CHAPTER 16 Adaptive Resonance Theory
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.
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 network’s adaptability (or plasticity), which causes prior learning to be eroded by more recent learning.
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.
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.
Overview Grossberg competitive network Basic ART architecture
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.
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 resetin Layer 2.
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.
ART Subsystems Layer 1 Comparison of input pattern and expectation. L1-L2 Connections (Instars) Perform clustering operation. Each row of W1:2 is a prototype pattern. Layer 2 Competition (Contrast enhancement) L2-L1 Connections (Outstars) Perform pattern recall (Expectation). Each column of W2:1 is a prototype pattern Orienting Subsystem Causes a reset when expectation does not match input pattern Disables current winning neuron
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
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.
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).
Steady State Analysis -- 1 • The response of neuron i in Layer 1: • Case 1: Layer 2 is inactive – eachIn steady state:If thenIf thenThe output of Layer 1 is the same as the input pattern
Steady State Analysis -- 2 • Case 2: Layer 2 is active – andIn steady state:Layer 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
Layer 1 Example • Let • Assume that Layer 2 is active and neuron 2 of Layer 2 wins the competition.
Layer 2 From the orienting subsystem
Layer 2 Operation excitatory input • Equation of operation of Layer 2:The rows of adaptive weights , after training, will represent the prototype patterns. on-center feedback adaptive instar inhibitory input off-surround feedback
Layer 2 Example • Let
t Response of Layer 2
Orienting Subsystem • Determine if there is a sufficient match between the L2-L1 expectation (a1) and the input pattern (p)
Orienting Subsyst. Operat. • Equation of operation of the Orienting Subsystem:excitatory input:inhibitory input: • Whenever the excitatory input is larger than the inhibitory input, the Orienting Subsystem will be driven on. inhibitory input excitatory input
Steady State Operation • Steady state:Let , then if , or if (vigilance)The condition that will cause a reset of Layer 2.
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
Orienting Subsystem Ex. • Suppose that • In this case a reset signalwill be sent to Layer 2,since is positive. t
Learning Law • Two separate learning laws:one 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.
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.
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.
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.
Fast Learning • For fast learning, we assume that the outputs of Layer 1 and Layer 2 remain constantuntil the weights reach steady state. • assume that and setCase 1:Case 2:Summary:
Learning Law: L2-L1 • Typical outstar learning:If neuron j in Layer 2 is active(has won the competition), then column j of is moved towarda1. • Fast learning: assume that andColumn 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.
ART1 Algorithm Summary 0. Initialization: The initial is set to all 1’s. 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 activatethe neuron in Layer 2 with the largest inputIn case of tie, the neuron with the smallest index is declared the winner.
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.
Algorithm Summary Cont. 7. Updaterowj of when resonance has occurred: 8. Updatecolumnj 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.
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:
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
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
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:
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.
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:
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.
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.
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:
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.
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
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:
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.
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.