Bayesian regularization of learning

Bayesian regularization of learning

Sergey Shumsky NeurOK Software LLC

Scientific methods

InductionF.Bacon

Machine

Models

Data

Deduction R.Descartes

Math. modeling

Outline Learning as ill-posed problem General problem: data generalization General remedy: model regularization

Bayesian regularization. Theory Hypothesis comparison Model comparison Free Energy & EM algorithm

Bayesian regularization. Practice Hypothesis testing Function approximation Data clustering

Outline Learning as ill-posed problem

General problem: data generalization General remedy: model regularization



Problem statement

Learning is inverse, ill-posed problem Model Data

Learning paradoxes Infinite predictions Finite data? How to optimize future predictions? How to select regular from casual in

data?

Regularization of learning Optimal model complexity

Well-posed problem

Solution is unique Solution is stable Hadamard (1900-s)

Tikhonoff (1960-s)

:h H h D

1 2

1 1

1 2

2 2

limD D

H h Dh h

H h D

lim limi ii i

H h H h h h

Learning from examples

Problem: Find hypothesis h, generating

observed data D in model H

Well defined if not sensitive to: noise in data (Hadamard) learning procedure (Tikhonoff)

:h H h D

Learning is ill-posed problem

Example: Function approximation

Sensitive tonoise in data

Sensitive tolearning procedure

Learning is ill-posed problem

Solution is non-unique

h x

h f

x x x x

Problem regularization Main idea: restrict solutions –

sacrifice precision to stability

: P hh H h D

arg minh H h D P h

How to choose?

Statistical Learning practice

Data Learning set

+ Validation set

Cross-validation:

Systematic approach to ensembles Bayes

+ +… +

Statistical Learning theory

Learning as inverse Probability Probability theory. H: h D

Learning theory. H: h D

, 1 hhN NN

h

NP D h H h h

N

,

,P D h H P h H

P h D HP D H

HBernoulli (1713)

Bayes (~ 1750)

Bayesian learning

D

h

,P h D H

,

,

,

P D h H P h H

P h D

P D

D

h

H

H

H P

P h H

P D HEvidence

Prior

Posterior

Coin tossing gameH

1P h

11, ,

2MP MPh h

MP

h hN Nh h h

N N N

,h h

P D h P h DP h D D P h N N N N

P D D

1 ,h

P D h P hP h D N P D h P h N N

P D

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

Monte Carlo simulations

100N

P h N t

10N

Bayesian regularization

Most Probable hypothesis

arg max log ,

arg min log , log

MPh P h D H

P D h H P h H

Learning error

212

2

η

ηlog ,

2η e

y hP D h H y h

P

x

x

Example: Function approximation

Regularization

Minimal Description Length

Most Probable hypothesis

2logL x P x

Code length for:

arg max log ,

arg min log , log

MPh P h D H

P D h H P h H

Data hypothesis

Example: Optimal prefix code

1

21

4 18

18

0 1

1110

110 111

P x

Rissanen (1978)

Data Complexity

Complexity K(D |H) = min L(h,D|

H)Code length L(h,D)= coded data L(D|h) + decoding program L(h)

Data D

Decoding:H h D

Kolmogoroff (1965)

Complex = Unpredictable

Prediction error ~ L(h,D)/L(D) Random data is uncompressible Compression = predictability

Program h: length L(h,D)

Data D

Decoding:H h D

Example: block coding

Solomonoff (1978)

Universal Prior All 2L programs with

length L are equiprobable

Data complexity

,

,2, , 2

L h D HL h D H

hP h D H P D H

P D H

Solomonoff (1960) Bayes (~1750)

logK D H L D H P D H

L(h,D)

D

H

Statistical ensemble Shorter description length

Proof:

Corollary: Ensemble predictions are superior to most probable prediction

,log 2 ,L h D H

MPhL D H L h D H

, ,log 2 log 2 ,MPL h D H L h D H

MPhL h D H

Ensemble prediction

MPh

1h

2h P h H

1h

MPh

P h H2h

Model comparison

D

h

,P h D H

, iP D h H

P D HEvidence

Posterior

Statistics: Bayes vs. Fisher

Fisher: max Likelihood

Bayes: max Evidence

arg max log ,

arg max log , log

MP

ML

h P h D H

P D h H P h H h

arg max log ,

arg max log , log

MP

ML

H P H D

P D H P H H

H

H H

arg max log , logMPh P D h H P h H

Historical outlook

20 – 60s of ХХ century Parametric statistics Asymptotic N

60 - 80s of ХХ century Non-Parametric statistics Regularization of ill-posed problems Non-asymptotic learning Algorithmic complexity Statistical physics of disordered systems

h

Fisher (1912)

Chentsoff (1962)

Tikhonoff (1963)

Vapnik (1968)

Kolmogoroff (1965)

h

Gardner (1988)

Statistical physics

Probability of hypothesis - microstate

Optimal model - macrostate

,

,

1, L h D H

L h D H

h

P h D H eZ

Z D H e

arg min logMLH L D H Z D H

Free energy

F = - log Z: Log of Sum

F = E – TS:

Sum of logs P = P{L}

, , log ,h

L D H P h D H L h D H P h D H

,log L h D H

hL D H e

EM algorithm. Main idea

Introduce independent P:

Iterations E-step:

М-step:

, log

log,

h

h

F h L h D H h

hL D H h L D H

P h D H

P P

PP

arg min ,

arg min ,

t t

t t

H

F H

H F H

P

P P

P

EM algorithm

Е-step Estimate Posterior for given Model

М-step Update Model for given Posterior

,t t th P D h H P D HP

arg min ,t

t

HH L D h H

P

Bayesian regularization: Examples

Hypothesis testing

Function approximation

Data clustering

yh

y

h(x) x

P(x|H)

x

Hypothesis testing Problem Noisy observations: y

Is theoretical value h0 true?

Model H:

2

2

0

2 exp2

2 exp2

y h

P

h hP h

Gaussian noise

Gaussian prior

yh0

00.2

0.40.6

0.8

1

0

2

4

6

8

107.5

8

8.5

9

9.5

10

10.5

11

11.5

Optimal model: Phase transition

1 2

1ML y

N

N

2 2, ln lnyL D N y NN N

,MLL D

y

y

Confidence finite infinite

Threshold effect

Student coefficient

Hypothesis h0 is true

Corrections to h0

22

2 1y

Nt y N

y

P(h)1Nt

yh

1Nt P(h)2

2

1NMP

N

th y

t

0P h h h

Function approximation

Problem Noisy data: y (x ) Find approximation h(x)

Model:

Noise

Prior

1

1

,

exp

exp W

y h

P Z E

P Z E

x w

w w

y

h(x)x

Optimal model

Free energy minimization

,

,

, ln ln ln

exp

x

e

e p

xp

D

W

W

D

Z d E E

L D Z Z Z

Z d E

Z dD E

w w

w w

w

w

,n nD n

E E y h w x w

Saddle point approximation

Function of best hypothesis

,ln exp

1ln

2

D W

W MP D MP

D MP W MP

Z d E E

E E

E E

w w w

w w

w w

MPw

ЕМ learning

Е-step. Optimal hypothesis

М-step. Optimal regularization

arg minMP ML W ML DE E w w w

, arg min , ,ML ML MPL D w

Laplace Prior

Pruned weights

Equisensitive weights

1

W

W iiE w

w

sgnD W DE E Et

w

w w w w

0D i iE w w

0i D iw E w

Clustering

Problem Noisy data: x

Find prototypes (mixture density approximation)

How many clusters?

Модель:

Noise

1

exp

M

m

m

P H P m P m

P m E

x x

x x h

P(x|H)

x

Optimal model

Free energy minimization

Iterations E-step:

М-step:

min ln ,n

n mL D H P m x

,min ln ln ,n

m nF m n m n P m P P x

arg min ,

arg min ,

t t

t t

H

F H

H F H

P

P P

P

ЕМ algorithm

Е-step:

М-step:

exp

exp

nm

n

nmm

P m EP m

P m E

x hx

x h

1

n n

n

m n

n

n

n

P m

P m

P m P mN

x xh

x

x

How many clusters?

Number of clusters M()

Optimal number of clusters

h(m)

1/

min ln ,m mmL D d P D h h

Simulations: Uniform data

Optimal model

0 10 20 30 4011

12

13

14

15

16

M

L D M

Simulations: Gaussian data

Optimal model

M0 10 20 30 40 50

-12.5

-12

-11.5

-11

-10.5

-10

-9.5 L D M

0 5 10 1520

25

30

35

40

45

Simulations: Gaussian mixture

Optimal model

M

L D M

Summary

Learning Ill-posed problem Remedy: regularization

Bayesian learning Built-in regularization (model assumptions) Optimal model = minimal Description Length

= minimal Free Energy

Practical issues Learning algorithms with built-in optimal

regularization - from first principles (opposite to cross validation)

Bayesian regularization of learning

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