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B219 Intelligent Systems Semester 1, 2003 Artificial Neural Networks (Ref: Negnevitsky, M. “Artificial Intelligence, Chapter 6) BPNN in Practice Week 3 Lecture Notes page 1 of 1 B219 Intelligent Systems Semester 1, 2003 The Hopfield Network § In this network, it was designed on analogy of brain’s memory, which is work by association. § For example, we can recognise a familiar face even in an unfamiliar environment within 100-200ms. § We can also recall a complete sensory experience, including sounds and scenes, when we hear only a few bars of music. § The brain routinely associates one thing with another. § Multilayer Neural Networks trained with backpropagation algorithm are used for pattern recognition problems. § To emulate the human memory’s associative characteristics, we use a recurrent neural network. Week 3 Lecture Notes page 2 of 2 B219 Intelligent Systems Semester 1, 2003 § A recurrent neural network has feedback loops from its outputs to its inputs. The presence of such loops has a profound impact on the learning capability of the network. § Single layer n-neuron Hopfield network § The Hopfield network uses McCulloch and Pitts neurons with the sign activation function as its computing element: § The current state of the Hopfield is determined by the current outputs of all neurons, y1, y2,…,yn. Week 3 Lecture Notes page 3 of 3 B219 Intelligent Systems Semester 1, 2003 Hopfield Learning Algorithm: § Step 1: Assign weights o Assign random connections weights with values wij = +1 or wij = −1 for all i ≠ j and 0 for i = j § Step 2: Initialisation o Initialise the network with an unknown pattern: xi = Oi ( k ), 0 ≤ i ≤ N − 1, where Oi (k ) is the output of node i at time t = k = 0 and xi is an element at input i of an input pattern, + 1 or - 1 § Step 3: Convergence o Iterate until convergence is reached, using the relation: N −1 Oi ( k + 1) = f ∑ wij Oi ( k ) , 0 ≤ j ≤ N − 1 i =0 where the function f(.) is a hard limiting nonlinearity. o Repeat the process until the node outputs remain unchanged. Week 3 Lecture Notes page 4 of 4 B219 Intelligent Systems Semester 1, 2003 o The node outputs then best represent the exemplar pattern that best matches the unknown input. § Step 4: Repeat for Next Pattern o Go back to step 2 and repeat for next xi , and so on. § Hopfield network can act as an error correction network. Type of Learning § Supervised Learning o the input vectors and the corresponding output vectors are given o the ANN learns to approximate the function from the inputs to the outputs Week 3 Lecture Notes page 5 of 5 B219 Intelligent Systems Semester 1, 2003 § Reinforcement Learning o the input vectors and a reinforcement signal are given o the reinforcement signal tells how good the true output was § Unsupervised Learning o only input are given o the ANN learns to form internal representations or codes for the input data that can then be used e.g. for clustering § From now we will look at unsupervised learning neural networks. Week 3 Lecture Notes page 6 of 6 B219 Intelligent Systems Semester 1, 2003 Hebbian Learning § In 1949, Donald Hebb proposed one of the key ideas in biological learning, commonly known as Hebb’s Law. § Hebb’s Law states that if neuron i is near enough to excite neuron j and repeatedly participates in its activation, the synaptic connection between these two neurons is strengthened and neuron j becomes more sensitive to stimuli from neuron i. § Hebb’s Law can be represented in the form of two rules: • If two neurons on either side of a connection are activated synchronously, then the weight of that connection is increased. • If two neurons on either side of a connection are activated asynchronously, then the weight of that connection is decreased. Week 3 Lecture Notes page 7 of 7 B219 Intelligent Systems Semester 1, 2003 § Hebbian learning implies that weights can only increase. To resolve this problem, we might impose a limit on the growth of synaptic weights. § It can be implemented by introducing a non-linear forgetting factor into Hebb’s Law: where ö is the following factor § Forgetting factor usually falls in the interval between 0 and 1, typically between 0.01 and 0.1, to allow only a little “forgetting” while limiting the weight growth. Week 3 Lecture Notes page 8 of 8 B219 Intelligent Systems Semester 1, 2003 Hebbian Learning Algorithm: § Step 1: Initialisation o Set initial synaptic weights and threshold to small random values, say in an interval [0,1]. § Step 2: Activation o Compute the neuron output at iteraction p where n is the number of neuron inputs, and èj is the threshold value of neuron j. § Step 3: Learning o Update the weights in the network: where Äwij(p) is the weight correction at iteration p. Week 3 Lecture Notes page 9 of 9 B219 Intelligent Systems Semester 1, 2003 o The weight correction is determine by the generalised activity product rule: § Step 4: Iteration Increase iteration p by one, go back to Step 2. Competitive Learning § Neurons compete among themselves to be activated § While in Hebbian Learning, several output neurons can be activated simultaneously, in competitive learning, only a single output neuron is active at any time. § The output neuron that wins the “competition” is called the winner-takes-all neuron. § In the late 1980s, Kohonen introduced a special call of ANN called self-organising maps. These maps are based on competitive learning. Week 3 Lecture Notes page 10 of 10 B219 Intelligent Systems Semester 1, 2003 Self-organising Map § Our brain is dominated by the cerebral cortex, a very complex structure of billions of neurons and hundreds of billions of synapses. § The cortex includes areas that are responsible for different human activities (motor, visual, auditory, etc), and associated with different sensory input. § Each sensory input is mapped into a corresponding area of the cerebral cortex. The cortex is a selforganising computational map in the human brain. Week 3 Lecture Notes page 11 of 11 B219 Intelligent Systems Semester 1, 2003 § Feature-mapping Kohonen model The Kohonen Network § The Kohonen model provides a topological mapping. It places a fixed number of input patterns from the input layer into a higher dimensional output or Kohonen layer. Week 3 Lecture Notes page 12 of 12 B219 Intelligent Systems Semester 1, 2003 § Training in the Kohonen network begins with the winner’s neighbourhood of a fairly large size. Then, as training proceeds, the neighbourhood size gradually decreases. § The lateral connections are used to create a competition between neurons. The neuron with the largest activation level among all neurons in the output layer becomes the winner. § The winning neuron is the only neuron that produces an output signal. The activity of all other neurons is suppressed in the competition. Week 3 Lecture Notes page 13 of 13 B219 Intelligent Systems Semester 1, 2003 § The lateral feedback connections produce excitatory or inhibitory effects, depending on the distance from the winning neuron. § This is achieved by the use of a Mexican hat function which describes synaptic weights between neurons in the Kohonen layer. § In the Kohonen network, a neuron learns by shifting its weights from inactive connections to actives ones. Only the winning neuron and its neighbourhood are allowed to learn. Week 3 Lecture Notes page 14 of 14 B219 Intelligent Systems Semester 1, 2003 Competitive Learning Algorithm § Step 1: Initialisation o Set initial synaptic weights to small random values, say in an interval [0, 1], and assign a small positive value to learning rate parameter á § Step 2: Activation and Similarity Matching o Activate the Kohonen network by applying the input vector X, and find the winner-takes-all (best matching) neuron jX at iteration p, using the minimum-distance Euclidean criterion where n is the number of neurons in the input layer, and m is the number of neurons in the Kohonen layer. Week 3 Lecture Notes page 15 of 15 B219 Intelligent Systems Semester 1, 2003 § Step 3: Learning o Update the synaptic weights where Äwij(p) is the weight correction at iteration p. o The weight correction is determined by the competitive learning rule: where á is the learning rate parameter, and Ëj(p) is the neighbourhood function central around the winner-takes-all neuron jx at iteration p. § Step 4: Iteration o Increase iteration p by one, go back to step 2 and continue until minimum-distance Euclidean criterion is satisfied, or no noticeable changes occur in the feature map Week 3 Lecture Notes page 16 of 16