Learning in One-Layer Networks Psych 209 January 9, 2020 - - PowerPoint PPT Presentation

Learning in One-Layer Networks

Psych 209 January 9, 2020

Input-output mapping

Simplest model of learning: input-output mapping

Input-output mapping

a1 a2 . . . an Input stimulus Input pattern b1 b2 . . . bn Output pattern Output behavior Pattern associator

Input-output mapping

How can we learn input-output associations? We know how to store (key, value) pairs on a computer hash address a1 a2 . . . an Input pattern b1 b2 . . . bn Output pattern

Input-output mapping

hash address a1 a2 . . . an b1 b2 . . . bn But location-based memory is brittle hash different address a1’ a2’ . . . an’ Similar input patterns z1 z2 . . . zn Totally different

• utput pattern

Pattern Associator

neuron (cell body) connection (axon)

Input pattern Output pattern Fully connected Content-addressable memory: keys directly “produce” values without location lookup Desirable properties

Forward Pass

Input pattern Output pattern Input 0 Output 0 Output 1 Input 1 Input 2 Input 3 w00 w01 w02 w03

Forward Pass

Input pattern Output pattern Input 0 Output 0 Output 1 Input 1 Input 2 Input 3 w00 w01 w02 w03

ai = X

j

wijaj

Forward Pass

Input pattern Output pattern Input 0 Output 0 Output 1 Input 1 Input 2 Input 3 w00 w01 w02 w03 w00 w01 w02 w03 w10 w11 w12 w13 inp0 inp1 inp2 inp3

• ut0
• ut1
• ut = W inp

Matrix Representation

w00 w01 w02 w03

How do we learn the weights?

w00 w01 w02 w03

How do we learn the weights?

w00 w01 w02 w03

Learning rules:

• 1. Hebb’s rule
• 2. Delta rule

Hebb’s Rule

• “Neurons that fire together wire together”

Hebb’s Rule

• “Neurons that fire together wire together”

1 1 1

• 1

1

• 1

1

• 1

input (u) target (v)

u ⊗ v

Hebb’s Rule

• “Neurons that fire together wire together”

1 1 1

• 1
• 1
• 1
• 1

1 1 1 1

• 1
• 1
• 1
• 1

1 1 1 1

• 1

1

• 1

1

• 1

input (u) target (v)

u ⊗ v

Storing a memory

Hebb’s Rule

2 2

• 2

2 2 2

• 2

2

• 1

1

• 1
• 1

1 1 1

• 1
• 1

1

• 1
• 1
• 1

1

• 1
• 1

inputs targets Storing multiple memories

UVT

U

VT

Hebb’s Rule

1

• 1

3

• 1
• 1
• 3

1 1 3 1 1 1 1

• 1
• 1

3

• 1
• 1

1

• 1
• 1

1

• 1
• 1

1 1 1

• 1

1

• 1

1 1

• 1
• 1
• 1
• 1

1

• 1
• 1
• 1

inputs targets

UVT

Storing multiple memories

U

VT

Hebb’s Rule

1

• 1

3

• 1
• 1
• 3

1 1 3 1 1 1 1

• 1
• 1

3 4 4 4

• 4

query

1

• 1

1

• 1

1 1 1

• 1

UVT

Retrieval threshold (clip to -1, 1)

Hebb’s Rule

1

• 1

3

• 1
• 1
• 3

1 1 3 1 1 1 1

• 1
• 1

3

• 1
• 1

1

• 1
• 1

1

• 1
• 1

1 1 1

• 1

1

• 1

1 1

• 1
• 1
• 1
• 1

1

• 1
• 1
• 1

inputs targets

UVT

U

VT

Observe that the input patterns here are an

• rthogonal set

Orthogonality

a ? b ( ) cos(θab) = 0 ( ) ha, bi = kakkbkcos(θab) = 0

Say we want to query the memory with input u_i: So if the inputs are orthogonal, retrieval will be correct up to a positive scale factor, which could be corrected by thresholding in our binary case.

Hebb’s Rule

uT

i UVT = [hui, u0i, . . . , hui, uii, . . . , hui, uni]VT

= [0, . . . , kuik2, . . . , 0]VT = kuik2vT

i

Hebb’s Rule

∆wij = ✏aiaj

• 1

1

aj

ai

• Incremental version:

• Incremental version:

Hebb’s Rule

∆wij = ✏

n

X

p=1

aipajp

aj

ai

Hebb’s Rule

∆w40 = ✏

3

X

p=0

a4pa0p

= ✏ + ✏ − ✏ − ✏ = 0

Hebb’s Rule

∆w40 = 0

∆w41 = 0

∆w42 = 2✏

∆w43 = 0

Hebb’s Rule

Delta Rule

0.5 0.5 1

• 1

1

• 1

input: [1, -1, 1, -1]

• utput: [0.5, 0.5]

target: [1, 1] error, or tss: (1-0.5)2 +(1-0.5)2 = 0.25 + 0.25 = 0.5

Delta Rule

0.5 0.5 1

• 1

1

• 1

input: [1, -1, 1, -1]

• utput: [0.5, 0.5]

target: [1, 1] error, or tss: (1-0.5)2 +(1-0.5)2 = 0.25 + 0.25 = 0.5

∆wij = ✏ @E @wij

E = 1 2 X

i

(yi − ˆ yi)2

Delta Rule

∆wij = −✏ @E @wij

E = 1 2 X

i

(yi − ˆ yi)2

Delta Rule

∂E ∂wij = ∂E ∂ˆ yi ∂ˆ yi ∂wij

∆wij = −✏ @E @wij

E = 1 2 X

i

(yi − ˆ yi)2

Delta Rule

∂E ∂wij = ∂E ∂ˆ yi ∂ˆ yi ∂wij

∂E ∂ˆ yi = 1 2 X

k

∂ ∂ˆ yi (yk − ˆ yk)2

∆wij = −✏ @E @wij

E = 1 2 X

i

(yi − ˆ yi)2

Delta Rule

∂E ∂wij = ∂E ∂ˆ yi ∂ˆ yi ∂wij

∂E ∂ˆ yi = 1 2 X

k

∂ ∂ˆ yi (yk − ˆ yk)2 = 1 2 ∂ ∂ˆ yi (yi − ˆ yi)2 = 1 2 ∗ 2(yi − ˆ yi) ∗ −1 = −(yi − ˆ yi)

∆wij = −✏ @E @wij

E = 1 2 X

i

(yi − ˆ yi)2

Delta Rule

∂E ∂wij = ∂E ∂ˆ yi ∂ˆ yi ∂wij

−(yi − ˆ yi)

xj

∆wij = −✏ @E @wij

E = 1 2 X

i

(yi − ˆ yi)2

Delta Rule

∂E ∂wij = ∂E ∂ˆ yi ∂ˆ yi ∂wij

= −(yi − ˆ yi)xj

∆wij = −✏ @E @wij

E = 1 2 X

i

(yi − ˆ yi)2

Delta Rule

∂E ∂wij = ∂E ∂ˆ yi ∂ˆ yi ∂wij

= −(yi − ˆ yi)xj

∆wij = ✏(yi − ˆ yi)xj

Update rule:

∆wij = −✏ @E @wij

• (…)

Homework question + lr demo

• These are linear models: outputs can only be expressed

as a linear combination of inputs

• Usually, we need nonlinear functions, e.g. to capture

mapping from images to class labels (not just a linear combination of pixels)

Linear Predictability Constraint