Introduction to Artificial Intelligence CSCI 3202: The Perceptron Algorithm

Greg Grudic Intro AI 1

Introduction to Artificial IntelligenceCSCI 3202:

The Perceptron Algorithm

Greg Grudic

Greg Grudic Intro AI 2

Questions?

Greg Grudic Intro AI 3

Binary Classification

• A binary classifier is a mapping from a set of d inputs to a single output which can take on one of TWO values

• In the most general setting

• Specifying the output classes as -1 and +1 is arbitrary!– Often done as a mathematical convenience

{ }inputs:output:

1, 1

d

yx Î ÂÎ - +

Greg Grudic Intro AI 4

A Binary Classifier

ClassificationModelx ˆ 1, 1y

Given learning data: ( ) ( )1 1, ,..., ,N Ny yx x

A model is constructed:

( )M x

Greg Grudic Intro AI 5

Linear Separating Hyper-Planes

1x

2x

01

0d

i ii

xb b=

+ £å0

1

0d

i ii

xb b=

+ >å

01

0d

i ii

xb b=

+ =å

1y=-

1y=+

Greg Grudic Intro AI 6

Linear Separating Hyper-Planes

• The Model:

• Where:

• The decision boundary:

( )0 1ˆ ˆ ˆˆ ( ) sgn ,..., T

dy M b b bé ù= = +ê úë ûx x

[ ] 1 if 0sgn

1 otherwiseA

Aì >ïï=íï -ïî

( )0 1ˆ ˆ ˆ,..., 0T

db b b+ =x

Greg Grudic Intro AI 7

Linear Separating Hyper-Planes

• The model parameters are:

• The hat on the betas means that they are estimated from the data

• Many different learning algorithms have been proposed for determining

( )0 1ˆ ˆ ˆ, ,..., db b b

( )0 1ˆ ˆ ˆ, ,..., db b b

Greg Grudic Intro AI 8

Rosenblatt’s Preceptron Learning Algorithm

• Dates back to the 1950’s and is the motivation behind Neural Networks

• The algorithm:– Start with a random hyperplane– Incrementally modify the hyperplane such that

points that are misclassified move closer to the correct side of the boundary

– Stop when all learning examples are correctly classified

( )0 1ˆ ˆ ˆ, ,..., db b b

Greg Grudic Intro AI 9

Rosenblatt’s Preceptron Learning Algorithm

• The algorithm is based on the following property:– Signed distance of any point to the boundary is:

• Therefore, if is the set of misclassified learning examples, we can push them closer to the boundary by minimizing the following

( ) ( )( )0 1 0 1ˆ ˆ ˆ ˆ ˆ ˆ, ,..., ,..., T

d i d ii M

D yb b b b b bÎ

=- +å x

( ) ( )0 1

0 1

2

1

ˆ ˆ ˆ,...,ˆ ˆ ˆ,...,

ˆ

Td T

dd

ii

dx

xb b b

b b bb

=

+= µ +æ ö÷ç ÷ç ÷ç ÷è øå

x

M

Greg Grudic Intro AI 10

Rosenblatt’s Minimization Function

• This is classic Machine Learning!• First define a cost function in model

parameter space

• Then find an algorithm that modifies such that this cost function is minimized

• One such algorithm is Gradient Descent

( )0 1 01

ˆ ˆ ˆ ˆ ˆ, ,...,d

d i k iki M k

D y xb b b b bÎ =

æ ö÷ç=- + ÷ç ÷ç ÷çè øå å( )0 1

ˆ ˆ ˆ, ,..., db b b

Greg Grudic Intro AI 11

Gradient Descent

( )0 1ˆ ˆ ˆ, ,..., dD b b b =

0b̂ = 1̂b =

Greg Grudic Intro AI 12

The Gradient Descent Algorithm

( )0 1ˆ ˆ ˆ, ,...,

ˆ ˆˆ

d

i ii

D b b bb b r

b

¶¬ -

¶

Where the learning rate is defined by: 0r >

Greg Grudic Intro AI 13

The Gradient Descent Algorithm for the Perceptron

0 0

11 1

ˆ ˆ

ˆ ˆ

ˆ ˆ

i

i i

i idd d

yy x

y x

b bb b r

b b

æ ö æ ö æ ö÷ ÷ç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷¬ - ç ÷ç ç÷ ÷ ç ÷÷ ÷ç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ç÷ ÷ çè øç ç÷ ÷ç ç ÷÷ ÷è ø è øç ç÷ ÷

MM M

( )0 1

0

ˆ ˆ ˆ, ,...,ˆ

d

ii M

Dy

b b b

b Î

¶=-

¶ å( )0 1

ˆ ˆ ˆ, ,...,, 1,...,ˆ

d

i iji Mj

Dy x j d

b b b

b Î

¶=- =

¶ å

0 0

11 1

ˆ ˆ

ˆ ˆ

ˆ ˆ

ii M

i ii M

d d i idi M

y

y x

y x

b bb b r

b b

Î

Î

Î

æ ö- ÷ç ÷æ ö æ ö ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷ -çç ç ÷÷ ÷ ç ÷ç ç÷ ÷ ÷çç ç÷ ÷¬ - ÷çç ç÷ ÷ ÷ç÷ ÷ç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷ ç ÷ç ç÷ ÷è ø è ø - ÷ç ÷ç ÷çè ø

åå

åM M M

Update One misclassified point at a time (online)

Update all misclassified points at once (batch)

Two Versions of the Perceptron Algorithm:

Greg Grudic Intro AI 14

The Learning Data

• Matrix Representation of N learning examples of d dimensional inputs

11 1 1

1

, d

N Nd N

x x yX Y

x x y

æ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç÷ ÷= =ç ç÷ ÷ç ç÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

KM O M M

L

( ) ( )1 1, ,..., ,N Ny yx xTraining Data:

Greg Grudic Intro AI 15

The Good Theoretical Properties of the Perceptron Algorithm

• If a solution exists the algorithm will always converge in a finite number of steps!

• Question: Does a solution always exist?

Greg Grudic Intro AI 16

Linearly Separable Data

• Which of these datasets are separable by a linear boundary?

+

a) b)

++

--

-

+

+

-

-

-

Greg Grudic Intro AI 17

Linearly Separable Data

• Which of these datasets are separable by a linear boundary?

+

a) b)

++

--

-

+

+

-

-

- NotLinearly

Separable!

Greg Grudic Intro AI 18

Bad Theoretical Properties of the Perceptron Algorithm

• If the data is not linearly separable, algorithm cycles forever!– Cannot converge!– This property “stopped” active research in this area between

1968 and 1984…• Perceptrons, Minsky and Pappert, 1969

• Even when the data is separable, there are infinitely many solutions– Which solution is best?

• When data is linearly separable, the number of steps to converge can be very large (depends on size of gap between classes)

Greg Grudic Intro AI 19

What about Nonlinear Data?

• Data that is not linearly separable is called nonlinear data

• Nonlinear data can often be mapped into a nonlinear space where it is linearly separable

Greg Grudic Intro AI 20

Nonlinear Models

• The Linear Model:

• The Nonlinear (basis function) Model:

• Examples of Nonlinear Basis Functions:

01

ˆ ˆˆ ( ) sgnd

i ii

y M xb b=

é ùê ú= = +ê úë ûåx

( )01

ˆ ˆˆ ( ) sgnk

i ii

y M b bf=

é ùê ú= = +ê úë ûåx x

( ) ( ) ( ) ( ) ( )2 21 1 2 2 3 1 2 4 55sinx x x x xff ff= = = =x x x x

Greg Grudic Intro AI 21

Linear Separating Hyper-Planes In Nonlinear Basis Function Space

1f

2f

01

0k

i ii

b bf=

+ £å0

1

0k

i ii

b bf=

+ >å

01

0k

i ii

b bf=

+ =å

1y=-

1y=+

Greg Grudic Intro AI 22

An Example

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1

x 2

: y=+1: y=-1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 = x1

2

2 = x

22

: y=+1: y=-1

F

Greg Grudic Intro AI 23

Kernels as Nonlinear Transformations

• Polynomial

• Sigmoid

• Gaussian or Radial Basis Function (RBF)

( ) ( )( ) ( )( ) 2

2

, ,

, tanh ,

1, exp2

k

i j i j

i j i j

i j i j

K q

K

K

x x x x

x x x x

x x x x

k q

s

= += +

æ ö÷ç= - - ÷ç ÷÷çè ø

Greg Grudic Intro AI 24

The Kernel Model

( )01

ˆ ˆˆ ( ) sgn ,N

i ii

y M Kx x xb b=

é ùê ú= = +ê úë ûå

( ) ( )1 1, ,..., ,N Ny yx xTraining Data:

The number of basis functions equals the number of training examples!

- Unless some of the beta’s get set to zero…

Greg Grudic Intro AI 25

Gram (Kernel) Matrix

( ) ( )

( ) ( )

1 1 1

1

, ,

, ,

N

N N N

K KK

K K

æ ö÷ç ÷ç ÷ç ÷=ç ÷ç ÷ç ÷ç ÷÷çè ø

x x x x

x x x x

KM O M

L

( ) ( )1 1, ,..., ,N Ny yx xTraining Data:

Properties:•Positive Definite Matrix

•Symmetric•Positive on diagonal

•N by N

Greg Grudic Intro AI 26

Picking a Model Structure?• How do you pick the Kernels?

– Kernel parameters• These are called learning parameters or

hyperparamters– Two approaches choosing learning paramters

• Bayesian– Learning parameters must maximize probability of correct

classification on future data based on prior biases• Frequentist

– Use the training data to learn the model parameters – Use validation data to pick the best hyperparameters.

• More on learning parameter selection later

( )0 1ˆ ˆ ˆ, ,..., db b b

Greg Grudic Intro AI 27

Perceptron Algorithm Convergence

• Two problems:– No convergence when data is not separable in

basis function space– Gives infinitely many solutions when data is

separable• Can we modify the algorithm to fix these

problems?