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!