2. Deep Learning is a kind of Representational LearningDeep Learning is a kind of Representational Learning
3. Deep Learning is a kind of Representational LearningDeep Learning is a kind of Representational Learning
picture source (https://www.deeplearningbook.org/)
7. Restricted Boltzmann MachinesRestricted Boltzmann Machines
An unsupervised greedy way to extract featuresAn unsupervised greedy way to extract features
發明:發明:
Smolensky, Paul (1986). Chapter 6: Information Processing in Dynamical Systems:
Foundations of Harmony Theory.
應⽤:應⽤:
降維:Hinton, G. E.; Salakhutdinov, R. R. (2006). Reducing the Dimensionality of
Data with Neural Networks. Science.
分類:Larochelle, H.; Bengio, Y. (2008). Classi cation using discriminative
restricted Boltzmann machines. ICML '08.
協同過濾:Salakhutdinov, R.; Mnih, A.; Hinton, G. (2007). Restricted Boltzmann
machines for collaborative ltering. ICML '07.
特徵學習:Coates, Adam; Lee, Honglak; Ng, Andrew Y. (2011). An analysis of
single-layer networks in unsupervised feature learning. International Conference
on Arti cial Intelligence and Statistics (AISTATS).
11. We need generative model!We need generative model!
Discriminative model:
Generative model:
p(Y |X)
p(X, Y )
12. Disentangle explanatory generative factorsDisentangle explanatory generative factors
to disentangle as many factors as possible, discarding as little information about the
data as is practical
x2
x3
x4
x5
z2
z1
x1
x2
x3
x4
x5
x1
z1
z2
18. To learn latent random variablesTo learn latent random variables
z x
N
θ
19. Introduce Bayesian theoremIntroduce Bayesian theorem
(z|x) =pθ
(x|z) (z)pθ pθ
(x)pθ
(x) = ∫ (x|z) (z)dzpθ pθ pθ
is intractable.(x)pθ
Variational inference: useVariational inference: use to approximateto approximate(z|x)qϕ (z|x)pθ
20. Kullback–Leibler divergenceKullback–Leibler divergence
Relative entropy, to measure the dissimilarity between two distributions.
Use data to approximate theoretical distributionp(X) q(X)
(p(X)||q(X)) = − p( ) log DKL ∑
i
xi
q( )xi
p( )xi
1. Asymmetry
2. Not distance
3.
4. and are equal
(p(X)||q(X)) ≥ 0DKL
(p(X)||q(X)) = 0DKL ⇔ p(X) q(X)
38. Achieve disentangled explainable generative factorAchieve disentangled explainable generative factor
Figure 6 in β-VAE: LEARNING BASIC VISUAL CONCEPTS WITH A CONSTRAINED
VARIATIONAL FRAMEWORK
39. What is the di erence between VAE andWhat is the di erence between VAE and -VAE?-VAE?β
VAE:
-VAE:
arg max L(θ, ϕ, x) = [log (x|z)] − ( (z|x)|| (z))E (z|x)qϕ
pθ DKL qϕ pθ
β
arg max L(θ, ϕ, x) = [log (x|z)] − β ( (z|x)|| (z))E (z|x)qϕ
pθ DKL qϕ pθ
L(θ, ϕ, x) = [log (x, z) − log (z|x)]E (z|x)qϕ
pθ qϕ
= ∫ (z|x)(log (x, z) − log (z|x))dzqϕ pθ qϕ
= ∫ (z|x)(log − log )dzqϕ
(x, z)pθ
(z)pθ
(z|x)qϕ
(z)pθ
= [log (x|z)] − ( (z|x)|| (z))E (z|x)qϕ
pθ DKL qϕ pθ
40. Why?Why?
The higher encourages learning a disentangled representation.
: encourage to learn good representations.
: constraint the capacity of
β
[log (x|z)]E (z|x)qϕ
pθ
( (z|x)|| (z))DKL qϕ pθ z
41. The information bottleneck methodThe information bottleneck method
arg max I (Z; Y ) − βI (X; Z)
: maximize mutual information between Z and Y.
: discard irrelevant information about Y from X.
I (Z; Y )
I (X; Z)
Learning is about forgetting irrelevant details.Learning is about forgetting irrelevant details.
44. Basic Information theoryBasic Information theory
EntropyEntropy
Information entropy, Shannon entropy
Measure the uncertainty of an event.
H(X) = E(I (X)) = − p( ) log p( )∑
i=1
n
xi xi
1. Nonnegativity:
2. Symmetry:
3. If and are independent random variable:
H(X) ≥ 0
H(X, Y ) = H(Y , X)
X Y H(X|Y ) = H(X)
45. EntropyEntropy
天氣預報100% 下⾬,0% 晴天:
天氣預報80% 下⾬,20% 晴天:
天氣預報50% 下⾬,50% 晴天:
1 lo 1 + 0 lo 0 = 0 + 0 = 0g2 g2
−0.8 lo 0.8 − 0.2 lo 0.2 = 0.258 + 0.464 = 0.722g2 g2
−0.5 lo 0.5 − 0.5 lo 0.5 = 0.5 + 0.5 = 1g2 g2
47. Conditional entropyConditional entropy
To measure how much information needed to describe the outcome of a random variable
Y given that the value of another random variable X is known.
H(Y |X) = p(x)H(Y |X = x)∑
x∈X
= − p(x) p(y|x) log p(y|x)∑
x∈X
∑
y∈Y
= − p(x, y) log ∑
x∈X ,y∈Y
p(x, y)
p(x)
48. Mutual informationMutual information
To measure how much information obtained about one random variable through
observing the other.
I (X; Y ) = H(X) − H(X|Y )
= H(Y ) − H(Y |X)
= H(X) + H(Y ) − H(X, Y )
= p(x, y) log ∑
x,y
p(x, y)
p(x)p(y)
1. Nonnegativity:
2. Symmetry:
I (X; Y ) ≥ 0
I (X; Y ) = I (Y ; X)
49. Relation to Kullback–Leibler divergenceRelation to Kullback–Leibler divergence
I (X; Y ) = (p(X, Y )||p(X)p(Y ))DKL
52. Cross entropyCross entropy
How much difference between two distributions.
H(q, p) = H(q) + (q||p)DKL
= − p(x) log q(x)∑
x
DKL(q∣p)
H (q)
H (q, p)
NOTION: notation confused with joint entropy.
53. Di erence between mutual information and cross entropyDi erence between mutual information and cross entropy
Mutual information
Measure the information share between two random variables.
Cross entropy
Measure the difference between two distributions.
54. Data processing inequality (DPI)Data processing inequality (DPI)
Let be a Markov chain, thenX → Y → Z
I (X; Y ) ≥ I (X; Z)
55. The neural network generates a successive Markov chainThe neural network generates a successive Markov chain
Treat the whole layer as a single random variableTi
Encoder Decoder
I (X; Y ) ≥ I ( ; Y ) ≥ I ( ; Y ) ≥. . . ≥ I ( ; Y ) ≥ I ( ; Y )T1 T2 Tm Y^
H(X) ≥ I (X; ) ≥ I (X; ) ≥. . . ≥ I (X; ) ≥ I (X; )T1 T2 Tm Y^
56. Codebook and volumeCodebook and volume
Let
: signal source with xed probability measure
: quantized codebook
: a soft partition of , with probability with
X p(x)
X^
p( |x)x^ X
p( ) = p(x)p( |x)x^ ∑
x
x^
57. What determines the quality of a quantization?What determines the quality of a quantization?
Rate, the average numbers of bits per message to encode the signal.
The information to transmit from to is bounded from belowX X^
I (X; )X^
58. Rate distortion theoryRate distortion theory
Bernd Girod: EE368b Image and Video Compression Rate Distortion Theory no. 1
Lossy compression
n Lower the bit-rate R by allowing some acceptable distortion
D of the signal.
Distortion D
Rate R
Lossless coding
D=0
59. Rate distortion theoryRate distortion theory
Bernd Girod: EE368b Image and Video Compression Rate Distortion Theory no. 2
Types of lossy compression problems
D
R
n Given maximum rate R,
minimize distortion D
n Given distortion D, minimize
rate R
D
R
Equivalent constrained optimization problems,
often unwieldy due to constraint.
60. Rate distortion theoryRate distortion theory
Def. rate distortion function as
R(D) = min I (X; )X^
w. r. t. E[d(x, )] ≤ Dx^
Apply Lagrange multiplier:
F (p( |x)) = I (X; ) + βE[d(x, )]x^ X^ x^
61. Information bottleneck methodInformation bottleneck method
, thenX → → YX^ I (X; ) ≥ I (X; Y )X^
Information bottleneck:
arg min L(x, ) = I (X; ) − βI ( ; Y )x^ X^ X^
We want this quantization to capture as much information about
tradeoff between compress the representation and preserve meaningful information.
Y
63. Opening the black box of Deep Neural Networks viaOpening the black box of Deep Neural Networks via
InformationInformation
64. IssuesIssues
1. The SGD layer dynamics in the Information plane.
2. The effect of the training sample size on the layers.
3. What is the bene t of the hidden layers?
4. What is the nal location of the hidden layers?
5. Do the hidden layers form optimal IB representations?
65. SetupSetup
standard DNN settings
tanh as activation function
sigmoid function in the nal layer
train with SGD and cross-entropy loss
7 fully connected hidden layers with widths: 12-10-7-5-4-3-2 neurons
66. Information planeInformation plane
Encoder Decoder
Given , plot point on the information plane.
Applied to the Markov chain of a k-layers of DNN, connected points form a unique
information path.
P (X; Y ) (I (X; T ), I (T ; Y ))
67. The dynamics of the training by Stochastic-Gradient-DecentThe dynamics of the training by Stochastic-Gradient-Decent
50 different randomized initializations with different randomized training samples
init − 400epochs − 9000epochs
The optimization process in the Information Plane (https://www.youtube.com/watch?
v=P1A1yNsxMjc)
68. The two optimization phases in the Information PlaneThe two optimization phases in the Information Plane
5% - 45% - 85% training samples5% - 45% - 85% training samples
Emperical risk minimization (ERM) phase (fast)
increase
layer learn the information while preserving the DPI order
Representation compression phase (slow)
decrease until convergence
layer lose irrelevant information (compression)
IY
IX
69. The drift and di usion phases of SGD optimizationThe drift and di usion phases of SGD optimization
Layer weight's gradient distributionsLayer weight's gradient distributions
70. The drift and di usion phases of SGD optimizationThe drift and di usion phases of SGD optimization
Drift phase
large gradient mean, small variance (high SNR)
increase and reduce the emperical error
ERM phase
Diffusion phase
small gradient mean, large uctuations (low SNR)
the gradients behave like Gaussian noise, weights evolve like Wiener
process
compression phase
Maximize the entropy of the weight distribution by addiing noise, known
as stochastic relaxation
compression by diffusion phase
attempts to interpret single weights or even single neurons in such networks can
be meaningless
IY
71. The computational bene t of the hidden layersThe computational bene t of the hidden layers
Train 6 different architecture with 1-6 hidden layers
72. The computational bene t of the hidden layersThe computational bene t of the hidden layers
1. Adding hidden layers dramatically reduces the number of training epochs for good
generalization.
2. The compression phase of each layer is shorter when it starts from a previous
compressed layer.
3. The compression is faster for the deeper (narrower and closer to the output)
layers.
4. Even wide hidden layers eventually compress in the diffusion phase. Adding extra
width does not help.
73. Convergence to the layers to the Information Bottleneck boundConvergence to the layers to the Information Bottleneck bound
74. Evolution of the layers with training sample sizeEvolution of the layers with training sample size
0 1 2 3 4 5 6 7 8 9
I(X;T)
0.3
0.4
0.5
0.6
0.7
I(T;Y)
4%
84%
Training data
75. with increasing training size the layers’ true label information (generalization) is
pushed up and gets closer to the theoretical IB bound for the rule distribution.
IY
76. Are our ndings general enough?Are our ndings general enough?
77. Hinton 的評論Hinton 的評論
Hinton 在聽完Tishby 的talk 之後,給Tishby 發了email:
“I have to listen to it another 10,000 times to really understand it,
but it’s very rare nowadays to hear a talk with a really original
idea in it that may be the answer to a really major puzzle.”
78. Caution!Caution!
No, information bottleneck (probably) doesn’t open the “black-box” of deep neural n
(https://severelytheoretical.wordpress.com/2017/09/28/no-information-bottlenec
black-box-of-deep-neural-networks/)
Tishby's 'Opening the Black Box of Deep Neural Networks via Information' received
(https://www.reddit.com/r/MachineLearning/comments/72eau7/d_tishbys_opening
On the Information Bottleneck Theory of Deep Learning [Harvard University] [ICLR
(https://openreview.net/forum?id=ry_WPG-A-)
79. Thank you for attentionThank you for attention
ReferenceReference
18. Information Theory of Deep Learning. Naftali Tishby
(https://www.youtube.com/watch?v=bLqJHjXihK8)