Competitive learningCollege voor cursus Connectionistische modellen

M Meeter

2

Unsupervised learning

To-be-learned patterns not wholly provided by modeller

Hebbian unsupervised learning

Competitive learning

3

The basic idea

© Rumelhart & Zipser, 1986

4

What’s it good for?

discovering structure in the input

discovering categories in the input Classification networks:

ART (Grossberg & Carpenter)

CALM (Murre & Phaf)

mapping inputs onto a topographic map Kohonen maps (Kohonen) CALM - Maps (Murre & Phaf)

5

Features of Competitive learning Two or more layers (no auto-association)

Competition between output nodes

Two phases: determining a winner learning

Weight normalisation

6

Two or more layers

Input must come from outside the inhibitory clusters

© Rumelhart & Zipser, 1986

7

Competition between output nodes At every presentation of an input pattern,

a winner is determined

Only winner is activated [activation at learning discrete: (0,1) ]

Hard Winner Take All: Find node with maximum input

max. ( wijaj )

Inhibition between nodes

8

Inhibition between nodes

Example: inhibition in CALM

V V

R

V

RR

A

E

Low

HighFlat

Strange

AE

Up Gaussian

Learning intermodular connections

9

Two phases

1. One node wins the competition

2. That node learns, others not

Nodes start off with random weights

No ‘correct’ output connected with inputs: unsupervised learning

10

Weight normalisation

Weights of winner node i changed wij = * aj

Weights add up to constant sum... wij = 1

rule of Rumelhart & Zipser:wij = g * ai / nk - g * wij

…or constant distance: (wij)

2 = 1

11

Geometric interpretation

Both weights & input patterns can be seen as vectors in a hyper space

Euclidian normalisation [ (wij)2 = 1]

all vectors on a sphere in space of n dimensions (n = number of inputs)

node with weight vector closest to input vector is winner

Linear normalisation [ wij = 1] all weights on a plane

12

Geometric interpretation II

Weight vectors move towards input in the hyper space

wij= g * ai/nk - g * wij

Output nodes move towards clusters in inputs

© Rumelhart & Zipser, 1986

13

Stable / unstable

Output nodes move towards clusters in inputs

If input not clustered...

...output nodes will continue moving through input space! © Rumelhart & Zipser, 1986

14

Statistical equivalents

Sarle (1994):Classification = k-means clustering

Kohonen = mapping continuous dimensions onto discrete ones

Statistical techniques usually more efficient...

...because statistical techniques use whole data set

15

Importance of competitive learning Supervised - unsupervised learning

Structure input sets not always given

Natural categories

16

Competitive learning in the brain Lateral inhibition feature of most parts of

the brain

… Implements winner-take-all ?

17

Part II

18

Map formation in the brain

Topographic maps omnipresent in the sensory regions of the brain

retinotopic maps: neurons ordered as the locations of their visual field on the retina

tonotopic maps: neurons ordered according to tone for which they are sensitive

maps in somatosensory cortex: neurons ordered according to body part for which they are sensitive

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

Map formation ubiquitous (also semantic maps?)

How do maps form? gradients in neurotransmitters pruning

22

Kohonen maps

Teuvo Kohonen first to show how maps can develop

Self Organising Maps (S.O.M.)

Demonstration: the ordering of colours (colours are vectors in a 3-dimensional space of brightness, hue, saturation).

23

Kohonen algorithm

Finding the activity bubble

Updating the weights for the nodes in the active bubble

24

Finding the activity bubble

Lateral inhibition

25

Finding activity bubble II

Find the winner

Activate all nodes in the neighbourhood of the winner

26

Updating the weights

Move weight vector of winner towards the input vector

Do the same for the active neighbourhood nodes

weight vectors of neigbouring nodes will start resembling each other

27

Simplest implementation

Weight vectors & input patterns all have length 1 (e.i., (wij)

2 = 1 ) Find node whose weight vector has mimimal

distance to the input vector:min. (aj - wij)2

Activate all nodes in neighbourhood radius Nt

Update weights of active nodes by moving weights towards the input vector:

wij = t * ( aj - wij)

wij(t+1) = wij(t) + t * ( aj - wij(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

Darn elegant

Pruning?

Speech recognition uses Hidden Markov Models

33

Summary

Prime example of unsupervised learning Two phases:

winner node is determined weights are updated of the winner only

Very good at discovering structure: discovering categories mapping the input onto a topographic

map Competitive learning important paradigm

in connectionism