SUPERSET LEARNING AND DATA IMPRECISIATION Eyke Hllermeier - - PowerPoint PPT Presentation

Eyke Hüllermeier

Intelligent Systems Group Department of Computer Science University of Paderborn, Germany

eyke@upb.de

SUPERSET LEARNING AND DATA IMPRECISIATION

TFML 2017, Krakow, 15-FEB-2017

O UTLI NE

2

PART 1 Superset learning PART 2 Optimistic loss minimization PART 3 Data imprecisiation

What it is about .... A general approach to superset learning .... Using superset learning for weighted learning ...

SUPERSET LEARNI NG

3

... is a specific type of weakly supervised learning, studied under different names in machine learning:

• learning from partial labels
• multiple label learning
• learning from ambiguously labeled examples
• ...

... also connected to learning from coarse data in statistics (Rubin, 1976; Heitjan and Rubin, 1991), missing values, data augmentation (Tanner and Wong, 2012).

SUPERSET LEARNI NG

4

• Consider a standard setting of supervised learning with instance

space X, output space Y, and hypothesis space H

• Output values yn 2 Y associated with training instances xn,

n = 1, . . . , N, are not necessarily observed precisely but only characterised in terms of supersets Yn 3 yn .

• Set of imprecise/ambiguous/coarse observations is denoted

O =

• (x1, Y1), . . . , (xN, YN)
• An instantiation of O, denoted D, is obtained by replacing each Yn

with a candidate yn ∈ Yn.

EXAM PLE: CLASSI FI CATI O N

5

Classes

EXAM PLE: CLASSI FI CATI O N

6

Classes

• ne of many

instantiations

EXAM PLE: REG RESSI O N

7

• ne of infinitely many

instantiations

DATA DI SAM BI G UATI O N

8

How to learn from (super)set-valued data?

DATA DI SAM BI G UATI O N

9

Classes

DATA DI SAM BI G UATI O N

10

Classes

DATA DI SAM BI G UATI O N

11

Classes

DATA DI SAM BI G UATI O N

12

DATA DI SAM BI G UATI O N

13

DATA DI SAM BI G UATI O N

14

MORE PLAUSIBLE LESS PLAUSIBLE A less plausible instantiation, because there is no LINEAR model with a good fit! A plausible instantiation that can be fitted reasonably well with a LINEAR model!

DATA DI SAM BI G UATI O N

15

PLAUSIBLE PLAUSIBLE A plausible instantiation that can be fitted quite well with a QUADRATIC model!

I t all depends on how y ou look at t he dat a!

A plausible instantiation that can be fitted quite well with a QUADRATIC model!

DATA DI SAM BI G UATI O N

16

−4 −2 2 4 6 8 −3 −2 −1 1 2 3 4 5 6 7 8

= { , }

assume both class distributions to be Gaussian

−4 −2 2 4 6 8 −3 −2 −1 1 2 3 4 5 6 7 8

DATA DI SAM BI G UATI O N

17

plausible instantiation

= { , }

assume both class distributions to be Gaussian

quadratic discriminant

−4 −2 2 4 6 8 −3 −2 −1 1 2 3 4 5 6 7 8

DATA DI SAM BI G UATI O N

18

implausible instantiation

= { , }

assume both class distributions to be Gaussian

Model identification and data disambiguation should be performed simultaneously:

identification disambiguation

DATA DI SAM BI G UATI O N

19

DATA MODEL

... quite natural from a Bayesian perspective: P(h, D) = P(h) P(D | h) = P(D) P(h | D)

O UTLI NE

20

PART 1 Superset learning PART 2 Optimistic loss minimization PART 3 Data imprecisiation

M AXI M UM LI KELI HO O D ESTI M ATI O N

21

Imprecise observation only depends on true data, not on the model.

precise DATA MODEL

generation

imprecise DATA

imprecisiation coarsening

Likelihood of a model h ∈ H: `(h) = P(O, D | h) = P(D | h) P(O | D, h) = P(D | h) P(O | D)

SUPERSET ASSUM PTI O N

22

Imprecise data is a superset, but no other assumption.

precise DATA MODEL

generation

imprecise DATA

imprecisiation ambiguation

G ENERALI ZED ERM

23

how well the (precise) model fits the imprecise data

We derive a principle of generalized empirical risk minimization with the empirical risk Remp(h) = 1 N

N

X

n=1

L∗ Yn, h(xn)

• and the optimistic superset loss (OSL) function

L∗(Y, ˆ y) = min

• L(y, ˆ

y) | y ∈ Y .

SPECI AL CASES

24

G ENERALI ZATI O N TO FUZZY DATA

25

1 1

interval fuzzy interval

G ENERALI ZATI O N TO FUZZY DATA

26

LO SS 1

α

α-cut

G ENERALI ZATI O N TO FUZZY DATA

27

LO SS RI SK

L∗∗(Y, ˆ y) = Z 1 L∗⇣ [Y ]α, ˆ y ⌘ dα

Remp(h) = 1 N

N

X

n=1

L∗∗⇣ Yn, h(xn) ⌘

G ENERALI ZATI O N TO FUZZY DATA

28

à Huber loss !

L∗∗(Y, ˆ y)

G ENERALI ZATI O N TO FUZZY DATA

29

à (generalized) Huber loss !

STRUCTURED O UTPUT PREDI CTI O N

30

Superset learning naturally applies to learning problems with structured outputs, which are often only partially specified and can then be associated with the set of all consistent completions.

LABEL RANKI NG

31

... is the problem to learn a model that maps instances to TOTAL ORDERS

• ver a fixed set of alternatives/labels:

D A C B

LABEL RANKI NG

32

(0,37,46,325,1,0)

... likes more ... reads more ... recommends more ...

A D C B

... is the problem to learn a model that maps instances to TOTAL ORDERS

• ver a fixed set of alternatives/labels:

LABEL RANKI NG

33

Tr aining dat a is t y pic ally inc om plet e!

(0,37,46,325,1,0)

A C

... is the problem to learn a model that maps instances to TOTAL ORDERS

• ver a fixed set of alternatives/labels:

LABEL RANKI NG

34

:

Tr aining dat a is t y pic ally inc om plet e!

set of linear extensions

(0,37,46,325,1,0)

... is the problem to learn a model that maps instances to TOTAL ORDERS

• ver a fixed set of alternatives/labels:

LABEL RANKI NG LO SSES

35

K E N D A L L S P E A R M A N

EXPERI M ENTAL STUDI ES

36

30 60 0.7 0.8 0.9

authorship 30 60 0.5 1 glass 30 60 0.7 0.8 0.9 iris 30 60 0.5 1 pendigits 30 60 0.5 1 segement 30 60 0.6 0.8 1 30 60 0.5 1 vovel 30 60 0.85 0.9 0.95 wine

• Cheng and H. (2015) compare an approach to label ranking based on

superset learning with a state-of-the-art label ranker based on the Plackett-Luce model (PL).

• Two missing label scenarios: missing at random, top-rank
• General conclusion: more robust toward incompleteness

O UTLI NE

37

PART 1 Superset learning PART 2 Optimistic loss minimization PART 3 Data imprecisiation

DATA I M PRECI SI ATI O N

38

So far: Observations are imprecise/incomplete, and we have to deal with that! Now: Deliberately turn precise into imprecise data, so as to modulate the influence of an observation on the learning process!

EXAM PLE W EI G HI NG

39

EXAM PLE W EI G HI NG

40

EXAM PLE W EI G HI NG

41

We suggest an alternative way of weighing examples, namely, via „data imprecisiation“ ... 1 1 1

full support for precise observation

EXAM PLE W EI G HI NG

42

EXAM PLE W EI G HI NG

43

weighing through „imprecisiation“

EXAM PLE W EI G HI NG

44

Different ways of (individually) discounting the loss function.

l o s s

In (Lu and H., 2015), we empirically compared standard locally weighted linear regression with this approach and essentially found no difference.

weighted loss OSL

EXAM PLE W EI G HI NG

45

1

certainly positive less certainly positive

We suggest an alternative way of weighing examples, namely, via „data imprecisiation“ ...

FUZZY M ARG I N LO SSES

46

G E N E R A L I Z E D H I N G E L O S S

w=1 w=3/4 w=1/2 w=1/4 w=0

FUZZY M ARG I N LO SSES

47

Different ways of (individually) discounting the loss function.

w=1 w=3/4 w=1/2 w=1/4 w=0 w=1 w=3/4 w=1/2 w=1/4 w=0

weighted loss OSL

THE HAT LO SS

48

DATA DI SAM BI G UATI O N

49

DATA DI SAM BI G UATI O N

50

DATA DI SAM BI G UATI O N

51

EXPERI M ENTS

52

Robust loss minimization techniques:

§ Robust truncated-hinge-loss support vector machines (RSVM) trains SVMs with the a truncated version of the hinge loss in order to be more robust toward outliers and noisy data (Wu and Liu, 2007). § One-step weighted SVM (OWSVM) first trains a standard SVM. Then, it weighs each training example based on its distance to the decision boundary and retrains using the weighted hinge loss (Wu and Liu, 2013). § Our approach (FLSVM) is the same as OWSVM, except for the weighted loss: instead of using a simple weighting of the hinge loss, we use the

• ptimistic fuzzy loss.

Non-convex optimization problem solved by concave-convex procedure (Yuille and Rangaraja, 2002).

EXPERI M ENTAL RESULTS

53

THEO RETI CAL FO UNDATI O NS

54

Under what conditions is (successful) learning in the superset setting actually possible?

THEO RETI CAL FO UNDATI O NS

55

THEO RETI CAL FO UNDATI O NS

56

systematic imprecisiation

THEO RETI CAL FO UNDATI O NS

57

non-systematicimprecisiation

THEO RETI CAL FO UNDATI O NS

58

Liu and Dietterich (2014) consider the ambiguity degree, which is defined as the largest probability that a particular distractor label co-occurs with the true label in multi-class classification: = sup n PY ∼Ds(x,y)(` 2 Y ) | (x, y) 2 X ⇥ Y, ` 2 Y, p(x, y) > 0, ` 6= y

THEO RETI CAL FO UNDATI O NS

59

Let ✓ = log(2/(1 + )) and dH the Natarajan dimension of H. Define n0(H, ✏, ) = 4 ✓✏ ✓ dH ✓ log(4dH + 2 log L + log ✓ 1 ✓✏ ◆◆ + log ✓1

• ◆

+ 1 ◆ . Then, in the realizable case, with probability at least 1 − , the model with the smallest empirical superset loss on a set of training data of size n > n0(H, ✏, ) has a generalisation error of at most ✏.

THEO RETI CAL FO UNDATI O NS

60

The balanced benefit condition: 0 ≤ η1 ≤ inf

h∈H

RS(h) R(h) ≤ sup

h∈H

RS(h) R(h) ≤ η2 ≤ 1 , where RS(h) is the expected superset loss of h. For sufficiently large sample size, R(ˆ h) ≤ R(h∗) + ∆(dH, ✏, , ⌘1, ⌘2) , with probability 1 − , where h∗ is the Bayes predictor and ˆ h the empirical (superset) risk minimizer; in general, ∆ cannot be made arbitrarily small.

SUM M ARY AND O UTLO O K

§ Method for superset learning based on optimistic loss minimization, performing simultaneous model identification and data disambiguation. § Our framework covers several existing methods as special cases but also supports the systematic development of new methods. § Completely generic principle (classification, regression, structured

• utput prediction, ...)

§ Example weighing via data imprecisiation (à „modeling data“) § Works for regression and classification, but seems to be even more interesting for other problems, including ranking, transfer learning, ... § More future work: Algorithmic solutions for specific instantiations of our framework, theoretical foundations.

61

REFERENCES

62

• E. Hüllermeier and W. Cheng (2015). Superset Learning Based on Generalized Loss
• Minimization. Proc. ECML/PKDD 2015.
• E. Hüllermeier (2014). Learning from Imprecise and Fuzzy Observations: Data

Disambiguation through Generalized Loss Minimization. International Journal of Approximate Reasoning, 55(7):1519-1534, 2014.

• S. Lu and E. Hüllermeier. Locally Weighted Regression through Data
• Imprecisiation. Workshop Computational Intelligence, Dortmund, 2015.

D.B. Rubin. Inference and missing data. Biometrika, 63(3):581–592, 1976.

• D. F. Heitjan and D. B. Rubin. Ignorability and coarse data. The Annals of Statistics,

19(4):2244–2253, 1991.