CMP784 DEEP LEARNING Lecture #03 Multi-layer Perceptrons Aykut - - PowerPoint PPT Presentation

Lecture #03 – Multi-layer Perceptrons

Aykut Erdem // Hacettepe University // Spring 2018

CMP784

DEEP LEARNING

Image: Jose-Luis Olivares

Breaking news!

• Practical 1 is out!

—Learning neural word embeddings —Due Friday, Mar. 16, 23:59:59

• Paper presentations and quizzes will start

next week!

− Discuss your slides with me 3-4 days prior to your presentation − submit your final slides by the night before the class. − We don’t have any code walker or demonstrator.

Previously on CMP784

• Learning problem
• Parametric vs. non-parametric models
• Nearest—neighbor classifier
• Linear classification
• Linear regression
• Capacity
• Hyperparameter
• Underfitting
• Overfitting
• Variance-Bias tradeoff
• Model selection
• Cross-validation

Lecture overview

• the perceptron
• the multi-layer perceptron
• stochastic gradient descent
• backpropagation
• shallow yet very powerful: word2vec
• Discl

sclaimer: Much of the material and slides for this lecture were borrowed from

—Hugo Larochelle’s Neural networks slides —Nick Locascio’s MIT 6.S191 slides —Efstratios Gavves and Max Willing’s UvA deep learning class —Leonid Sigal’s CPSC532L class —Richard Socher’s CS224d class —Dan Jurafsky’s CS124 class

A Brief History of Neural Networks

Image: VUNI Inc.

today

The Perceptron

The Perceptron

x0 x1 x2 xn 1 ∑ inputs bias weights w0 w1 w2 wn b … sum non-linearity

Perceptron Forward Pass

• Neuron pre-activation

(or input activation)

• Neuron output activation:

where

w are the weights (parameters) b is the bias term g(·) is called the activation function

• a(x) = b + P

i wixi = b + w>x

P

• P
• h(x) = g(a(x)) = g(b + P

i wixi)

• x0

x1 x2 xn 1 ∑ inputs bias weights w0 w1 w2 wn b … sum non-linearity

Output Activation of The Neuron

Bi t ri s ed

(x a(x

(from Pascal Vincent’s slides) Image credit: Pascal Vincent

• P
• h(x) = g(a(x)) = g(b + P

i wixi)

Bias only changes the position of the riff Range is determined by g(·)

x0 x1 x2 xn 1 ∑ inputs bias weights w0 w1 w2 wn b … sum non-linearity

Linear Activation Function

• P
• h(x) = g(a(x)) = g(b + P

i wixi)

• No nonlinear transformation
• No input squashing

x0 x1 x2 xn 1 ∑ inputs bias weights w0 w1 w2 wn b … sum non-linearity tion

• {
• g(a) = a

Sigmoid Activation Function

x0 x1 x2 xn 1 ∑ inputs bias weights w0 w1 w2 wn b … sum non-linearity

• g(a) = sigm(a) =

1 1+exp(a)

s

• utput between 0 and 1
• utput between 0 and 1
• P
• h(x) = g(a(x)) = g(b + P

i wixi)

• Squashes

the neuron’s

• utput

between 0 and 1

• Always

positive

• Bounded
• Strictly

Increasing

Perceptron Forward Pass

2 3

• 1

5 1 ∑ inputs bias weights 0.1 0.5 2.5 0.2 3.0 sum non-linearity

• P
• h(x) = g(a(x)) = g(b + P

i wixi)

Perceptron Forward Pass

2 3

• 1

5 1 ∑ inputs bias weights 0.1 0.5 2.5 0.2 3.0 sum non-linearity

• h(x) = g(a

(x

• xi)

(2*0.1) + (3*0.5) + (-1*2.5) + (5*0.2) + (1*3.0)

Perceptron Forward Pass

2 3

• 1

5 1 ∑ inputs bias weights 0.1 0.5 2.5 0.2 3.0 sum non-linearity

h(x) = g(3.2) = σ(3.2) 1 1 + e−3.2 = 0.96

Hyperbolic Tangent (tanh) Activation Function

x0 x1 x2 xn 1 ∑ inputs bias weights w0 w1 w2 wn b … sum non-linearity

• P
• h(x) = g(a(x)) = g(b + P

i wixi)

• Squashes the

neuron’s output between

• 1 and 1
• Can be positive
• r negative
• Bounded
• Strictly

Increasing

h(a) = exp(a)exp(a)

exp(a)+exp(a) = exp(2a)1 exp(2a)+1

• g(a) = tanh(a) = exp(a)exp(a)

exp(a)+exp(a)

Rectified Linear (ReLU) Activation Function

x0 x1 x2 xn 1 ∑ inputs bias weights w0 w1 w2 wn b … sum non-linearity

• P
• h(x) = g(a(x)) = g(b + P

i wixi)

• Bounded below

by 0 (always non-negative)

• Not upper

bounded

• Strictly

increasing

• Tends to

produce units with sparse activities

• g(a) = reclin(a) = max(0, a)

Decision Boundary of a Neuron

• Could do binary classification:

—with sigmoid, one can interpret neuron as estimating p(y = 1 | x) —also known as logistic regression classifier —if activation is greater than 0.5, predict 1 —otherwise predict 0

Same idea can be applied to a tanh activation

Image credit: Pascal Vincent

(from Pascal Vincent’s slides)

han

Decision boundary is linear

Capacity of Single Neuron

• Can solve linearly separable problems

1 1 1 1 1 1

OR (x1, x2)

AND (x1, x2)

AND (x1, x2) (x1

, x2) , x2) , x2)

(x1 (x1

Capacity of Single Neuron

• Can not solve non-linearly separable problems
• Need to transform the input into a better representation
• Remember basis functions!

1 1

?

XOR (x1, x2)

(x1

1 1

XOR (x1, x2) AND (x1, x2)

AND (x1, x2)

, x2)

Perceptron Diagram Simplified

x0 x1 x2 xn 1 ∑ inputs bias weights w0 w1 w2 wn b … sum non-linearity

Perceptron Diagram Simplified

x0 x1 x2 xn

inputs …

• utput

Multi-Output Perceptron

• Remember multi-way classification

—We need multiple outputs (1 output per class) —We need to estimate conditional probability p(y = c|x) —Discriminative Learning

• Softmax activation function at the output

—Strictly positive —sums to one

• Predict class with the highest estimated class conditional probability.

x0 x1 x2 xn

inputs …

• utput
• n
• |
• o(a) = softmax(a) =

h

exp(a1) P

c exp(ac) . . .

exp(aC) P

c exp(ac)

i>

Multi-Layer Perceptron

Single Hidden Layer Neural Network

• Hidden layer pre-activation:
• Hidden layer activation:
• Output layer activation:

x0 x1 xn h1 inputs … hidden layer h2 h0 hn

• n
• utput

layer

• a(x) = b(1) + W(1)x

⇣ a(x)i = b(1)

i

+ P

j W (1) i,j xj

⌘

• h(x) = g(a(x))
• (x) = o

⇣ b(2) + w(2)h(1)x ⌘

>

Multi-Layer Perceptron (MLP)

• Consider a network with L hidden

layers.

—layer pre-activation for k>0 —hidden layer activation from 1 to L: —output layer activation (k=L+1)

x0 x1 xn h1 inputs … hidden layer h2 h0 hn

• n
• utput

layer

• a(k)(x) = b(k) + W(k)h(k1)(x) (
• h(k)(x) = g(a(k)(x))
• h(L+1)(x) = o(a(L+1)(x)) = f(x)
• (h(0)(x) = x)

Deep Neural Network

x0 x1 xn h1 inputs … hidden layer h2 h0 hn

• n
• utput

layer h1 h2 h0 hn …

Capacity of Neural Networks

• Consider a single layer neural network

Image credit: Pascal Vincent

R´ eseaux de neurones

• 1.5

• .4
• 1

x1 x2 x

• 1

• 1

• 1

• 1

• 1

• 1

• 1

• 1

• 1

y1 y2 z zk

wkj wji

x1 x2 x1 x2 x1 x2 y1 y2

sortie k entr´ ee i cach´ ee j biais

Input Hidden Output bias

(from Pascal Vincent’s slides)

Capacity of Neural Networks

• Consider a single layer neural network

Image credit: Pascal Vincent

y1 y2 y4 y3 y3 y4 y2 y1 x1 x2 z1 z1 x1 x2

(from Pascal Vincent’s slides)

Capacity of Neural Networks

• Consider a single layer neural network

Image credit: Pascal Vincent (from Pascal Vincent’s slides)

Universal Approximation

• Universal Approximation Theorem (Hornik, 1991):

—“a single hidden layer neural network with a linear output unit can approximate any continuous function arbitrarily well, given enough hidden units’’

• This applies for sigmoid, tanh and many other activation functions.
• However, this does not mean that there is learning algorithm that can

find the necessary parameter values.

Applying Neural Networks

Example Problem: Will my flight be delayed?

Example Problem: Will my Flight be Delayed?

x0 x1 h2 h1

h0 Predicted: 0.05 [-20, 45] Temperature: -20 F Wind Speed: 45 mph

Example Problem: Will my flight be delayed?

Example Problem: Will my Flight be Delayed?

x0 x1 h2 h1

h0 Predicted: 0.05 [-20, 45] Predicted: 0.05

[-20, 45]

x0 x1 h1 h2 h0

Example Problem: Will my flight be delayed?

Example Problem: Will my Flight be Delayed?

x0 x1 h2 h1

h0 Predicted: 0.05 [-20, 45] Predicted: 0.05 Actual: 1

[-20, 45]

x0 x1 h1 h2 h0

Quantifying Loss

Predicted: 0.05 Actual: 1

[-20, 45]

x0 x1 h1 h2 h0

Predicted

`(f(x(i); ✓), y(i))

Actual

Total Loss

[ [-20, 45], [80, 0], [4, 15], [45, 60], ]

x0 x1 h1 h2 h0

Predicted Actual

J(✓) = 1 N X

i

`(f(x(i); ✓), y(i))

Input Predicted Actual

[ 0.05 0.02 0.96 0.35 ] [ 1 1 1 ]

Total Loss

[ [-20, 45], [80, 0], [4, 15], [45, 60], ]

x0 x1 h1 h2 h0

Predicted Actual

J(✓) = 1 N X

i

`(f(x(i); ✓), y(i))

Input Predicted Actual

[ 0.05 0.02 0.96 0.35 ] [ 1 1 1 ]

Binary Cross Entropy Loss

[ [-20, 45], [80, 0], [4, 15], [45, 60], ]

x0 x1 h1 h2 h0

Input Predicted Actual

[ 0.05 0.02 0.96 0.35 ] [ 1 1 1 ]

Jcross entropy(θ) = 1 N X

i

y(i) log(f(x(i); θ)) + (1 − y(i)) log(1 − f(x(i); θ)))

• For classification problems with a softmax output layer.
• Maximize log-probability of the correct class given an input

Binary Cross Entropy Loss

[ [-20, 45], [80, 0], [4, 15], [45, 60], ]

x0 x1 h1 h2 h0

Input Predicted Actual

[ 0.05 0.02 0.96 0.35 ] [ 1 1 1 ]

JMSE(θ) = 1 N X

i

⇣ f(x(i); θ) − y(i)⌘2

Training Neural Networks

arg min

θ

1 T X

t

l(f(x(t); θ), y(t)) + λΩ(θ)

Training

• Learning is cast as optimization.

—For classification problems, we would like to minimize classification error —Loss function can sometimes be viewed as a surrogate for what we want to optimize (e.g. upper bound)

Loss function Regularizer

Training Neural Networks: Objective

MIT 6.S191 | Intro to Deep Learning | IAP 2017

loss function

=

Training Neural Networks: Objective

MIT 6.S191 | Intro to Deep Learning | IAP 2017

Loss is a function of the model’s parameters

Loss is a function of the model’s parameters

MIT 6.S191 | Intro to Deep Learning | IAP 2017

Loss is a function of the model’s parameters

MIT 6.S191 | Intro to Deep Learning | IAP 2017

How to minimize loss?

Loss is a function of the model’s parameters

MIT 6.S191 | Intro to Deep Learning | IAP 2017

Start at random point

How to minimize loss?

Compute:

How to minimize loss?

Move in direction opposite

• f gradient to new point

How to minimize loss?

Move in direction opposite

• f gradient to new point

How to minimize loss?

Repeat!

This is called Stochastic Gradient Descent (SGD)

Repeat!

Stochastic Gradient Descent (SGD)

• Initialize θ randomly
• For N Epochs
• For each training example (x, y):
• Compute Loss Gradient:
• Update θ with update rule:

• Initialize θ randomly
• For N Epochs

○ For each training example (x, y): ■ Compute Loss Gradient: ■ Update θ with update rule:

Stochastic Gradient Descent (SGD)

MIT 6.S191 | Intro to Deep Learning | IAP 2017

• Initialize θ randomly
• For N Epochs

○ For each training example (x, y): ■ Compute Loss Gradient: ■ Update θ with update rule:

Stochastic Gradient Descent (SGD)

MIT 6.S191 | Intro to Deep Learning | IAP 2017

𝜄(𝑢+1) = 𝜄(𝑢) − 𝜃𝑢𝛼𝜄ℒ

Why is it Stochastic Gradient Descent?

• Initialize θ randomly
• For N Epochs
• For each training example (x, y):
• Compute Loss Gradient:
• Update θ with update rule:

• Initialize θ randomly
• For N Epochs

○ For each training example (x, y): ■ Compute Loss Gradient: ■ Update θ with update rule:

Stochastic Gradient Descent (SGD)

MIT 6.S191 | Intro to Deep Learning | IAP 2017 Only an estimate of true gradient!

𝜄(𝑢+1) = 𝜄(𝑢) − 𝜃𝑢𝛼𝜄ℒ

𝜄(𝑢+1) = 𝜄(𝑢) − 𝜃𝑢𝛼𝜄ℒ

• Why is it Stochastic Gradient Descent?
• Initialize θ randomly
• For N Epochs
• For each training batch {(x0, y0),…, (xB, yB)}:
• Compute Loss Gradient:
• Update θ with update rule:

• Initialize θ randomly
• For N Epochs

○ For each training example (x, y): ■ Compute Loss Gradient: ■ Update θ with update rule:

Stochastic Gradient Descent (SGD)

MIT 6.S191 | Intro to Deep Learning | IAP 2017 More accurate estimate!

• Initialize θ randomly
• For N Epochs

○ For each training batch {(x0, y0), … , (xB, yB)}: ■ Compute Loss Gradient: ■ Update θ with update rule:

Minibatches Reduce Gradient Variance

MIT 6.S191 | Intro to Deep Learning | IAP 2017

More accurate estimate!

Advantages:

• More accurate estimation of gradient

⎯ Smoother convergence ⎯ Allows for larger learning rates

• Minibatches lead to fast training!

⎯ Can parallelize computation + achieve significant speed increases on GPU’s

θ)

Training epoch = Iteration of all examples

Stochastic Gradient Descent (SGD)

• Algorithm that performs updates after each example

—initialize —for N iterations

—for each training example or batch

• To apply this algorithm to neural network training, we need:

—the loss function —a procedure to compute the parameter gradients: —the regularizer (and the gradient )

ze:

• θ ⌘ {W(1), b(1), . . . , W(L+1), b(L+1)}

mple

• r
• (x(t), y(t))
• •

r 8

• ∆ = rθl(f(x(t); θ), y(t)) λrθΩ(θ)
• P r
• θ θ + α ∆

Training epoch = Iteration over all all examples

ent: ,

• Ω(θ)

,

r

• rθΩ(θ)

ion:

• l(f(x(t); θ), y(t))

𝜄(𝑢+1) = 𝜄(𝑢) − 𝜃𝑢𝛼𝜄ℒ

• tions

θ)

Training epoch = Iteration of all examples

Stochastic Gradient Descent (SGD)

• Algorithm that performs updates after each example

—initialize —for N iterations

—for each training example or batch

• To apply this algorithm to neural network training, we need:

—the loss function —a procedure to compute the parameter gradients: —the regularizer (and the gradient )

ze:

• θ ⌘ {W(1), b(1), . . . , W(L+1), b(L+1)}

mple

• r
• (x(t), y(t))
• •

r 8

• ∆ = rθl(f(x(t); θ), y(t)) λrθΩ(θ)
• P r
• θ θ + α ∆

Training epoch = Iteration over all all examples

ent: ,

• Ω(θ)

,

r

• rθΩ(θ)

ion:

• l(f(x(t); θ), y(t))

𝜄(𝑢+1) = 𝜄(𝑢) − 𝜃𝑢𝛼𝜄ℒ

• tions

What is a neural network again?

• A family of parametric, non-linear and hierarchical representation learning

functions

• ⎯ x: input, θl: parameters for layer l, al = hl(x, θl): (non-)linear function
• Given training corpus {X, Y} find optimal parameters

aL(x; θ1,...,L) = hL(hL−1(. . . h1(x, θ1), θL−1), θL)

✓∗ ← arg min

θ

X

(x,y)∈(X,Y )

`(y, aL(x; ✓1,...,L))

Neural network models

• A neural network model is a series of hierarchically connected functions
• The hierarchy can be very, very complex

h1(xi; θ)

Input

h2(xi; θ) h3(xi; θ) h4(xi; θ) h5(xi; θ) Loss

Forward connections (Feedforward architecture)

Neural network models

• A neural network model is a series of hierarchically connected functions
• The hierarchy can be very, very complex

Input Interweaved connections (Directed Acyclic Graph architecture – DAGNN)

h1(xi; θ) h2(xi; θ) h3(xi; θ) h4(xi; θ) h5(xi; θ) Loss h2(xi; θ) h4(xi; θ)

Neural network models

• A neural network model is a series of hierarchically connected functions
• The hierarchy can be very, very complex

Input

h2(xi; θ) h3(xi; θ) h4(xi; θ) h5(xi; θ) Loss

Loopy connections (Recurrent architecture, special care needed)

h1(xi; θ)

Neural network models

• A neural network model is a series of hierarchically connected functions
• The hierarchy can be very, very complex

Functions → Modules

h1(xi; θ)

Input

h2(xi; θ) h3(xi; θ) h4(xi; θ) h5(xi; θ) Loss h1(xi; θ)

Input

h2(xi; θ) h3(xi; θ) h4(xi; θ) h5(xi; θ) Loss

Input

h1(xi; θ) h2(xi; θ) h3(xi; θ) h4(xi; θ) h5(xi; θ) Loss h2(xi; θ) h4(xi; θ)

What is a module

• A module is a building block for our network
• Each module is an object/function ! = h(x; ") that

⎯ Contains trainable parameters (") ⎯ Receives as an argument an input $ ⎯ And returns an output ! based on the activation function h(...)

• The activation function should be (at least) first order

differentiable (almost) everywhere

• For easier/more efficient backpropagation

→ store module input

⎯ easy to get module output fast ⎯ easy to compute derivatives

Input

h1(xi; θ) h2(xi; θ) h3(xi; θ) h4(xi; θ) h5(xi; θ) Loss h2(xi; θ) h4(xi; θ)

Anything goes or do special constraints exist?

• A neural network is a composition of modules (building blocks)
• Any architecture works
• If the architecture is a feedforward cascade, no special care
• If acyclic, there is right order
• f computing the forward

computations

• If there are loops, these

form recu current connections (revisited later)

What is a module

• Simply compute the activation of each module in the

network

• We need to know the precise function behind each

module h!(... )

• Recursive operations
• One module’s output is another’s input
• Steps
• Visit modules one by one starting from the data input
• Some modules might have several inputs from multiple modules
• Compute modules activations wi

with th th the right t order

• Make sure all the inputs computed at the right time

Input

h1(xi; θ) h2(xi; θ) h3(xi; θ) h4(xi; θ) h5(xi; θ) Loss h2(xi; θ) h4(xi; θ) where

al = hl(xl; θ) al = xl+1 xl = al−1

• r

What is a module

• Simply compute the gradients of each module

for our data

• We need to know the gradient formulation of each

module !h"(#";$") w.r.t. their inputs #" and parameters $"

• We need the forward computations first
• Their result is the sum of losses for our input data
• Then take the reverse network (reverse connections)

and traverse it backwards

• Instead of using the activation functions, we use

their gradients

• The whole process can be described very neatly and concisely

with the backpropagation algorithm

h1(xi; θ) h2(xi; θ) h3(xi; θ) h4(xi; θ) h5(xi; θ) h2(xi; θ) h4(xi; θ)

dLoss(Input)

Again, what is a neural network again?

• d

⎯ x: input, θl: parameters for layer l, al = hl(x, θl): (non-)linear function

• Given training corpus {X, Y} find optimal parameters
• To use any gradient descent based optimization

we need the gradients

• How to compute the gradients for such a complicated function enclosing other

functions, like "# (... )?

aL(x; θ1,...,L) = hL(hL−1(. . . h1(x, θ1), θL−1), θL)

✓∗ ← arg min

θ

X

(x,y)∈(X,Y )

`(y, aL(x; ✓1,...,L))

✓ θt+1 = θt − ηt ∂L ∂θt ◆

∂L ∂θl , l = 1, . . . , L

Again, what is a neural network again?

• d

⎯ x: input, θl: parameters for layer l, al = hl(x, θl): (non-)linear function

• Given training corpus {X, Y} find optimal parameters
• To use any gradient descent based optimization

we need the gradients

• How to co

compute the grad adients s for su such ch a a co complicat cated funct ction encl closi sing other funct ctions, s, like ke "# (... )? ?

aL(x; θ1,...,L) = hL(hL−1(. . . h1(x, θ1), θL−1), θL)

✓∗ ← arg min

θ

X

(x,y)∈(X,Y )

`(y, aL(x; ✓1,...,L))

✓ θt+1 = θt − ηt ∂L ∂θt ◆

∂L ∂θl , l = 1, . . . , L