of patterns NP constituting a batch; • the number of batches NB to be extracted from the training set; • the number of times the generic batch is presented to the network during learn- ing. The assumption is that the training set is suﬃciently rich so that it contains all the relevant information necessary for learning. The requirement coincides with the hypothesis that the training data have been generated by a fully exciting input signal, such as white noise, which is able to excite all the process dynamics. White noise is a persistently exciting input signal and is used for the driving component of moving average (MA), autoregressive (AR) and autoregressive moving average (ARMA) models. 2.9 General Framework for the Training of Recurrent Networks by Gradient-Descent-Based Algorithms In this section we summarise some of the important concepts mentioned earlier. 2.9.1 Adaptive Versus Nonadaptive Training The training of a network makes use of two sequences, the sequence of inputs and the sequence of corresponding desired outputs. If the network is ﬁrst trained (with a train- ing sequence of ﬁnite length) and subsequently used (with the ﬁxed weights obtained from training), this mode of operation is referred to as non-adaptive (Nerrand et al. 1994). Conversely, the term adaptive refers to the mode of operation whereby the net- work is trained permanently throughout its application (with a training sequence of inﬁnite length). Therefore, the adaptive network is suitable for input processes which exhibit statistically non-stationary behaviour, a situation which is normal in the ﬁelds of adaptive control and signal processing (Bengio 1995; Haykin 1996a; Haykin and
  17. FUNDAMENTALS 25 Li 1995; Khotanzad and Lu 1990; Narendra and Parthasarathy 1990; Nerrand et al. 1994). 2.9.2 Performance Criterion, Cost Function, Training Function The computation of the coeﬃcients during training aims at ﬁnding a system whose operation is optimal with respect to some performance criterion which may be either qualitative, e.g. (subjective) quality of speech reconstruction, or quantitative, e.g. maximising signal to noise ratio for spatial ﬁltering. The goal is to deﬁne a positive training function which is such that a decrease of this function through modiﬁcations of the coeﬃcients of the network leads to an improvement of the performance of the system (Bengio 1995; Haykin and Li 1995; Nerrand et al. 1994; Qin et al. 1992). In the case of non-adaptive training, the training function is deﬁned as a function of all the data of the training set (in such a case, it is usually termed as a cost function). The minimum of the cost function corresponds to the optimal performance of the system. Training is an optimisation procedure, conventionally using gradient-based methods. In the case of adaptive training, it is impossible, in most instances, to deﬁne a time-independent cost function whose minimisation leads to a system that is optimal with respect to the performance criterion. Therefore, the training function is time dependent. The modiﬁcation of the coeﬃcients is computed continually from the gradient of the training function. The latter involves the data pertaining to a time window of ﬁnite length, which shifts in time (sliding window) and the coeﬃcients are updated at each sampling time. 2.9.3 Recursive Versus Nonrecursive Algorithms A nonrecursive algorithm employs a cost function (i.e. a training function deﬁned on a ﬁxed window), whereas a recursive algorithm makes use of a training function deﬁned on a sliding window of data. An adaptive system must be trained by a recursive algorithm, whereas a non-adaptive system may be trained either by a nonrecursive or by a recursive algorithm (Nerrand et al. 1994). 2.9.4 Iterative Versus Noniterative Algorithms An iterative algorithm performs coeﬃcient modiﬁcations several times from a set of data pertaining to a given data window, a non-iterative algorithm makes only one (Nerrand et al. 1994). For instance, the conventional LMS algorithm (2.31) is thus a recursive, non-iterative algorithm operating on a sliding window. 2.9.5 Supervised Versus Unsupervised Algorithms A supervised learning algorithm performs learning by using a teaching signal, i.e. the desired output signal, while an unsupervised learning algorithm, as in blind signal processing, has no reference signal as a teaching input signal. An example of a super- vised learning algorithm is the delta rule, while unsupervised learning algorithms are,
  18. 26 MODULARITY WITHIN NEURAL NETWORKS Module 1 Module 2 Module N Input Output Figure 2.7 A cascaded realisation of a general system for example, the reinforcement learning algorithm and the competitive rule (‘winner takes all’) algorithm, whereby there is some sense of concurrency between the elements of the network structure (Bengio 1995; Haykin and Li 1995). 2.9.6 Pattern Versus Batch Learning Updating the network weights by pattern learning means that the weights of the network are updated immediately after each pattern is fed in. The other approach is to take all the data as a whole batch, and the network is not updated until the entire batch of data is processed. This approach is referred to as batch learning (Haykin and Li 1995; Qin et al. 1992). It can be shown (Qin et al. 1992) that while considering feedforward networks (FFN), after one training sweep through all the data, the pattern learning is a ﬁrst- order approximation of the batch learning with respect to the learning rate η. There- fore, the FFN pattern learning approximately implements the FFN batch learning after one batch interval. After multiple sweeps through the training data, the dif- ference between the FFN pattern learning and FFN batch learning is of the order6 O(η 2 ). Therefore, for small training rates, the FFN pattern learning approximately implements FFN batch learning after multiple sweeps through the training data. For recurrent networks, the weight updating slopes for pattern learning and batch learn- ing are diﬀerent 7 (Qin et al. 1992). However, the diﬀerence could also be controlled by the learning rate η. The diﬀerence will converge to zero as quickly as η goes to zero 8 (Qin et al. 1992). 2.10 Modularity Within Neural Networks The hierarchical levels in neural network architectures are synapses, neurons, layers and neural networks, and will be discussed in Chapter 5. The next step would be combinations of neural networks. In this case we consider modular neural networks. Modular neural networks are composed of a set of smaller subnetworks (modules), each performing a subtask of the complete problem. To depict this problem, let us recourse to the case of linear adaptive ﬁlters described by a transfer function in the 6 In fact, if the data being processed exhibit highly stationary behaviour, then the average error calculated after FFN batch learning is very close to the instantaneous error calculated after FFN pattern learning, e.g. the speech data can be considered as being stationary within an observed frame. That forms the basis for use of various real-time and recursive learning algorithms, e.g. RTRL. 7 It can be shown (Qin et al. 1992) that for feedforward networks, the updated weights for both pattern learning and batch learning adapt at the same slope (derivative dw/dη) with respect to the learning rate η. For recurrent networks, this is not the case. 8 In which case we have a very slow learning process.
  19. FUNDAMENTALS 27 Module 1 Module 2 Σ Input Output Module N Figure 2.8 A parallel realisation of a general system z-domain H(z) as M b(k)z −k k=0 H(z) = N . (2.44) −k 1+ a(k)z k=1 We can rearrange this function either in a cascaded manner as max{M,N } 1 − βk z −1 H(z) = A , (2.45) 1 − αk z −1 k=1 or in a parallel manner as N Ak H(z) = , (2.46) 1 − αk z −1 k=1 where for simplicity we have assumed ﬁrst-order poles and zeros of H(z). A cascaded realisation of a general system is shown in Figure 2.7, whereas a parallel realisation of a general system is shown in Figure 2.8. We can also combine neural networks in these two conﬁgurations. An example of cascaded neural network is the so-called pipelined recurrent neural network, whereas an example of a parallel realisation of a neural network is the associative Gaussian mixture model, or winner takes all network. Taking into account that neural networks are nonlinear systems, we talk about nested modular architectures instead of cascaded architectures. The nested neural scheme can be written as F (W, X) = Φ wn Φ vi Φ · · · Φ uj Xj · · · , (2.47) n i j where Φ is a sigmoidal function. It corresponds to a multilayer network of units that sum their inputs with ‘weights’ W = {wn , vi , uj , . . . } and then perform a sigmoidal
  20. 28 MODULARITY WITHIN NEURAL NETWORKS 1 0.75 second nonlinear pass first nonlinear pass 0.8 0.7 0.6 0.65 0.4 0.6 0.2 0.55 0 0.5 −10 −5 0 5 10 −10 −5 0 5 10 argument argument 0.68 0.665 fourth nonlinear pass 0.67 third nonlinear pass 0.66 0.66 0.65 0.64 0.655 0.63 0.62 0.65 −10 −5 0 5 10 −10 −5 0 5 10 argument argument Figure 2.9 Eﬀects of nesting sigmoid nonlinearities: ﬁrst, second, third and fourth pass transformation of this sum. Its motivation is that the function F (W, X) = Φ wn Φ uj Xj (2.48) n j can approximate arbitrarily well any continuous multivariate function (Funahashi 1989; Poggio and Girosi 1990). Since we use sigmoid ‘squashing’ activation functions, modular structures con- tribute to a general stability issue. The eﬀects of a simple scheme of nested sigmoids are shown in Figure 2.9. From Figure 2.9 we see that pure nesting successively reduces the range of the output signal, bringing this composition of nonlinear functions to the ﬁxed point of the employed nonlinearity for suﬃciently many nested sigmoids. Modular networks possess some advantages over classical networks, since the overall complex function is simpliﬁed and modules possibly do not have hidden units which speeds up training. Also, input data might be decomposable into subsets which can be fed to separate modules. Utilising modular neural networks has not only compu- tational advantages but also development advantages, improved eﬃciency, improved interpretability and easier hardware implementation. Also, there are strong sugges- tions from biology that modular structures are exploited in cognitive mechanisms (Fiesler and Beale 1997).


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