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Framework For PDP Models
Psych 85-419/719
Jan 18, 2001
What A Model Is
A set of processing unitsA pattern of connectivity
A propagation rule net to generate input a
a A transfer function ff
.. To generate output o
o
A learning rule to change the connections
A training environment
cat dog
meows barks
.. With weights
w
w
w
w
Formally...
• Consider unit j communicating with unit i at time t.
• neti(t) = g(wj,i,oj(t-1)) (propagation rule)
• ai(t) = netj(t) (activation function)
• oi(t)= f(ai(t)) (transfer function)
The Propagation Rule net
• Computes input to unit ai
– From outputs of other units oj
– And the weights from those units wj,i
• Ex: weighted sumai = SUMj(wj,I * oj)
• Ex: sigma pi unitai = PRODUCTj(wj,I * oj)
Weighted Sum:Matrix Algebra
.1 .5 1.0
.1 .5 1.0
2
1.5
-3.2-6
2 -6 -31 .2 5X =
-5.85.2
-5.8 5.2
The Transfer Functionf: o=f(a)
Simple linear:
o=k * a
-2
-1
0
1
2
-1 -0.5 0 0.5 1
k=1
k=1/2
k=2
Transfer Functions:Clipped Linear
• o = zero (if a < zero)• o = 1 (if a > 1)• o = a (otherwise)
-2
-1
0
1
2
-2 -1 0 1 2
Transfer Functions:Threshold
• o = 1 (if a > thresh)• o = zero otherwise
thresh
Transfer Functions:Logistic
• o = 1/(1+e-a)
-1
-0.5
0
0.5
1
-4.0 -2.0 0.0 2.0 4.0
Transfer Functions:Hyperbolic Tangent
• o = TANH(a)
-1
-0.5
0
0.5
1
-4.0 -2.0 0.0 2.0 4.0
Transfer Functions With and Without Memory
• Memoryless: The output is a function of its activation at that moment in time. Stateless.
• With memory: The output is a function not only of the activation at time t, but its own output at time t-1.
• Which one is more like real neurons?
• Can you build a unit with memory out of units without memory?
w
Time In PDP Networks
• Feedforward networks– Activity is passed from input to output in one pass
• Discrete time networks– At each time sample, each unit computes its output
based on output of other units at previous time step
• Continuous time networks– Activation ramps up gradually over time
Discrete Time Example
1
0.1
11
0.10.1
0.1
0.2
0.1
0.2
0.2
0.3
0.2
Continuous Time Example
1
0.1
1-0.5
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 10 20 30 40 50 60 70 80 90
Learning in Models
• “Knowledge is in the weights!”– Grow new connections– Prune existing connections– Modify existing connections
• Examples?– Developmental biology– Learning skills– Unlearning (brain damage: stroke, Alzheimer’s, etc.)
Types of Learning
• Supervised learning– Input and target outputs are specified– Learning involves teaching network what correct
output should be
• Reinforcement learning– Actual target isn’t specified. Reward is given
• Unsupervised learning– No teaching signal available– “discover” interesting things about environment
Training Data (Environment)
• Specifies input to network
• Can specify targets (with supervised), or reward (with reinforcement)
• Inputs can be sampled by network, or pre-specified
• Sequential, random, or weighted random
An Example: Content Addressable Memory
• Humans can recall items from memory based on partial information; a subset of that memory– “What was the name of that guy in my math
class who always wore the Misfits t-shirt?”
• Graceful degradation: you don’t lose all information as you forget things
The Challenge
• Standard data structures, as used by computers, don’t tend to have these properties
• Why would human memory have these properties in the first place?
• A simple model to demonstrate how a neural system gives rise to these behaviors
An Example Model:Interactive Activation
• The theory: processing units representing features of the world are interconnected
• Their dynamics are such that for coherent memories, they maintain each other’s state
• Partial information can bootstrap further memory through the weights between units
An Example Model: Interactive Activation (details)
• Unit output zero if activity below threshold
• Equal to difference between activity and threshold if above threshold
• Decay term: units tended to decay, lacking proper excitation
• See PDP1, pages 71-72 for more details
Jets and Sharks
The Jets and Sharks Network
Nice Properties of this Model• Content addressability
– “Who is a Shark in their 20’s?”
• Graceful degradation– “I know Al is a burglar in his 30’s… is he in the Jets or
the Sharks?”
• Default assignment– “What could Lance’s job be?”
• Spontaneous generalization– “What is a member of the Jets like?”
For Next Time
• Read for class: PDP2, Chapter 14, pages 7-38 only
• Also read material handed out today
• Homework 1 will be handed out. Don’t fall behind on the reading!