Polynomial learning from Laurent Miclet, Jose Oncina and Tim - - PDF document

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Polynomial learning from positive and negative examples

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Acknowledgements

• Laurent Miclet, Jose Oncina

and Tim Oates for previous versions of these slides.

• Rafael Carrasco, Paco Casacuberta, Rémi

Eyraud, Philippe Ezequel, Henning Fernau, Thierry Murgue, Franck Thollard, Enrique Vidal, Frédéric Tantini,...

• List is necessarily incomplete. Excuses

to those that have been forgotten. http://eurise.univ-st-etienne.fr/~cdlh/slides

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Outline

• 1. The problem
• 2. Notations
• 3. Models
• 4. Proof techniques
• 5. Conclusion

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1 The problem:

• In a general way to learn a language

(belonging to some class L) from examples and perhaps from:

– counter-examples – queries to an oracle – specific knowledge

• Once the program is written we would

like:

– to say it is correct – to prove that no correct program can be written

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What does ‘correct’ mean?

• We need a goal:

– L is a target (unknown). The harder L is the harder it is going to be to learn.

• Learn what?

– find a representation of L – find some reasonable approximation of L

(what is a reasonable approximation?)

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Representation of L

• We are going to have to fix some

representation of L;

• We denote by r(L) this ideal

representation of L;

• And

∫r(L)∫ is the size of this representation;

• Or

at least some polynomial measure of the number of bits needed to encode r(L).

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How long can it take?

• Ideally:

with p(∫r(L)∫) examples we are sure to learn/find...

• Interesting:

with p(∫r(L)∫) examples drawn according to some distribution D, we will be nearly sure of finding a grammar/ classifier that will be nearly always right… according to D (PAC model: Valiant 84)

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What about efficiency?

• We can try to bound

– global time – update time – errors before converging – queries – good examples needed

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2 General Notations

C H r(L) h r(L)≡h r(L)≈h

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The examples

Σ*

1 x

r( L)

x

r(L )

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The classes C and H

• sets of examples
• representations of these sets
• the computation of r(L)(x)

(and h(x)) must take place in time polynomial in ⏐x⏐

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How do we consider a finite set?

Σ* D Σn D≤n Pr<ε

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3 Some models/paradigms

• Identification in the limit
• PAC learnability
• PAC predictability
• Learning with a teacher

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• and...

– Identification in the limit with probability 1 – Identification PAC – Simple PAC – PAC Simple – Different teaching models – …

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Protocol

We have a presentation of some language L:

∀x∈L, x

appears in the presentation (learning from text : positive presentation)

∀x∈Σ*

<x, L(x)> appears in the presentation (learning from informant)

3.1 Identification in the limit (Gold 67,78)

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x1 x2 h1 h2 xn hn xi hi ≡ hn … ≡ r(L)

C is identifiable in the limit iff ∀L∈C, ∀ presentation

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Main results (Gold 67)

• No

super-finite class is identifiable from text;

• any

recursively denumerable class is identifiable from an informant.

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Main results in GI

• All well known classes of

languages can be identified from complete presentations

• No usual class of languages

can be identified from positive presentations

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3.2 PAC learning

(Valiant 84, Pitt 89)

• C a set of languages
• H a set of hypothesis
• ε>0 and δ >0
• L∈C
• h∈H

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h is AC (approximately correct)* iff PrD[h(x)≠L(x)]< ε

* For some specific ε

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f h

Errors: we want (1(L)⊕1(h))<ε

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h is PAC* (probably approximately correct) iff PrD[h(x)≠L(x)]<ε with probability at least 1-δ

* For some specific ε and δ

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The oracle EX

• The examples may cost, but…
• (X, D) set of examples.
• Denote by n

the size of an example.

• EX(L,D) returns in time at most

O(n) a pair <x,L(x)>.

• simplifying… EX

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The class C is PAC learnable by H iff there exists an algorithm (maybe probabilistic) a that uses EX to

• btain

∀ε>0 and δ>0, ∀L∈C and for any distribution D

• ver Σ*, a PAC hypothesis h ∈ H.

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The class C is polynomially PAC-learnable by H if C is PAC-learnable by H and if for any L∈C, a returns a PAC solution in time polynomial in 1/ε, 1/δ, ∫r(L)∫, n.

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3.3 PAC Prediction

The class C is polynomially PAC-predictable if there is a class H such that C is polynomially PAC-learnable by H.

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Some observations

• Different variants

– PAC-identifiable: ε=0 – EX-pos, EX-neg

• the case C

is PAC-learnable (by C): this is the usual case for positive results, but is not that useful in the negative case.

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PAC and GI

• PAC learning DFA is still an open

problem but it is believed to be impossible because

– intractability of minimum consistency problem (Gold 78) – hardness

• f

prediction due to cryptographic limitations (Kearns & Valiant 89) – hardness of learning with equivalence queries (Pitt 89, Angluin 87)

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3.4 Active Learning

• Idea:

the learner can interrogate a master (an

• racle)

­ the

• racle

must answer correctly ­ the oracle may choose the worse of the correct answers

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Active learning and GI

­ see the 2 lectures on the subject ­ with poor queries, cannot learn anything ­ with strong queries, can learn DFA

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3.5 Learning from a Teacher

• Idea:

– the teacher can choose some good examples. – All examples are given at the beginning. – To avoid cheating (collusion) these examples will be mixed with others, less useful.

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Intermediate Model

Identification from a characteristic sample

• Algorithm must be polynomial and ...

… every concept admits a polynomial characteristic sample

• Related to learning from a teacher: a

set of models for the harder classes. Goldman, Mathias, ...

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Identification in the limit from polynomial time and data

1) Given a sample <X+, X->, of size m, ϕ returns h in H consistent with <X+, X-> in time in O(p(m)). 2) For any r(L) of size n, there exists a characteristic sample <CS+, CS-> of size at most q(n), with which, given <X+, X->, with CS+⊆X+, CS-⊆X-, ϕ returns h equivalent to f.

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Identification in the limit from polynomial time and data.

a

h ≡ L

• f size q(∫r(L)∫)

<X+, X-> h [in time p(║X+║+║X-║)] <CS+,CS-> ⊆ <X+,X-> L

a

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A theorem by Gold (1978)

• DFA are identifiable in the

limit from polynomial time and data

• alternative

results and algorithms:

– Trakhenbrot & Barzdin 73 – Oncina & García 92 – Lang 92

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By morphisms the result may extend to:

• Even linear grammars (Takada 88 &

94; Sempere & García 94, Mäkinen 96)

• Total

subsequential functions (Oncina, García & Vidal 93 )

• Context-free

grammars from skeletons (Sakakibara 90)

• Tree automata (Knuutila 94)

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4 Hardness proofs

– algorithmic proofs – ‘information theoretic’ proofs

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4.1 Algorithmic proofs

• We prove that learning some

grammar would solve some hard problem.

• Usually:
• RSA
• RP-complete problem

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4.2 ‘Information theoretic’ proofs

• We prove that there cannot be

enough information to learn.

• Examples

– Approximate fingerprints, Angluin – Polynomial characteristic sets, cdlh

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Negative Results

• For any alphabet Σ of size at

least 2, the following classes are not identifiable in the limit from polynomial time and data (cdlh 97):

– CFG(Σ), Context-Free Grammars; – LIN(Σ), Linear Grammars; – NFA(Σ), Non-Deterministic Finite Automata.

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Proof

• Let G

be a class for which the equivalence problem (g1≡g2?) is undecidable.

• Then for any polynomial p()

there exist g1 and g2 inseparable by strings

• f

length <p(∫g1∫ +∫g2∫).

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• Suppose

g1 and g2 are learnable with polynomial characteristic samples CS1 and CS2.

• What grammar (function) will

be inferred from CS1∪CS2?

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Conclusion

• For identification in the limit

from a complete presentation

– nearly everything is inferable

• For identification in the limit

from a positive presentation

– nearly nothing is inferable

• For PAC-prediction

– nearly nothing is inferable

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State of the art

DFA Context-free Identification in the limit yes yes PAC no no

• Poly. Identif.

yes no Simple PAC yes ?

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Open problems

• Minimum Description Length provides

an alternative convergence principle. Relate it to Identification in the limit.

• Relate identification in the limit as

defined by Gold in 78 and in 67.

• Improve the results of Honavar

& Parekh 98.