Multilayer Perceptrons Lecture 11labs.seas.wustl.edu/bme/...MultiLayerPerceptrons.pdf · 4!...

Multilayer Perceptrons!!

Lecture 11!

Chain Rule!g Letʼs say we have two functions f(x) and g(x)"

g What is the derivative of f(g(x))?"!

f (x) = x 5

g(x) = (x 2 +1)f (g(x)) = (x 2 +1)5

df (x)dx

= 5x 4

df (g)dg

dg(x)dx

df (g(x))dx

=df (g)dg

"dg(x)dx

= 5(x 2 +1)4 " 2x

Chain Rule!

y = esinx2

= esin x2

!cos x2 !2x

= esin x2

= cos x2

=dydu!dudv!dvdx

v = h(x)

u = g(v) =sin(v)

y = f(u)

Multilayer Perceptron!g Graph of a multilayer perceptron with two

hidden layers"

Input Layer

First Hidden Layer

Second Hidden Layer

Output Layer

Signal Flow Graph (Output Neuron j)!

yo=+1 [Bias-term]

yi(n) wji(n) vj(n) yj(n)

φj(vj(n))

dj(n) [desired output value]

-1 ej(n)

(error at node j)

v j (n) = w ji(n)yi(n)i=0

wj0(n)

yo=+1 [Bias-term]

φj(vj(n))

-1 ej(n)

(error at node j)

y j (n) =" j (v j (n))

wj0(n)

yo=+1 [Bias-term]

φj(vj(n))

-1 ej(n)

(error at node j)

y j (n) =" j (v j (n))

e j (n) = d j (n) " y j (n)

wj0(n)

Goal: Minimize Total Error at the O/P!

φj(vj(n))

-1 ej(n)

y j (n) =" j (v j (n))

e j (n) = d j (n) " y j (n)

wj0(n)

E(n) =12

2 (n)j"O /P#

Update Weights!

φj(vj(n))

-1 ej(n)

y j (n) =" j (v j (n))

e j (n) = d j (n) " y j (n)

wj0(n)

w ji(n +1) = w ji(n) "#$E(n)$w ji(n)

Small step against the direction of the gradient to minimize error

Update for Weights!

"g where"

w ji(n +1) = w ji(n) "#$E(n)$w ji(n)

E(n) =12

2 (n)j"O /P#

e j (n) = d j (n) $ y j (n)y j (n) =% j (v j (n))

Update for Weights!

""g Applying chain rule we get:""

w ji(n +1) = w ji(n) "#$E(n)$w ji(n)

"E(n)"w ji(n)

="E(n)"e j (n)

#"e j (n)"y j (n)

#"y j (n)"v j (n)

#"v j (n)"w ji(n)

E(n) =12

2 (n)j"O /P#

Update for Weights!g Applying chain rule we get:""

"E(n)"w ji(n)

="E(n)"e j (n)

#"e j (n)"y j (n)

#"y j (n)"v j (n)

#"v j (n)"w ji(n)

"E(n)"e j (n)

"e j (n)12

2 (n)j$O /P%

* + + = e j (n)

E(n) =12

2 (n)j"O /P#

E(n) =12

2 (n)j"O /P#

"E(n)"w ji(n)

="E(n)"e j (n)

#"e j (n)"y j (n)

#"y j (n)"v j (n)

#"v j (n)"w ji(n)

"E(n)"e j (n)

"e j (n)12

2 (n)j$O /P%

* + + = e j (n)

"e j (n)"y j (n)

"y j (n)d j (n) , y j (n)( ) = ,1

"E(n)"w ji(n)

="E(n)"e j (n)

#"e j (n)"y j (n)

#"y j (n)"v j (n)

#"v j (n)"w ji(n)

"E(n)"e j (n)

"e j (n)12

2 (n)j$O /P%

* + + = e j (n)

"e j (n)"y j (n)

"y j (n)d j (n) , y j (n)( ) = ,1

"y j (n)"v j (n)

"v j (n)- j (v j (n))( ) =- j '(v j (n))

E(n) =12

2 (n)j"O /P#

"E(n)"w ji(n)

="E(n)"e j (n)

#"e j (n)"y j (n)

#"y j (n)"v j (n)

#"v j (n)"w ji(n)

"E(n)"e j (n)

"e j (n)12

2 (n)j$O /P%

* + + = e j (n)

"e j (n)"y j (n)

"y j (n)d j (n) , y j (n)( ) = ,1

"y j (n)"v j (n)

"v j (n)- j (v j (n))( ) =- j '(v j (n))

"v j (n)"w ji(n)

"w ji(n)w ji(n)yi(n)

* + = yi(n)

E(n) =12

2 (n)j"O /P#

Update for Weights!g Applying chain rule we get:"

"E(n)"w ji(n)

="E(n)"e j (n)

#"e j (n)"y j (n)

#"y j (n)"v j (n)

#"v j (n)"w ji(n)

"E(n)"w ji(n)

= e j (n)# ($1)# % j '(v j (n))# yi(n)

Update for Weights!g Applying chain rule we get:"

g Delta rule for updating weights"!

"E(n)"w ji(n)

="E(n)"e j (n)

#"e j (n)"y j (n)

#"y j (n)"v j (n)

#"v j (n)"w ji(n)

"E(n)"w ji(n)

= e j (n)# ($1)# % j '(v j (n))# yi(n)

w ji(n +1) = w ji(n) "#$E(n)$w ji(n)

% w ji(n +1) = w ji(n) + #learningrate

! & e j (n)& ' j '(v j (n))local gradient

" # $ $ % $ $ & yi(n)

input!

Update for Weights!g Delta rule for updating weights"

g  For output neurons this rule can be directly applied"

w ji(n +1) = w ji(n) "#$E(n)$w ji(n)

! & ' j (n)local gradient" # $

& yi(n)input!

' j (n) =$E(n)$v j (n)

= e j (n)& ( j)(v j (n))

How to update weights for hidden layers?!g Delta rule for updating weights"

g Credit-assignment problem:"n  Even though the hidden neurons are not directly

accessible they share responsibility for the error"n  How to penalize or reward hidden neurons?"

w ji(n +1) = w ji(n) "#$E(n)$w ji(n)

! & ' j (n)local gradient" # $

& yi(n)input!

Signal-flow at hidden node h!

yh(n) wjh(n) vj(n) yj(n)

φj(vj(n))

dj(n)]

-1 ej(n)

wj0(n)

yi(n) whi(n) vh(n)

φh(vh(n))

wh0(n)

yM2(n)

How to update weights for hidden layers?!g Local gradient of hidden neuron ʻhʼ:"

"h (n) = #$E(n)$vh (n)

= #$E(n)$yh (n)

%$yh (n)$vh (n)

"h (n) = #$E(n)$vh (n)

= #$E(n)$yh (n)

%$yh (n)$vh (n)

"h (n) = #$E(n)$yh (n)

$vh (n)(&h (vh (n)))

"h (n) = #$E(n)$vh (n)

= #$E(n)$yh (n)

%$yh (n)$vh (n)

"h (n) = #$E(n)$yh (n)

$vh (n)(&h (vh (n)))

"h (n) = #$E(n)$yh (n)

% (&h'(vh (n)))

"g We know"

"h (n) = #$E(n)$yh (n)

% (&h'(vh (n)))

E(n) =12

e j2(n)

j"O /P#

$E(n)$yh (n)

= e j (n)j"O /P#

$e j (n)$yh (n)

"g We know"

g Again applying chain rule:"

"h (n) = #$E(n)$yh (n)

% (&h'(vh (n)))

"E(n)"yh (n)

= e j (n)j#O /P$

"e j (n)"yh (n)

"E(n)"yh (n)

= e j (n)j#O /P$

"e j (n)"v j (n)

%"v j (n)"yh (n)

"h (n) = #$E(n)$yh (n)

% (&h'(vh (n)))

"E(n)"yh (n)

= e j (n)j#O /P$

"e j (n)"v j (n)

%"v j (n)"yh (n)

"h (n) = #$E(n)$yh (n)

% (&h'(vh (n)))

"E(n)"yh (n)

= e j (n)j#O /P$

"e j (n)"v j (n)

%"v j (n)"yh (n)

"e j (n)"v j (n)

"v j (n)(d j (n) # y j (n))

"h (n) = #$E(n)$yh (n)

% (&h'(vh (n)))

"E(n)"yh (n)

= e j (n)j#O /P$

"e j (n)"v j (n)

%"v j (n)"yh (n)

"e j (n)"v j (n)

"v j (n)(d j (n) # y j (n))

"e j (n)"v j (n)

"v j (n)(d j (n) #$ j (v j (n))) = #$ j

%(v j (n))

"h (n) = #$E(n)$yh (n)

% (&h'(vh (n)))

"E(n)"yh (n)

= e j (n)j#O /P$

"e j (n)"v j (n)

%"v j (n)"yh (n)

"e j (n)"v j (n)

"v j (n)(d j (n) # y j (n))

"e j (n)"v j (n)

"v j (n)(d j (n) #$ j (v j (n))) = #$ j

%(v j (n))

"v j (n)"yh (n)

"yh (n)w jh (n)yh (n)

+ , = w jh (n)

"h (n) = #$E(n)$yh (n)

% (&h'(vh (n)))

"h (n) =#h$(vh (n)) e j (n)% # j (v j (n))

local gradient! " # # $ # #

% w jh (n)j&O /P'

"h (n) =#h$(vh (n)) " j (n)% w jh (n)

j&O /P'

"h (n) =#h$(vh (n)) " j (n)% w jh (n)

j&O /P'

whi(n +1) = whi(n) + "learningrate

! # $h (n)local gradient" # $ # yi(n)

input!

Back-propagation of errors!

δj(n)

φj’(vj(n)) ej(n) wjh(n)

δM(n)

φ1’(v1(n))

φM’(vM(n))

δ1(n)

δh(n)

"h (n) =#h$(vh (n)) " j (n)% w jh (n)

j&O /P'

Intuition: weight the error at each output node by the connection weights of the hidden node to the output node and assign that as the error caused by the hidden node

Back Propagation Algorithm!g Output node"

g Hidden node"

w ji(n +1) = w ji(n) + "learningrate

! # $ j (n)local gradient" # $

# yi(n)input!

"h (n) =#h$(vh (n)) " j (n)% w jh (n)

j&O /P'

whi(n +1) = whi(n) + "learningrate

! # $h (n)local gradient" # $ # yi(n)

input!

" j (n) = e j (n)# $ j%(v j (n))

An Example!g  Lets assume a simple MLP with one hidden

layer"

x0=1 y0=1

Input Layer

Hidden Layer

Output Layer

An Example!g  Begin with random assignment of weights"

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

Input Layer

Hidden Layer

Output Layer

An Example!g  Let input x=[0,1] and the desired output be

d=[1,0]; η=0.1"

u11= -1

u22= 1

w11= 1

x0=1 y0=1

w20= 1

Input Layer

Hidden Layer

Output Layer

An Example!g Forward pass: Hidden Layer"

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

Input Layer

Hidden Layer

Output Layer

v1 = u10x0+u11x1+u12x2 v2 = u20x0+u21x2+u22x2

An Example!g Forward pass"

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

Input Layer

Hidden Layer

Output Layer

v1 = u10x0+u11x1+u12x2 v2 = u20x0+u21x2+u22x2

An Example!g Forward pass: Lets assume identity

activation function: φj(x)=x "

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

Input Layer

Hidden Layer

Output Layer

y1=φ(v1) y2=φ(v2)

An Example!g Forward pass: Lets assume identity

activation function: φj(x)=x "

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

Input Layer

Hidden Layer

Output Layer

y1=φ(v1) y2=φ(v2)

"'(x) =1[ ]

An Example!g Forward pass: Output layer"

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

ov1 = w10y0+w11y1+w12y2 ov2 = w20y0+w21y2+w22y2

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

ov1 = w10y0+w11y1+w12y2 ov2 = w20y0+w21y2+w22y2

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

o1=φ(ov1) o2=φ(ov2)

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

Desired o/p d1=1 d2=0

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

e1=d1-o1

e2=d2-o2

An Example!g Backward pass: Output layer"

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

w ji(n +1) = w ji(n) +"e j (n)yi(n) [output]

u ji(n +1) = u ji(n) +"xi(n) e j (n)# w jh (n) [hidden]j$O /P%

An Example!g Backward pass: "

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

" j (n) = e j (n)# $ j%(v j (n))

"h (n) =$h%(vh (n)) " j (n)# w jh (n)

j&O /P'

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

δ1=-1

δ2=-2

" j (n) = e j (n)# $ j%(v j (n))

"h (n) =$h%(vh (n)) " j (n)# w jh (n)

j&O /P'

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

δ1=-1

δ2=-2

" j (n) = e j (n)# $ j%(v j (n))

"h (n) =$h%(vh (n)) " j (n)# w jh (n)

j&O /P'

δ1h=?

δ2h=?

u11= -1

v22= 1

δ1w11

x0=1 y0=1

δ2w20

δ1=-1

δ2=-2

" j (n) = e j (n)# $ j%(v j (n))

"h (n) =$h%(vh (n)) " j (n)# w jh (n)

j&O /P'

δ1h=?

δ2h=?

u11= -1

v22= 1

δ1w11

x0=1 y0=1

δ2w20

δ1=-1

δ2=-2

" j (n) = e j (n)# $ j%(v j (n))

"h (n) =$h%(vh (n)) " j (n)# w jh (n)

j&O /P'

δ1h=1

δ2h=-2

u11= -1

v22= 1

w11= 1

x0=1 y0=1

w20= 1

δ1=-1

δ2=-2

" j (n) = e j (n)# $ j%(v j (n))

"h (n) =$h%(vh (n)) " j (n)# w jh (n)

j&O /P'

δ1h=1

δ2h=-2

u11= -1

v22= 1

w11= 0.9

x0=1 y0=1

w20= 0.8

δ1=-1

δ2=-2

" j (n) = e j (n)# $ j%(v j (n))

"h (n) =$h%(vh (n)) " j (n)# w jh (n)

j&O /P'

δ1h=1

δ2h=-2

u11= -1

v22= 1

w11= 0.9

x0=1 y0=1

w20= 0.8

δ1=-1

δ2=-2

" j (n) = e j (n)# $ j%(v j (n))

"h (n) =$h%(vh (n)) " j (n)# w jh (n)

j&O /P'

δ1h=1

δ2h=-2

u11= -1

v22= 0.8

w11= 0.9

x0=1 y0=1

w20= 0.8

δ1=-1

δ2=-2

" j (n) = e j (n)# $ j%(v j (n))

"h (n) =$h%(vh (n)) " j (n)# w jh (n)

j&O /P'

δ1h=1

δ2h=-2

An Example!g Again forward pass: "

o1=1.66

o2=0.32

u11= -1

v22= 0.8

w11= 0.9

x0=1 y0=1

w20= 0.8

y1=1.2 x1=0

y2=1.6

Notice the error has reduced

What does each layer do?!

1st layer draws linear boundaries!

2nd layer combines the boundaries!

3rd layer can generate arbitrarily complex boundaries!

Multilayer Perceptrons Lecture 11labs.seas.wustl.edu/bme/...MultiLayerPerceptrons.pdf · 4!...

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