Information in the Weights

1. Information in the Weights
Mark Chang
2020/06/10

2. Outline
• Traditional Machine Learning v.s. Deep Learning
• Basic Concepts in Information Theory
• Information in the Weights

3. Traditional Machine Learning v.s. Deep
Learning
• VC Bound
• Generalization in Deep Learning
• PAC-Bayesian Bound for Deep Learning

4. VC Bound
• What cause over-fitting?
• Too many parameters -> over fitting ?
• Too many parameters -> high VC Dimension -> over fitting ?
• … ?
h
h h
under fitting over fittingappropriate fitting
too few
parameters
adequate
parameters
too many
parameters
training data
testing data

5. VC Bound
• Over-fitting is caused by high VC Dimension
• For a given dataset (n is constant), search for the best VC Dimension
d=n (shatter)
d (VC Dimension)
over-fittingerror
best VC Dimension
numbef of
training instances
VC Dimension
(model complexity)✏(h)  ˆ✏(h) +
r
8
n
log(
4(2n)d
)
training error
testing error

6. VC Dimension
• VC Dimension of linear model:
• O(W)
• W = number of parameters
• VC Dimension of fully-connected
neural networks:
• O(LW log W)
• L = number of layers
• W = number of parameters
• VC Dimension is independent from data distribution, and only
depends on model

7. Generalization in Deep Learning
• Considering a toy example:
neural networks
input:
780
hidden:
600
d ≈ 26M
dataset
n = 50,000
d >> n, but testing error < 0.1

8. Generalization in Deep Learning
• However, when you are solving your problem …
neural networks
input:
780
hidden:
600
testing error = 0.6
over fitting !!
your
dataset
n = 50,000
10 classes
testing error = 0.6
over fitting !!
reduce
VC Dimension
neural networks
input:
780
hidden:
200
…
reduce
VC Dimension

9. Generalization in Deep Learning
d (VC Dimension)
error
✏(h)  ˆ✏(h) +
r
8
n
log(
4(2n)d
)
d=n
over-fitting
over-parameterization
model with
extremely high VC-
Dimension

10. Generalization in Deep Learning
ICLR2017

11. Generalization in Deep Learning
1 0 1 0 2
random noise features
shatter !
deep
neural
networks
(Inception)
feature
:label: 1 0 1 0 2
original dataset (CIFAR)
0 1 1 2 0
random label
ˆ✏(h) ⇡ 0 ˆ✏(h) ⇡ 0 ˆ✏(h) ⇡ 0

12. Generalization in Deep Learning
1 0 1 0 2
random noise features
deep
neural
networks
(Inception)
feature
:label: 1 0 1 0 2
original dataset (CIFAR)
0 1 1 2 0
random label
ˆ✏(h) ⇡ 0 ˆ✏(h) ⇡ 0 ˆ✏(h) ⇡ 0
✏(h) ⇡ 0.14 ✏(h) ⇡ 0.9 ✏(h) ⇡ 0.9

13. Generalization in Deep Learning
• Testing error depends on data distribution
• However, VC-Bound does not depend on data distribution
1 0 1 0 2
random noise
features
original dataset
feature
:label: 1 0 1 0 2
random label
0 1 1 2 0
✏(h) ⇡ 0.14 ✏(h) ⇡ 0.9

14. Generalization in Deep Learning

15. Generalization in Deep Learning
• high sharpness -> high testing error

16. PAC-Bayesian Bound for Deep Learning
UAI 2017

17. PAC-Bayesian Bound for Deep Learning
• Deterministic Model • Stochastic Model (Gibbs Classifier)
data ✏(h)
hypothesis
h
error
data
✏(Q)
= Eh⇠Q(✏(h))
distribution of
hypothesis
hypothesis
h
Gibbs
error
sampling

18. PAC-Bayesian Bound for Deep Learning
• Considering the sharpness of local minimums
single
hypothesis
h
distribution of
hypothesis
Q
flat minimum:
• low
• low
ˆ✏(h)
ˆ✏(Q)
sharp minimum:
• low
• high
ˆ✏(h)
ˆ✏(Q)

19. KL(ˆ✏(Q)k✏(Q)) 
KL(QkP) + log(n
)
n 1
PAC-Bayesian Bound for Deep Learning
• With 1-ẟ probability, the following inequality (PAC-Bayesian Bound) is
satisfied
number of
training instances
KL divergence
between P and Q
Distribution of model
before training (prior)
Distribution of models
after training (posterior)
KL divergence
between testing error
& training error

20. PAC-Bayesian Bound for Deep Learning
✏(Q)  ˆ✏(Q) +
s
KL(Q||P) + log(n
) + 2
2n 1
high ✏(Q)  ˆ✏(Q) +
s
KL(Q||P) + log(n
) + 2
2n 1
, high
under fitting over fittingappropriate fitting
✏(Q)  ˆ✏(Q) +
s
KL(Q||P) + log(n
) + 2
2n 1
moderate ✏(Q)  ˆ✏(Q) +
s
KL(Q||P) + log(n
) +
2n 1
, moderate ✏(Q)  ˆ✏(Q) +
s
KL(Q||P) + log(n
2n 1
low ✏(Q)  ˆ✏(Q) +
s
, high
low KL(Q||P) moderate KL(Q||P) high KL(Q||P)
training data
testing data
P
Q
P
Q
P
Q
KL(ˆ✏(Q)k✏(Q)) 
KL(QkP) + log(n
)
n 1
lowKL(ˆ✏(Q)k✏(Q)) moderae KL(ˆ✏(Q)k✏(Q)) high KL(ˆ✏(Q)k✏(Q))

21. PAC-Bayesian Bound for Deep Learning
• PAC-Bayesian Bound is data-dependent
KL(ˆ✏(Q)k✏(Q)) 
KL(QkP) + log(n
)
n 1
• High VC Dimension, but clean data
-> low KL (Q||P)
• High VC Dimension, and noisy data
-> high KL(Q||P)
P
Q Q
P

22. PAC-Bayesian Bound for Deep Learning
• Data : M-NIST (binary classification, class0 : 0~4, class1 : 5~9)
• Model : 2 layer or 3 layer NN
Training
Testing
VC Bound
Pac-Bayesian
Bound
0.028
0.034
26m
0.161
0.027
0.035
56m
0.179
Varying the width of
hidden layer (2 layer NN)
600 1200
0.027
0.032
121m
0.201
0.028
0.034
26m
0.161
0.028
0.033
66m
0.186
Varying the number of
hidden layers
2 3 4
0.028
0.034
26m
0.161
0.112
0.503
26m
1.352
original random
original M-NIST
v.s. random label

23. PAC-Bayesian Bound for Deep Learning
• PAC-Bayesian Bound is Data Dependent
clean data:
->small KL(Q||P)
->small ε(Q)
noisy data:
->large KL(Q||P)
->large ε(Q)
feature:
label:
original M-NIST
0 0 0 1 1
random label
1 0 1 0 1
KL(ˆ✏(Q)k✏(Q)) 
KL(QkP) + log(n
)
n 1

24. Basic Concepts in Information Theory
• Entropy
• Joint Entropy
• Conditional Entropy
• Mutual Information
• Cross Entropy

25. Entropy
• The uncertainty of a random variable X
H(X) =
X
x2X
p(x) log p(x)
x 1 2
P(X=x) 0.5 0.5
x 1 2
P(X=x) 0.9 0.1
x 1 2 3 4
P(X=x) 0.25 0.25 0.25 0.25
H(X) = 2 ⇥ 0.5 log(0.5) = 1
H(X) = 0.9 log(0.9) 0.1 log(0.1) = 0.469
H(X) = 4 ⇥ 0.25 log(0.25) = 2

26. Joint Entropy
• The uncertainty of a joint distribution involving two random variables
X, Y
H(X, Y ) =
X
x2X,y2Y
p(x, y) log p(x, y)
P(X,Y) Y=1 Y=2
X=1 0.25 0.25
X=2 0.25 0.25
H(X, Y ) = 4 ⇥ 0.25 log(0.25) = 2

27. Conditional Entropy
• The uncertainty of a random variable Y given the value of another
random variable X
H(Y |X) =
X
x2X
p(x)
X
y2Y
p(y|x) log p(y|x)
P(X,Y) Y=1 Y=2
X=1 0.4 0.4
X=2 0.1 0.1
P(X,Y) Y=1 Y=2
X=1 0.4 0.1
X=2 0.1 0.4
X & Y are independent Y is a stochasit function of X
H(X, Y ) = 1.722
H(Y |X) = 0.722
H(X, Y ) = 1.722
H(Y |X) = 1
P(X,Y) Y=1 Y=2
X=1 0.5 0
X=2 0 0.5
Y is a deterministic function of X
H(X, Y ) = 1
H(Y |X) = 0

28. Conditional Entropy
Information Diagram
H(X)H(Y )
H(X, Y )
H(Y |X)
H(Y |X) =
X
x2X
p(x)
X
y2Y
p(y|x) log p(y|x)
=
X
x2X
p(x)
X
y2Y
p(y|x)(log p(x, y) log p(x))
=
X
x2X,y2Y
p(x, y) log p(x, y) +
X
x2X
p(x) log p(x)
= H(X, Y ) H(X)

29. Conditional Entropy
P(X,Y) Y=1 Y=2
X=1 0.4 0.4
X=2 0.1 0.1
P(X,Y) Y=1 Y=2
X=1 0.4 0.1
X=2 0.1 0.4
X & Y are independent Y is a stochasit function of X
H(X, Y ) = 1.722
H(X) = 0.722, H(Y ) = 1
H(Y |X) = H(X, Y ) H(X)
= 1 = H(Y )
H(X, Y ) = 1.722
H(X) = 1, H(Y ) = 1
H(Y |X) = H(X, Y ) H(X)
= 0.722
P(X,Y) Y=1 Y=2
X=1 0.5 0
X=2 0 0.5
Y is a deterministic function of X
H(X, Y ) = 1
H(X) = 1, H(Y ) = 1
H(Y |X) = H(X, Y ) H(X)
= 0
H(X)
H(X, Y )
H(Y |X) = H(Y )
H(Y )
H(X, Y )
H(X)
H(Y |X) = 0
H(Y |X)
H(X, Y )

30. Mutual Information
• The mutual dependence between two variables X, Y
I(X; Y ) =
X
x2X,y2Y
p(x, y) log
p(x, x)
p(x)p(y)
P(X,Y) Y=1 Y=2
X=1 0.4 0.4
X=2 0.1 0.1
P(X,Y) Y=1 Y=2
X=1 0.4 0.1
X=2 0.1 0.4
X & Y are independent
I(X; Y ) = 0 I(X; Y ) = 0.278
Y is a stochasit function of X
P(X,Y) Y=1 Y=2
X=1 0.5 0
X=2 0 0.5
Y is a deterministic function of X
I(X, Y ) = 1

31. Mutual Information
H(X)H(Y )
H(X, Y )
Information
Diagram
I(X; Y )
I(X; Y ) =
X
x2X,y2Y
p(x, y) log
p(x, x)
p(x)p(y)
=
X
x2X,y2Y
p(x, y)( log p(x) log p(y) + log p(x, x))
=
X
x2X
p(x) log p(x)
X
y2Y
p(y) log p(y) +
X
x2X,y2Y
p(x, y) log p(x, x))
= H(X) + H(Y ) H(X, Y )

32. Mutual Information
P(X,Y) Y=1 Y=2
X=1 0.4 0.4
X=2 0.1 0.1
P(X,Y) Y=1 Y=2
X=1 0.4 0.1
X=2 0.1 0.4
X & Y are independent
H(Y )
H(X, Y )
H(X)
I(X; Y )
H(X)
H(X, Y )
H(Y )
I(X; Y ) = 0
Y is a stochasit function of X
P(X,Y) Y=1 Y=2
X=1 0.5 0
X=2 0 0.5
Y is a deterministic function of X
H(X, Y ) = 1.722
H(X) = 0.722, H(Y ) = 1
I(X; Y ) = H(X) + H(Y )
H(X, Y ) = 0
H(X, Y ) = 1.722
H(X) = 1, H(Y ) = 1
I(X; Y ) = H(X) + H(Y )
H(X, Y ) = 0.278
H(X, Y ) = 1
H(X) = 1, H(Y ) = 1
I(X; Y ) = H(X) + H(Y )
H(X, Y ) = 1
H(Y )
H(X, Y )
H(X)
I(X; Y )

33. Entropy, Joint Entropy , Conditional Entropy &
Mutual Information
H(X)
H(Y ) H(Z)
H(X|Y, Z)
H(Y |X, Z) H(Z|X, Y )
I(X; Z|Y )I(X; Y |Z)
I(Y ; Z|X)
I(X; Y ; Z)

34. Cross Entropy
Hp,q(X) =
X
x2X
p(x) log q(x)
=
X
x2X
p(x) log p(x)
X
x2X
p(x) log q(x) log p(x)
= Hp(X) + KL p(X)kq(X)

35. Information in the Weights
• Cause of Over-fitting
• Information in the Weights as a Regularizer
• Bounding the Information in the Weights
• Connection with Flat Minimum
• Connection with PAC-Bayesian Bound
• Experiments

36. Information in the Weights

37. Cause of Over-fitting
• Training loss (Cross-Entropy):
Hp,q(y|x, w) = Ex,y
⇥
p(y|x, w) log q(y|x, w)
⇤
p : probability density function of data
q : probability density function predicted by model
x : input feature of training data
y : label of training data
w : weights of model
✓ : latent parameters of data distribution

38. Hp,q(y|x, w) = Hp(y|x, w) + Ex,wKL p(y|x, w)kq(y|x, w)
Cause of Over-fitting
the uncertainty of y given w and x
Hp(x)
Hp(y)
Hp(y|x, w)
Hp(w)

39. Cause of Over-fitting
• lower
-> lower uncertainty of y given w and x -> lower training error
• ex: given x as , and a fixed w
8
>><
>>:
p(y = 1|x, w) = 0.9
p(y = 2|x, w) = 0.1
p(y = 3|x, w) = 0.0
p(y = 4|x, w) = 0.0
8
>><
>>:
p(y = 1|x, w) = 0.3
p(y = 2|x, w) = 0.3
p(y = 3|x, w) = 0.2
p(y = 4|x, w) = 0.2
higher Hp(y|x, w)lower Hp(y|x, w)
Hp(y|x, w)
Hp,q(y|x, w) = Hp(y|x, w) + Ex,wKL p(y|x, w)kq(y|x, w)

40. Cause of Over-fitting
Hp,q(y|x, w) = Hp(y|x, w) + Ex,wKL p(y|x, w)kq(y|x, w)
Hp(y|x, w) = Hp(y|x, ✓) + I(y; ✓|x, w) I(y; w|x, ✓)
✓ : latent parameters of (training & testing) data distribution
Hp(y|x)Hp(ytest|xtest)
Hp(✓)

41. Cause of Over-fitting
Hp(y|x)Hp(ytest|xtest)
Hp(✓)
✓ : latent parameters of (training & testing) data distribution
1
2
3
x y
3
2
x y
3
2
x y
3
Hp(y|x, ✓)
Ip(y; ✓|x)
useful information in
training data
noisy information
and outlier in
training data
noise and outlier
in testing data
normal samples not in
training data

42. Cause of Over-fitting
Hp(y|x, w) = Hp(y|x, ✓) + I(y; ✓|x, w) I(y; w|x, ✓)
Hp(x)
Hp(y)
Hp(✓)
Hp(y|x, ✓) the uncertainty of
y given w and x
Hp(y|x, w)
Hp(w)
I(y; ✓|x, w)
noisy information
and outlier in training data
useful information not
learned by weights
noisy information
and outlier learned by
weights
I(y; w|x, ✓)

43. Cause of Over-fitting
Hp(x)
Hp(y)
Hp(✓)
Hp(y|x, ✓)
Hp(w)
noisy information
and outlier in training data
Hp(y|x, w) = Hp(y|x, ✓) + I(y; ✓|x, w) I(y; w|x, ✓)

44. Cause of Over-fitting
• lower -> less noise and outlier in training data. low Hp(y|x, ✓)
lower Hp(y|x, ✓) higher Hp(y|x, ✓)
Hp(y|x)Hp(✓)
3
2
x y
1
3
Hp(y|x, w) = Hp(y|x, ✓) + I(y; ✓|x, w) I(y; w|x, ✓)
Hp(y|x)Hp(✓)
3
2
x y
2
3

45. Cause of Over-fitting
Hp(y|x, w) = Hp(y|x, ✓) + I(y; ✓|x, w) I(y; w|x, ✓)
Hp(x)
Hp(y)
Hp(✓)
Hp(w)
I(y; ✓|x, w)
useful information not
learned by weights

46. Cause of Over-fitting
• lower
-> more useful information learned by weights
-> lower
-> lower testing error
Hp(✓)
Hp(w1)
Hp(w2)
I(y; ✓|x, w)
Hp(y|x)
Hp(ytest|xtest)
I(y; ✓|x, w2) < I(y; ✓|x, w1)
) Hp(ytest|xtest, w2) < Hp(ytest|xtest, w1)
Hp(ytest|xtest, w)
Hp(y|x, w) = Hp(y|x, ✓) + I(y; ✓|x, w) I(y; w|x, ✓)

47. Cause of Over-fitting
Hp(y|x, w) = Hp(y|x, ✓) + I(y; ✓|x, w) I(y; w|x, ✓)
Hp(x)
Hp(y)
Hp(✓)
Hp(w)
noisy information
and outlier learned by
weights
I(y; w|x, ✓)

48. w
• Cause of over fitting: weights memorize the noisy informationin training data.
Cause of Over-fitting
High VC Dimension, but clean data
-> few noise to memorize
High VC Dimension, and noisy data
-> much noise to memorize
training data
testing data
Hp(y|x, w) = Hp(y|x, ✓) + I(y; ✓|x, w) I(y; w|x, ✓)
ww

49. • is unknown, cannot be compute
• , the information in the weight, is an upper bound of
Information in the Weights as a Regularizer
I(y; w|x, ✓)  I(y, x; w|✓) = I(D; w|✓)  I(D; w)
I(y; w|x, ✓)
I(y; w|x, ✓)I(D; w)
D
Hp(x)
Hp(y)
Hp(✓)
I(y; w|x, ✓) Hp(x, y) = Hp(D)
I(y, x; w|✓) = I(D; w|✓)I(D; w)
Hp(w)
I(y; w|x, ✓)

50. Information in the Weights as a Regularizer
• The actual data distribution p is unknown
• Estimate by
• New loss function : as a regularizerIp(D; w) ⇡ Iq(D; w)
L (q(w|D)) = Hp,q(y|x, w) + Iq(D; w)
I(D; w) Iq(D; w)

51. Connection with Flat Minimum
• Flat minimum has low information in the weight
kH k⇤ : nuclear norm of the Hessian at the local minimum
Flat minimum -> low neclear norm of the Hessian -> low information
Iq(w; D) 
1
2
K
⇥
log k ˆwk2
2 + log kH k⇤ K log(
K2
2
)
⇤

52. Connection with PAC-Bayesian Bound
• Given a prior distribution , we have:p(w)
distribution of weights
before training (prior)
distributionof weights
after training on dataset D
(posterior)
Iq(w, D) = EDKL(q(w|D)kq(w))
 EDKL(q(w|D)kq(w)) + EDKL(q(w|D)kp(w))
 EDKL(q(w|D)kp(w))

53. Connection with PAC-Bayesian Bound
• Loss function with the regularizer :
• PAC Bayesian Bound :
L (q(w|D)) = Hp,q(y|x, w) + Iq(D; w)
 Hp,q(y|x, w) + EDKL(q(w|D)kp(w))
Ip(D; w) ⇡ Iq(D; w)
ED
⇥
Ltest
(q(w|D))
⇤

Hp,q(y|x, w) + LmaxED
⇥
KL(q(w|D)kp(w))
⇤
n(1 1
2 )
Ltest : test error of the network with weights q(w|D)
Lmax : maximum per-sample loss function

54. Experiments
• Random Labels
Dataset size
Information complexity
L (q(w|D)) = Hp,q(y|x, w) + Iq(D; w)

55. Experiments
Dataset size
Information complexity
• Real Labels
L (q(w|D)) = Hp,q(y|x, w) + Iq(D; w)

56. Experiments
• information in the weights v.s. percentage of corrupted labels