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Self Organization: Competitive Learning

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1 Self Organization: Competitive Learning
CS/CMPE 537 – Neural Networks

2 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Introduction Competition for limited resources is a way of diversifying and optimizing a distributed system E.g. a sports tournament, evolution theory Local competition leads to optimization at local and global levels without the need for global control to assign system resources Neurons compete among themselves to decide the one (winner-takes-all) or ones that are activated Through competitive learning the output neurons form a topological map unto the input neurons Sefl-organizing feature maps, where the networks learns the input and output maps in a geometrical sense CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

3 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Competition Competition is intrinsically a nonlinear operation Two types of competition Hard competition means that only one neuron wins the resources Soft competition means that there is a clear winner, but its neighbors also share a small percentage of the system resources CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

4 Winner-Takes-All Network
Inputs xi (i = 1, p); output is yi = 1 if neuron i wins the competition yi = 0 otherwise The winning criterion can be any rule, as long as one clear winner is found (e.g. largest output, smallest output) The winner-takes-all network has one layer with p neurons, self-excitatory connections, and lateral inhibitory connections to all other neurons When a input is applied, the network evolves until the output of the winning neuron tends to 1 while the outputs of the other neurons tends to 0 Winner-takes-all network acts as a selector CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

5 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Instar Network (1) p p Instar network is made up of McCulloch-Pitts neurons Instar neuron is able to recognize a single vector pattern based on a modified Hebbian rule that automatically accomplishes weight normalization for normalized inputs CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

6 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Instar Network (2) Learning rule wji(n + 1) = wji(n) + ηyj(n)[xi(n) – wji(n)] y is the hard-limiter output of either 0 or 1 The rule modifies the weights for neuron j if neuron j is active (wins). Otherwise, the weights remain unchanged. This rule is similar to Oja’s rule, except that the weights are not multiplied by the output CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

7 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Instar Network (3) The effect of the instar rule is to move the weight closer to the input in a straight line proportional to η A trained instar neuron provides a response of 1 for data samples located near the training pattern (the bias controls the size of the neighborhood) CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

8 Competitive Rule for Winner-Takes-All Network
The hard-limiter in the instar network acts as a gate that controls the update of weights If a winner-takes-all network is used, the gating mechanism is not needed, as the winning neuron is decided separately Update rule wj*i(n + 1) = wj*i(n) + η[xi(n) – wj*i(n)] j* = winning neuron CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

9 Criterion for Competition (1)
What criterion should be used to determine the winning neuron? The criterion should have some geometrical relevance Inner product A neuron wins if the input is closest to the neuron (i.e. its weights) However, this only works when the input is normalized Winning neuron is the one with the largest inner product CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

10 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS

11 Criterion for Competition (2)
Euclidean distance ||x – w|| = [Σi (xi – wi)2]1/2 Winning neuron is the one with the smallest Euclidean distance Less efficient than inner product L1 norm or Manhattan distance |x – w| = Σi (xi – wi) Winning neuron is the one with the smallest L1 norm CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

12 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Clustering The competitive rule allows a single-layer linear network to group and represent data samples that lie in a neighborhood of the input space Each neighborhood is represented by a single output neuron The weights of the neuron represent the center of the cluster Thus, a single-layer linear competitive network performs clustering Clustering is a ‘divide and conquer’ process where an input space is divided into groups of proximal patterns Vector quantization – an example of clustering CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

13 k-Means Clustering Algorithm (1)
Classical algorithm for data clustering Cluster data vectors xi (i = 1, N) into k cluster Ci (i = 1, k) such that the distance between the vectors and their respective cluster centers ti (i = 1, k) is minimized Algorithm Randomly assign vectors to class Ci Find center of each cluster Ci ti = (1/Ni) ΣnЄCi xn Reassign the vectors to their nearest cluster, i.e., assign vector i to cluster k if ||xi – tk||2 is minimum for all k Repeat the last two steps until no changes occur CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

14 k-Means Clustering Algorithm (2)
Performance measure or function J = Σi=1 kΣnЄCi ||xn – ti||2 This measure is minimized in the k-means algorithm Estimating J with its instantaneous value, then taking derivative wrt ti δJ(n)/ δti = -η[x(n) – ti(n)] and, the change becomes (to minimize J) Δti(n) = η[x(n) – ti(n)] This is identical to the competitive update rule, when ti is replaced by wi Thus, the single-layer linear competitive network performs k-means clustering of the input data CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

15 k-Means Clustering - Example
Cluster [0, 1, 3, 6, 8. 9, 10, 11] into 2 groups. N = 8, k = 2 C1 = [0, 1, 3, 6]; C2 = [8, 9, 10, 11] t1 = 10/4 = 2.5; t2 = 38/4 = 9.5 C1 = [0, 1, 3]; C2 = [6, 8, 9, 10, 11] t1 = 4/3 = 1.33; t2 = 44/5 = 8.8 No change. Thus, these are the final groupings and clusters CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

16 Determining the Number of Clusters
The number of clusters k (the number of output neurons) has to be decided apriori How is k determined ? There is no easy answer without prior knowledge of the data that indicate the number of clusters If k is greater than the number of clusters in data, then natural clusters are split into smaller regions If k is less than the number of clusters in data, then more than one natural cluster is grouped together An iterative solution Start with a small k, cluster and find value of J Increment k by 1, cluster and find value J Repeat until J starts to increase CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

17 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Soft Competition Hard competition = winner-takes-all. That is, the winner neuron is active while the rest are inactive Soft competition = winner-shares-reward. That is, the winning neuron activates its neighboring neurons as well The winning neuron has the highest output, while its neighboring neurons’ outputs decrease with increase in distance Consequently, weights of all active neurons are modified, not just that of the winning neuron Soft competition is common in biological neural systems CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

18 Kohonen Self-Organizing Maps (1)
Kohonen SOMs are single-layer competitive networks with a soft competition rule They perform a mapping from a continuous input space to a discrete output space, preserving the topological properties of the input Topological map points close to each other in the input space are mapped to the same or neighboring neurons in the output space There is topological ordering or spatial significance to the neurons in the output and input layer CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

19 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Kohonen SOMs (2) CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

20 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Kohonen SOMs (3) 1-D SOM has output neurons arranged as a string, with each neuron having two adjacent neighbors 2-D SOM has output neurons arranged as an 2-D array, with more flexible definition of neighborhoods CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

21 Kohonen Learning Algorithm (1)
Competitive rule wj(n + 1) = wj(n) + hj,i*(n) η(n) [x(n) – wj(n)] hj,j* = neighborhood function for winning neuron i* A common neighboring function is the Gaussian hj,i*(n) = exp(- dj,i*2 / 2σ(n)2) CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

22 Kohonen Learning Algorithm (2)
Initialization: Choose random values for the initial weight vectors. Make sure the weights for each neuron are different and of small magnitude Sampling and similarity matching: select an input vector x. Find the best-matching (winning) neuron at time n, using the minimum-distance Euclidean criterion. Updating: adjust the weight vectors using the Kohonen learning rule Continuation: Repeat the previous two steps until no noticeable changes occur in the feature map are observed CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

23 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Example (1) Create a 1-D feature map of a 2-D input space. The input vectors x lie on a unit circle. No. of inputs = 2 (defined by the problem, 2-D) No. of outputs = 20 (our choice, 1-D linear array or lattice) Randomly initialize the weights so that each neuron has a different weight (in this example weights are initialized along the x-axis, i.e, the y component is zero) See the MATLAB code for details CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

24 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Example (2) Consider the input x = [ ]T Assume the current weights are W = [ 0 0:19] = [0 0; 0 1; 0 2; 0 3;…;0 19] Winning neuron is the one with smallest norm from the input vector Winning neuron is 2 (weight vector w2 = [0 1]T) Norm = ( )1/2 = (this is the smallest distance from the weight vectors) Update of weights wj(n + 1) = wj(n) + hj,j*(n) η(n) [x(n) – wj(n)] Assume Gaussian neighborhood function with variance of 0.5 CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

25 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Example (3) Neuron 1 w1(n + 1) = w1(n) + h1,2*(n) η(n) [x(n) – w1(n)] = [0 0] ([ ] – [0 0]) = [ ] Neuron 2 (winning neuron) w2(n + 1) = w2(n) + h2,2*(n) η(n) [x(n) – w2(n)] = [0 1] + (1)([ ] – [0 1]) = [ ] CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

26 CS/CMPE 537 - Neural Networks (Sp 2004/2005) - Asim Karim @ LUMS
Example (4) Neuron 3 w3(n + 1) = w3(n) + h3,2*(n) η(n) [x(n) – w3(n)] = [0 2] ([ ] – [0 2]) = [ ] Neuron 4 w4(n + 1) = w4(n) + h4,2*(n) η(n) [x(n) – w4(n)] = [0 3] ([ ] – [0 3]) = [ ] Neuron 5 and beyond There update will be negligible because the Gaussian tends to zero with increasing distance CS/CMPE Neural Networks (Sp 2004/2005) - Asim LUMS

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