Adaptive

1
CHAPTER 16

• Adaptive
• Resonance
• Theory

2
Objectives

• There is no guarantee that, as more inputs are
  applied to the competitive network, the weight
  matrix will eventually converge.
• Present a modified type of competitive learning,
  called adaptive resonance theory (ART), which is
  designed to overcome the problem of learning
  stability.

3
Theory Examples

• A key problem of the Grossberg network and the
  competitive network is that they do NOT always
  from stable clusters (or categories).
• The learning instability occurs because of the
  networks adaptability (or plasticity), which
  causes prior learning to be eroded by more recent
  learning.

4
Stability / Plasticity

• How can a system be receptive to significant new
  patterns and yet remain stable in response to
  irrelevant patterns?
• Grossberg and Carpenter developed the ART to
  address the stability/plasticity dilemma.
• The ART networks are based on the Grossberg
  network of Chapter 15.

5
Key Innovation

• The key innovation of ART is the use of
  expectations.
• As each input is presented to the network, it is
  compared with the prototype vector that is most
  closely matches (the expectation).
• If the match between the prototype and the input
  vector is NOT adequate, a new prototype is
  selected. In this way, previous learned memories
  (prototypes) are not eroded by new learning.

6
Overview
Grossberg competitive network
Basic ART architecture
7
Grossberg Network

• The L1-L2 connections are instars, which performs
  a clustering (or categorization) operation. When
  an input pattern is presented, it is multiplied
  (after normalization) by the L1-L2 weight matrix.
• A competition is performed at Layer 2 to
  determine which row of the weight matrix is
  closest to the input vector. That row is then
  moved toward the input vector.
• After learning is complete, each row of the L1-L2
  weight matrix is a prototype pattern, which
  represents a cluster (or a category) of input
  vectors.

8
ART Networks -- 1

• Learning of ART networks also occurs in a set of
  feedback connections from Layer 2 to Layer 1.
  These connections are outstars which perform
  pattern recall.
• When a node in Layer 2 is activated, this
  reproduces a prototype pattern (the expectation)
  at layer 1.
• Layer 1 then performs a comparison between the
  expectation and the input pattern.
• When the expectation and the input pattern are
  NOT closely matched, the orienting subsystem
  causes a reset in Layer 2.

9
ART Networks -- 2

• The reset disables the current winning neuron,
  and the current expectation is removed.
• A new competition is then performed in Layer 2,
  while the previous winning neuron is disable.
• The new winning neuron in Layer 2 projects a new
  expectation to Layer 1, through the L2-L1
  connections.
• This process continues until the L2-L1
  expectation provides a close enough match to the
  input pattern.

10
ART Subsystems
Layer 1 Comparison of input pattern and
expectation. L1-L2 Connections (Instars) Perform
clustering operation. Each row of W12 is a
prototype pattern. Layer 2 Competition (Contrast
enhancement) L2-L1 Connections (Outstars) Perform
pattern recall (Expectation). Each column of
W21 is a prototype pattern Orienting
Subsystem Causes a reset when expectation does
not match input pattern Disables current winning
neuron
11
Layer 1
12
Layer 1 Operation

• Equation of operation of Layer 1
• Output of Layer 1

Excitatory input Input pattern L1-L2
expectation
Inhibitory input Gain control from L2
13
Excitatory Input to L1

• The excitatory input
• Assume that the jth neuron in Layer 2 has won the
  competition, i.e.,
• The excitatory input to Layer 1 is the sum of the
  input pattern and the L2-L1 expectation.

14
Inhibitory Input to L1

• The inhibitory input the gain control
• The inhibitory input to each neuron in Layer 1 is
  the sum of all of the outputs of Layer 2.
• The gain control to Layer 1 will be one when
  Layer 2 is active (one neuron has won the
  competition), and zero when Layer 2 is inactive
  (all neurons having zero output).

15
Steady State Analysis -- 1

• The response of neuron i in Layer 1
• Case 1 Layer 2 is inactive eachIn steady
  stateIf thenIf thenThe
  output of Layer 1 is the same as the input
  pattern

16
Steady State Analysis -- 2

• Case 2 Layer 2 is active andIn
  steady stateLayer 1 is to combine the input
  vector with the expectation from Layer 2. Since
  both the input and the expectation are binary
  pattern, we will use a logic AND operation to
  combine the two vectors. if either or
  is equal to 0 ? if both and
  are equal to 1 ?

17
Layer 1 Example

• Let
• Assume that Layer 2 is active and neuron 2 of
  Layer 2 wins the competition.

18
Response of Layer 1
19
Layer 2
From the orienting subsystem
20
Layer 2 Operation
excitatory input

• Equation of operation of Layer 2The rows
  of adaptive weights , after training, will
  represent the prototype patterns.

on-center feedback
adaptive instar
inhibitory input
off-surround feedback
21
Layer 2 Example

• Let

22
Response of Layer 2
23
Orienting Subsystem

• Determine if there is a sufficient match between
  the L2-L1 expectation (a1) and the input pattern
  (p)

24
Orienting Subsyst. Operat.

• Equation of operation of the Orienting
  Subsystemexcitatory inputinhibitory
  input
• Whenever the excitatory input is larger than the
  inhibitory input, the Orienting Subsystem will be
  driven on.

excitatory input
inhibitory input
25
Steady State Operation

• Steady stateLet , then
  if , or if
  (vigilance)The condition that
  will cause a reset of Layer 2.

26
Vigilance Parameter

• . The term ? is called the
  vigilance parameter and must fall in the range
• If ? is close to 1, a reset will occur unless
  is close to
• If ? is close to 0, need not be close to to
  present a reset.
• , whenever Layer 2 is active.The
  orienting subsystem will cause a reset when there
  is enough of a mismatch between and

27
Orienting Subsystem Ex.

• Suppose that
• In this case a reset signalwill be sent to Layer
  2,since is positive.

t
28
Learning Law

• Two separate learning lawsone for the L1-L2
  connections,(instar) and another for
  L2-L1connections (outstar).
• Both L1-L2 connections and L2-L1 connections
  are updated at the same time.Whenever the input
  and theexpectation have an adequate match.
• The process of matching, and subsequentadaptation
  , is referred to as resonance.

29
Subset / Superset Dilemma

• Suppose that ,so that the
  prototype patterns are
• If the output of Layer 1 isthen the input to
  Layer 2 will be
• Both prototype vectors have the same inner
  product with a1, even though the 1st prototype is
  identical to a1 and the 2nd prototype is
  not.This is called subset/superset dilemma.

30
Subset / Superset Solution

• One solution to the subset/superset dilemma is to
  normalize the prototype patterns.
• The input to Layer 2 will then be
• The first prototype has the largest inner product
  with a1. The first neuron in Layer 2 will be
  active.

31
Learning Law L1-L2

• Instar learning with competition
• When neuron i of Layer 2 is active, the ith row
  of , , is moved in the direction of
  a1. The learning law is that the elements of
  compete, and thereforeis normalized.

32
Fast Learning

• For fast learning, we assume that the outputs of
  Layer 1 and Layer 2 remain constant until the
  weights reach steady state.
• assume that and setCase 1Case
  2Summary

33
Learning Law L2-L1

• Typical outstar learningIf neuron j in Layer 2
  is active (has won the competition), then column
  j of is moved toward a1.
• Fast learning assume that and?Column
  j of converges to the output of Layer 1,
  a1, which is a combination of the input pattern
  and the appropriate prototype pattern. The
  prototype pattern is modified to incorporate the
  current input pattern.

34
ART1 Algorithm Summary

• 0. Initialization The initial is set to
  all 1s. Every elements of the initial is
  set to .
• 1. Present an input pattern to the network.Since
  Layer 2 is NOT active on initialization, the
  output of Layer 1 is .
• 2. Compute the input to Layer 2, , and
  activate the neuron in Layer 2 with the largest
  inputIn case of tie, the neuron with the
  smallest index is declared the winner.

35
Algorithm Summary Cont.

• 3. Compute the L2-L1 expectation (assume that
  neuron j of Layer 2 is activated)
• 4. Layer 2 is active. Adjust the Layer 1 output
  to include the L2-L1 expectation
• 5. Determine the degree of match between the
  input pattern and the expectation (Orienting
  Subsystem)
• 6. If , then set , inhibit it until
  an adequate match occurs (resonance), and return
  to step 1.If , then continue with step 7.

36
Algorithm Summary Cont.

• 7. Update row j of when resonance has
  occurred
• 8. Update column j of
• 9. Remove the input pattern, restore all
  inhibited neurons in Layer 2, and return to step
  1.
• The input patterns continue to be applied to the
  network until the weights stabilize (do not
  change).
• ART1 network can only be used for binary input
  patterns.

37
Solved Problem P16.5

• Train an ART1 network using the parameters
  and , and choosing (3
  categories), and using the
• following three input vectors
• Initial weights
• 1-1 Compute the Layer 1 response

38
P16.5 Continued

• 1-2 Compute the input to Layer 2 Since all
  neurons have the same input, pick the first
  neuron as winner.
• 1-3 Compute the L2-L1 expectation

39
P16.5 Continued

• 1-4 Adjust the Layer 1 output to include the
  expectation
• 1-5 Determine the match degree Therefore
  (no reset)
• 1-6 Since , continued with step 7.
• 1-7 Resonance has occurred, update row 1 of

40
P16.5 Continued

• 1-8 Update column 1 of
• 2-1 Compute the new Layer 1 response
• (Layer 2 inactive)
• 2-2 Compute the input to Layer 2
• Since neurons 2 and 3 have the same
  input, pick the second neuron as winner

41
P16.5 Continued

• 2-3 Compute the L2-L1 expectation
• 2-4 Adjust the Layer 1 output to include the
  expectation
• 2-5 Determine the match degree Therefore
  (no reset)
• 2-6 Since , continued with step 7.

42
P16.5 Continued

• 2-7 Resonance has occurred, update row 2 of
• 2-8 Update column 2 of
• 3-1 Compute the new Layer 1 response
• 3-2 Compute the input to Layer 2

43
P16.5 Continued

• 3-3 Compute the L2-L1 expectation
• 3-4 Adjust the Layer 1 output to include the
  expectation
• 3-5 Determine the match degree Therefore
  (no reset)
• 3-6 Since , continued with step 7.

44
P16.5 Continued

• 3-7 Resonance has occurred, update row 1 of
• 3-8 Update column 2 of
• This completes the training, since if you apply
  any of the three patterns again they will not
  change the weights. These patterns have been
  successfully clustered.

45
Solved Problem P16.6

• Repeat Problem P16.5, but change the vigilance
  parameter to .
• The training will proceed exactly as in Problem
  P16.5, until pattern p3 is presented.
• 3-1 Compute the Layer 1 response
• 3-2 Compute the input to Layer 2

46
P16.6 Continued

• 3-3 Compute the L2-L1 expectation
• 3-4 Adjust the Layer 1 output to include the
  expectation
• 3-5 Determine the match degree Therefore
  (reset)
• 3-6 Since , set , inhibit it
  until an adequate match occurs (resonance),
  and return to step 1.

47
P16.6 Continued

• 4-1 Recompute the Layer 1 response
• (Layer 2 inactive)
• 4-2 Compute the input to Layer 2
• Since neuron 1 is inhibited, neuron 2 is
  the winner
• 4-3 Compute the L2-L1 expectation
• 4-4 Adjust the Layer 1 output to include the
  expectation

48
P16.6 Continued

• 4-5 Determine the match degree Therefore
  (reset)
• 4-6 Since , set , inhibit it
  until an adequate match occurs (resonance),
  and return to step 1.
• 5-1 Recompute the Layer 1 response
• 5-2 Compute the input to Layer 2
• Since neurons 1 2 are inhibited,
• neuron 3 is the winner

49
P16.6 Continued

• 5-3 Compute the L2-L1 expectation
• 5-4 Adjust the Layer 1 output to include the
  expectation
• 5-5 Determine the match degree Therefore
  (no reset)
• 5-6 Since , continued with step 7.

50
P16.6 Continued

• 5-7 Resonance has occurred, update row 3 of
• 5-8 Update column 2 of
• This completes the training, since if you apply
  any of the three patterns again they will not
  change the weights. These patterns have been
  successfully clustered.

51
Solved Problem P16.7

• Train an ART1 network using the following input
  vectors.
• Present the vectors in the order p1-p2-p3-p1-p4.
  Use the
• parameters and , and choose
  (three
• categories). Train the network until the weights
  have
• converged.
• The initial matrix is an
  matrix of 1s.
• The initial matrix is an
  matrix, with equal to
• blue square ? 1
  white square ? 0
• 5?5 grids ?
  25-dimensional vectors

52
P16.7 Continued

• Training sequence p1-p2-p3-p1-p4
• ? resonance
• v reset