Competitive learning College voor cursus Connectionistische modellen M Meeter

2 Unsupervised learning To-be-learned patterns not wholly provided by modeller u Hebbian unsupervised learning u Competitive learning

3 The basic idea © Rumelhart & Zipser, 1986

4 What’s it good for? n discovering structure in the input n discovering categories in the input u Classification networks: ART (Grossberg & Carpenter) CALM (Murre & Phaf) n mapping inputs onto a topographic map u Kohonen maps (Kohonen) u CALM - Maps (Murre & Phaf)

5 Features of Competitive learning n Two or more layers (no auto-association) n Competition between output nodes n Two phases: u determining a winner u learning n Weight normalisation

6 Two or more layers Input must come from outside the inhibitory clusters © Rumelhart & Zipser, 1986

7 Competition between output nodes n At every presentation of an input pattern, a winner is determined n Only winner is activated [activation at learning discrete: (0,1) ] u Hard Winner Take All: Find node with maximum input max.  ( w ij a j ) u Inhibition between nodes

8 Inhibition between nodes n Example: inhibition in CALM

9 Two phases 1.One node wins the competition 2.That node learns, others not n Nodes start off with random weights n No ‘correct’ output connected with inputs: unsupervised learning

10 Weight normalisation n Weights of winner node i changed  w ij =  * a j n Weights add up to constant sum...  w ij = 1 rule of Rumelhart & Zipser:  w ij = g * a i / n k - g * w ij n …or constant distance:  (w ij ) 2 = 1

11 Geometric interpretation n Both weights & input patterns can be seen as vectors in a hyper space n Euclidian normalisation [  (w ij ) 2 = 1] u all vectors on a sphere in space of n dimensions (n = number of inputs) u node with weight vector closest to input vector is winner n Linear normalisation [  w ij = 1] u all weights on a plane

12 Geometric interpretation II n Weight vectors move towards input in the hyper space  w ij = g * a i /n k - g * w ij n Output nodes move towards clusters in inputs © Rumelhart & Zipser, 1986

13 Stable / unstable n Output nodes move towards clusters in inputs n If input not clustered......output nodes will continue moving through input space! © Rumelhart & Zipser, 1986

14 Statistical equivalents n Sarle (1994): Classification = k-means clustering Kohonen = mapping continuous dimensions onto discrete ones u Statistical techniques usually more efficient... u...because statistical techniques use whole data set

15 Importance of competitive learning n Supervised - unsupervised learning n Structure input sets not always given n Natural categories

16 Competitive learning in the brain n Lateral inhibition feature of most parts of the brain … Implements winner-take-all ?

17 Part II

18 Map formation in the brain n Topographic maps omnipresent in the sensory regions of the brain u retinotopic maps: neurons ordered as the locations of their visual field on the retina u tonotopic maps: neurons ordered according to tone for which they are sensitive u maps in somatosensory cortex: neurons ordered according to body part for which they are sensitive u maps in motor cortex: neurons ordered according to muscles they control

19 Somatosensory maps © Kandel, Schwartz & Jessell, 1991

20 Somatosensory maps II © Kandel, Schwartz & Jessell, 1991

21 Speculations n Map formation ubiquitous (also semantic maps?) n How do maps form? u gradients in neurotransmitters u pruning

22 Kohonen maps n Teuvo Kohonen first to show how maps can develop n Self Organising Maps (S.O.M.) n Demonstration: the ordering of colours (colours are vectors in a 3-dimensional space of brightness, hue, saturation).

23 Kohonen algorithm n Finding the activity bubble n Updating the weights for the nodes in the active bubble

24 Finding the activity bubble Lateral inhibition

25 Finding activity bubble II n Find the winner n Activate all nodes in the neighbourhood of the winner

26 Updating the weights n Move weight vector of winner towards the input vector n Do the same for the active neighbourhood nodes  weight vectors of neigbouring nodes will start resembling each other

27 Simplest implementation n Weight vectors & input patterns all have length 1 (e.i.,  (w ij ) 2 = 1 ) n Find node whose weight vector has mimimal distance to the input vector: min.  (a j - w ij ) 2 n Activate all nodes in neighbourhood radius N t n Update weights of active nodes by moving weights towards the input vector:  w ij =  t * ( a j - w ij ) w ij (t+1) = w ij (t) +  t * ( a j - w ij (t) )

28 Results of Kohonen © Kohonen, 1982

29 Influence of neighbourhood radius © Kohonen, 1982 Larger neighbourhood size leads to faster learning

30 Results II: the phonological typewriter © Kohonen, 1988

31 Phonological typewriter II © Kohonen, 1988

32 Kohonen conclusions n Darn elegant n Pruning? n Speech recognition uses Hidden Markov Models

33 Summary n Prime example of unsupervised learning n Two phases: u winner node is determined u weights are updated of the winner only n Very good at discovering structure: u discovering categories u mapping the input onto a topographic map n Competitive learning important paradigm in connectionism