Deep Learning

Report
Deep Learning
Bing-Chen Tsai
1/21
1
outline
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




Neural networks
Graphical model
Belief nets
Boltzmann machine
DBN
Reference
2
Neural networks
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Supervised learning


The training data consists of input information with their
corresponding output information.
Unsupervised learning

The training data consists of input information without their corresponding
output information.
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Neural networks
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Generative model

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Model the distribution of input as well as output ,P(x , y)
Discriminative model

Model the posterior probabilities ,P(y | x)
P(x,y1)
P(x,y2)
P(y1|x)
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P(y2|x)
Neural networks

x1
What is the neural?

Linear neurons y  b   x i w i
x2
w1
w2
1
b
y
i

Binary threshold neurons z = b + å xi wi
y 
0 otherwise
i

Sigmoid neurons z  b   x i w i

1
y 
i
1
Stochastic binary neurons z  b 

i
5
1 if z³0
e
xi wi
 z
1
p ( y  1) 
1
 e
 z
Neural networks

Two layer neural networks (Sigmoid neurons)
Back-propagation
Step1:
Randomly initial weight
Determine the output vector
Step2:
Evaluating the gradient
of an error function
Step3:
Adjusting weight,
Repeat The step1,2,3
until error enough low
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Neural networks
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Back-propagation is not good for deep learning

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It requires labeled training data.
 Almost data is unlabeled.
The learning time is very slow in networks with multiple hidden
layers.
 It is very slow in networks with multi hidden layer.
It can get stuck in poor local optima.
 For deep nets they are far from optimal.
Learn P(input) not P(output | input)

What kind of generative model should we learn?
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outline






Neural networks
Graphical model
Belief nets
Boltzmann machine
DBN
Reference
8
Graphical model
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A graphical model is a probabilistic model for which graph
denotes the conditional dependence structure between
random variables probabilistic model
In this example: D
depends on A, D depends
on B, D depends on C, C
depends on B, and C
depends on D.
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Graphical model

A
Directed graphical model
 , , ,  =         (|, )
B
C
D
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Undirected graphical model
 , , ,  =
A
C
B
D
1
∗ φ , ,  ∗ (, , )

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outline
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


Neural networks
Graphical model
Belief nets
Boltzmann machine
DBN
Reference
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Belief nets
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A belief net is a directed acyclic graph composed of
stochastic variables
stochastic hidden causes
Stochastic binary neurons
z  b

i
xi wi
1
p ( y  1) 
1 e
 z
It is sigmoid belief nets
visible
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Belief nets

we would like to solve two problems


The inference problem: Infer the states of the unobserved variables.
The learning problem: Adjust the interactions between variables to
make the network more likely to generate the training data.
stochastic hidden causes
visible
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Belief nets
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It is easy to generate sample P(v | h)
It is hard to infer P(h | v)

stochastic hidden causes
Explaining away
visible
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Belief nets
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Explaining away
H1
H2
H1 and H2 are independent, but they
can become dependent
when we observe an effect that they
can both influence
 1    2   
V
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Belief nets
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Some methods for learning deep belief nets
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Monte Carlo methods

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
But its painfully slow for large, deep belief nets
Learning with samples from the wrong distribution
Use Restricted Boltzmann Machines
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outline






Neural networks
Graphical model
Belief nets
Boltzmann machine
DBN
Reference
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Boltzmann Machine
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
It is a Undirected graphical model
The Energy of a joint configuration
hidden
j
i
-E(v, h) =
å vibi +
iÎvis
p(v, h) =
e
å
kÎhid
-E(v, h)
å e-E(u, g)
hk bk + å vi v j wij + å vi hk wik + å hk hl wkl
i< j
i, k
e-E(v, h)
å
h
p(v) =
åu, g e-E(u, g)
u, g
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k<l
visible
Boltzmann Machine
v h -E
e-E
p(v, h ) p(v)
An example of how weights
define a distribution
h1
+2
v1
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-1
h2
+1
v2
Boltzmann Machine

A very surprising fact
¶log p(v)
= si s j
¶wij
Derivative of log
probability of one
training vector, v
under the model.
v
- si s j
Expected value of
product of states at
thermal equilibrium
when v is clamped on
the visible units
Dwij µ
si s j
data
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- si s j
model
Expected value of
product of states at
thermal equilibrium
with no clamping
model
Boltzmann Machines
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
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Restricted Boltzmann Machine
We restrict the connectivity to make learning easier.
 Only one layer of hidden units.
 We will deal with more layers later
 No connections between hidden units
Making the updates more parallel
visible
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Boltzmann Machines
the Boltzmann machine learning algorithm for an RBM

j
j
j
<vi h j>¥
<vi h j>0
i
t=0
j
i
i
t=1
i
t=2
Dwij = e ( <vi h j >0 - <vi h j>¥ )
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t = infinity
Boltzmann Machines

Contrastive divergence: A very surprising short-cut
j
<vi h j>0
i
t=0
data
j
<vi h j>1
This is not following the gradient of the
log likelihood. But it works well.
i
t=1
reconstruction
Dwij = e ( <vi h j >0 - <vi h j>1 )
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outline






Neural networks
Graphical model
Belief nets
Boltzmann machine
DBN
Reference
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DBN


It is easy to generate sample P(v | h)
It is hard to infer P(h | v)

stochastic hidden causes
Explaining away
visible

Use RBM to initial weight can get good optimal
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DBN

Combining two RBMs to make a DBN
Then train
this RBM
h2
W2
h1
Compose the
two RBM
models to
make a single
DBN model
W2
h1
copy binary state for each v
W1
h1
Train this
RBM first
h2
v
W1
It’s a deep belief nets!
v
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DBN
etc.
W
T
h2

Why we can use RBM to initial belief nets weights?

An infinite sigmoid belief net that is equivalent to an RBM
W
v2
W
T
h1
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Inference in a directed net with replicated weights
 Inference is trivial. We just multiply v0 by W transpose.
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The model above h0 implements a complementary prior.
Multiplying v0 by W transpose gives the
product of the likelihood term and the prior term.
W
v1
W
h0
W
v0
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T
DBN
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Complementary prior
X1
X2
X3
X4
A Markov chain is a sequence of variables X1;X2; : : : with the Markov
property
  1 , … , −1 = ( |−1 )
A Markov chain is stationary if the transition probabilities do not
depend on time
  =  ′ −1 =  =   →  ′
( → ′) is called the transition matrix.
If a Markov chain is ergodic it has a unique equilibrium distribution
  =  → ∞  =    → ∞
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DBN
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Most Markov chains used in practice satisfy detailed balance
∞ ()( → ′) = ∞ (′)(′ → )
e.g. Gibbs, Metropolis-Hastings, slice sampling. . .
Such Markov chains are reversible
X1
X2
X3
X4
∞ 1  1 → 2  2 → 3 (3 → 4 )
X1
X2
X3
X4
 1 ← 2  2 ← 3  3 ← 4 ∞ (4 )
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DBN
  = 1 +1 = (  +1 + )
  = 1  = ( + )
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DBN

Combining two RBMs to make a DBN
Then train
this RBM
h2
W2
h1
Compose the
two RBM
models to
make a single
DBN model
W2
h1
copy binary state for each v
W1
h1
Train this
RBM first
h2
v
W1
It’s a deep belief nets!
v
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Reference
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Deep Belief Nets,2007 NIPS tutorial ,
G . Hinton
https://class.coursera.org/neuralnets-2012001/class/index
Machine learning 上課講義
http://en.wikipedia.org/wiki/Graphical_mod
el
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