Structure and Predictions

Structure and Predictions

Machine Learning: Jordan Boyd-Graber

University of Maryland

PERCEPTRON: SLIDES ADAPTED FROM LIANG HUANG

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How do we set the feature weights?

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Goal is to minimize errors

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Want to reward features that lead to right answers

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Penalize features that lead to wrong answers

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Problem: predictions are correlated

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Perceptron Algorithm

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Rather than just counting up how often we see events?

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We’ll use this for intuition in 2D case

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Perceptron Algorithm

1: ^w ~ 1 ← ~ ⁰

2: for t ← ¹ . . . T do

3: Receive x _t

4: ˆ y _t ← ^sgn ( w _{~ t} · ~ ^x t )

5: Receive y _t

6: if ˆ y _t 6 = y _t then

7: ^w ~ t + 1 ← ~ ^w t + y _t ~ ^x t

8: else

9: ^w ~ t + 1 ← ^w t

return w _{T + 1}

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Binary to Structure

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Binary to Structure

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Binary to Structure

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Generic Perceptron

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perceptron is the simplest machine learning algorithm

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online-learning: one example at a time

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learning by doing

ƒ

find the best output under the current weights

ƒ

update weights at mistakes

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2D Example

Initially, weight vector is zero:

~

w ₁ = _〈 0 , 0 〉 ⁽¹⁾

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Observation 1

x ₁ = _〈− 2 , 2 〉 ⁽²⁾ ˆ

y ₁ = 0 (3)

y ₁ = + 1 (4)

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Update 1

~

w _{t + 1} ← ~ ^w t + y _t ^x ~ t (5)

~

w ₂ ← ⁽⁶⁾

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Update 1

~

w _{t + 1} ← ~ ^w t + y _t ^x ~ t (5)

~

w ₂ ← 〈 ⁰ , 0 〉 + _〈− 2 , 2 〉 ⁽⁶⁾ (7)

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Update 1

~

w _{t + 1} ← ~ ^w t + y _t ^x ~ t (5)

~

w ₂ ← 〈 ⁰ , 0 〉 + _〈− 2 , 2 〉 ⁽⁶⁾

~

w ₂ = _〈− 2 , 2 〉 ⁽⁷⁾

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Observation 2

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Observation 2

x ₂ = _〈− 2 , − ³ 〉 ⁽⁸⁾ ˆ

y ₂ = + 4 + ₋ 6 = ₋ 2 (9)

y ₂ = ₋ 1 (10)

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Update 2

~

w _{t + 1} ← ~ ^w t (11)

~

w ₂ ← ⁽¹²⁾

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Update 2

~

w _{t + 1} ← ~ ^w t (11)

~

w ₂ ← 〈− ² , 2 〉 ⁽¹²⁾

(13)

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Update 2

~

w _{t + 1} ← ~ ^w t (11)

~

w ₂ ← 〈− ² , 2 〉 ⁽¹²⁾

~

w ₂ = _〈− 2 , 2 〉 ⁽¹³⁾

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Observation 3

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Observation 3

x ₃ = _〈 2 , − ¹ 〉 ⁽¹⁴⁾ ˆ

y ₃ = ₋ 4 + ₋ 2 = ₋ 6 (15)

y ₃ = + 1 (16)

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Update 3

~

w _{t + 1} ← ~ ^w t + y _t ^x ~ t (17)

~

w ₃ ← ⁽¹⁸⁾

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Update 3

~

w _{t + 1} ← ~ ^w t + y _t ~ ^x t (17)

~

w ₃ ← 〈− ² , 2 〉 + _〈 2 , − ¹ 〉 ⁽¹⁸⁾ (19)

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Update 3

~

w _{t + 1} ← ~ ^w t + y _t ~ ^x t (17)

~

w ₃ ← 〈− ² , 2 〉 + _〈 2 , − ¹ 〉 ⁽¹⁸⁾

~

w ₃ = _〈 0 , 1 〉 ⁽¹⁹⁾

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Observation 4

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Observation 4

x ₄ = _〈 1 , − ⁴ 〉 ⁽²⁰⁾ ˆ

y ₄ = ₋ 4 (21) y ₄ = ₋ 1 (22)

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Update 4

~

w ₄ ← ⁽²³⁾

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Update 4

~

w ₄ ← ~ ^w 3 (23)

(24)

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Update 4

~

w ₄ ← ~ ^w 3 (23)

~

w ₄ = _〈 0 , 1 〉 ⁽²⁴⁾

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Observation 5

x ₅ = _〈 2 , 2 〉 ⁽²⁵⁾ ˆ

y ₅ = 2 (26) y ₅ = + 1 (27)

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Update 5

~

w ₅ ← ⁽²⁸⁾

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Update 5

~

w ₅ ← ~ ^w 4 (28)

(29)

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Update 5

~

w ₅ ← ~ ^w 4 (28)

~

w ₅ = _〈 0 , 1 〉 ⁽²⁹⁾

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Observation 6

x ₆ = _〈 2 , 2 〉 ⁽³⁰⁾ ˆ

y ₆ = 2 (31) y ₆ = + 1 (32)

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Update 6

~

w ₆ ← ⁽³³⁾

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Update 6

~

w ₆ ← ~ ^w 5 (33)

(34)

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Update 6

~

w ₆ ← ~ ^w 5 (33)

~

w ₆ = _〈 0 , 1 〉 ⁽³⁴⁾

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Structured Perceptron

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Perceptron Algorithm

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POS Example

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