Complexity of Teaching by a Restricted Number of Examples Hayato - - PowerPoint PPT Presentation

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Complexity of Teaching by a Restricted Number of Examples

Hayato Kobayashi and Ayumi Shinohara Tohoku University, Japan

1

• COLT2009. Montreal, Canada. 21 June.

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Background

• Computational teaching theory

– Aims to bring out the nature of teaching – which is inextricably linked to learning

• Teachability [Shinohara and Miyano 1991]
• Teaching dimension [Goldman and Kearns 1991]

…

• Expected teaching dimension [Balbach 2005]
• Recursive teaching dimension [Zilles et al. 2008]

2

• COLT2009. Montreal, Canada. 21 June.

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Illustrative problem: Phone-a-friend lifeline

• Millionaire (=“Who Wants to Be a Millionaire?”)

– Challenge multiple-choice questions – Win a cash award depending on the number of correct answers – Get help from the three lifelines during the game

• Lifelines

– Phone-a-friend

• Will give you advice from friends

– 50:50

• Removes two incorrect answers

– Ask the Audience

• Lets you see the answers of audience

3

• COLT2009. Montreal, Canada. 21 June.

Censored Censored “Who Wants to Be a Millionaire?” Censored

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Classical model (1/3)

• Millionaire (one correct choice)

Learner (challenger) Teaching Teacher (friend) Concept class (available answers) C = {{A}, {B}, {C}, {D}} When was the COLT conference first held? A: 1983 C: 1992 B: 1988 D: 1997

4

• COLT2009. Montreal, Canada. 21 June.

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Classical model (1/3)

• Millionaire (one correct choice)

Learner (challenger) Teaching Teacher (friend) S = {(A, False), (B, True), (C, False), (D, False)} Concept class (available answers) C = {{A}, {B}, {C}, {D}} Target concept (correct answer) c = {B} When was the COLT conference first held? A: 1983 C: 1992 B: 1988 D: 1997 B: 1988

5

• COLT2009. Montreal, Canada. 21 June.

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Classical model (1/3)

• Millionaire (one correct choice)

Learner (challenger) Teaching Teacher (friend) S = {(A, False), (B, True), (C, False), (D, False)} Concept class (available answers) C = {{A}, {B}, {C}, {D}}

Teaching set

Target concept (correct answer) c = {B} When was the COLT conference first held? A: 1983 C: 1992 B: 1988 D: 1997 B: 1988

6

• COLT2009. Montreal, Canada. 21 June.

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Classical model (1/3)

• Millionaire (one correct choice)

Learner (challenger) Teaching Teacher (friend) S = {(A, False), (B, True), (C, False), (D, False)} Concept class (available answers) C = {{A}, {B}, {C}, {D}}

Teaching set

Target concept (correct answer) c = {B} When was the COLT conference first held? A: 1983 C: 1992 B: 1988 D: 1997 B: 1988 Teaching Dimension TD(c, C) := |Minimum teaching set| In this case, TD(c, C)=1

7

• COLT2009. Montreal, Canada. 21 June.

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Classical model (2/3)

• Millionaire 2.0 (two correct choices)

Learner (challenger) Teaching Teacher (friend) Concept class (available answers) C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} Which are the two cities where the Olympic games were held in Canada? A: Montreal C: Ottawa B: Calgary D: Vancouver

8

• COLT2009. Montreal, Canada. 21 June.

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Classical model (2/3)

• Millionaire 2.0 (two correct choices)

Learner (challenger) Teaching Teacher (friend) S = {(A, True), (B, True)} Concept class (available answers) C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} Target concept (correct answer) c = {A, B} Which are the two cities where the Olympic games were held in Canada? A: Montreal C: Ottawa B: Calgary D: Vancouver B: Calgary A: Montreal

9

• COLT2009. Montreal, Canada. 21 June.

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Classical model (2/3)

• Millionaire 2.0 (two correct choices)

Learner (challenger) Teaching Teacher (friend) S = {(A, True), (B, True)} Concept class (available answers) C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}}

Minimum teaching set

Target concept (correct answer) c = {A, B} Which are the two cities where the Olympic games were held in Canada? A: Montreal C: Ottawa B: Calgary D: Vancouver B: Calgary A: Montreal Teaching Dimension TD(c, C) := |Minimum teaching set| In this case, TD(c, C)=2

10

• COLT2009. Montreal, Canada. 21 June.

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Classical model (3/3)

• Generalized Millionaire (unknown # of correct choices)

Teaching S = {(A, True), (B, True), (C, True), (D, True)} Concept class (available answers) C = 2{A,B,C,D} Target concept (correct answer) c = {A, B, C, D} Which choices are correct specialties of Canada? A: Maple tea C: Maple dressing B: Maple butter D: Maple mustard Learner (challenger) Teacher (friend)

11

• COLT2009. Montreal, Canada. 21 June.

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Classical model (3/3)

• Generalized Millionaire (unknown # of correct choices)

Teaching S = {(A, True), (B, True), (C, True), (D, True)} Concept class (available answers) C = 2{A,B,C,D} Target concept (correct answer) c = {A, B, C, D} Which choices are correct specialties of Canada? A: Maple tea C: Maple dressing B: Maple butter D: Maple mustard Learner (challenger) Teacher (friend) A: Maple tea

12

• COLT2009. Montreal, Canada. 21 June.

B: Maple butter C: Maple dressing D: Maple mustard

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Classical model (3/3)

• Generalized Millionaire (unknown # of correct choices)

Teaching S = {(A, True), (B, True), (C, True), (D, True)} Concept class (available answers) C = 2{A,B,C,D} Minimum teaching set Target concept (correct answer) c = {A, B, C, D} Which choices are correct specialties of Canada? A: Maple tea C: Maple dressing B: Maple butter D: Maple mustard Learner (challenger) Teacher (friend) A: Maple tea

13

• COLT2009. Montreal, Canada. 21 June.

B: Maple butter C: Maple dressing D: Maple mustard Teaching Dimension TD(c, C) := |Minimum teaching set| In this case, TD(c, C)=4

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Our contributions

• When # of examples < teaching dimension
• Formal proofs that

– Special teaching strategies are necessary

• A subset of a teaching set is not always optimal

– Smart teachers dare to tell a lie

• Inconsistent examples are more useful
• Exact analyses of optimal teaching errors and
• ptimally incremental teachabilities for concept

classes of

– Mn

+ : Monotone monomials

– Mn’ : Monomials without the empty concept – Mn : Monomials

14

• COLT2009. Montreal, Canada. 21 June.

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Our model

• Restriction: # of examples ≦ k

– Phone-a-friend lifeline: 30 sec. – This presentation: 25 min. – Lectures in our univ.: 90 min.

Teaching Concept class (available answers) C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} Target concept (correct answer) c = {A, B} Learner (challenger) Teacher (friend) Millionaire 2.0 (two correct choices)

15

• COLT2009. Montreal, Canada. 21 June.

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Our model

• Restriction: # of examples ≦ k

– Phone-a-friend lifeline: 30 sec. – This presentation: 25 min. – Lectures in our univ.: 90 min.

Teaching Concept class (available answers) C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} Target concept (correct answer) c = {A, B} Learner (challenger) Teacher (friend) Millionaire 2.0 (two correct choices) “The question is which two are …“ (He used 29 sec.)

16

• COLT2009. Montreal, Canada. 21 June.

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Our model

• Restriction: # of examples ≦ k

– Phone-a-friend lifeline: 30 sec. – This presentation: 25 min. – Lectures in our univ.: 90 min.

Teaching Concept class (available answers) C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} S = {(A, True)} Target concept (correct answer) c = {A, B} “A is True. B is“ Learner (challenger) Teacher (friend) Millionaire 2.0 (two correct choices) “The question is which two are …“ (He used 29 sec.)

17

• COLT2009. Montreal, Canada. 21 June.

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Our model

• Restriction: # of examples ≦ k

– Phone-a-friend lifeline: 30 sec. – This presentation: 25 min. – Lectures in our univ.: 90 min.

Teaching Concept class (available answers) C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} S = {(A, True)} Target concept (correct answer) c = {A, B} “A is True. B is“ Learner (challenger) Teacher (friend) Millionaire 2.0 (two correct choices) “The question is which two are …“ (He used 29 sec.) (Complexity) Optimal Teaching Error

18

• COLT2009. Montreal, Canada. 21 June.

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Optimal Teaching Error

h A B C D Err(c, h) {A,B} T T F F 0/4 {A,C} T F T F 2/4 {A,D} T F F T 2/4 {B,C} F T T F 2/4 {B,D} F T F T 2/4 {C,D} F F T T 4/4

Millionaire 2.0 (two correct choices) C = {{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}} c = {A, B} OptTSets1(c, C) = { {(A, True)}, {(B, True)} {(C, False)}, {(D, False)} } OptTSets2(c, C) = MinTSets(c, C) = { {(A, True), (B, True)}, {(C, False), (D, False)} }

c =

| } , , , { | | | | | | | : ) , ( D C B A h c h c X h c h c Err      

c Definition

) , ( max min : ) , (

) , ( | :|

h c Err C c OptTErr

C S CONS h k S S k  



Worst case error

) , ( max min arg : ) , (

) , ( | :|

h c Err C c OptTSets

C S CONS h k S S k  



k-optimal teaching sets achieving the optimal teaching error

19

• COLT2009. Montreal, Canada. 21 June.

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Optimal Teaching Error

h A B C D Err(c, h) {A,B} T T F F 0/4 {A,C} T F T F 2/4 {A,D} T F F T 2/4 {B,C} F T T F 2/4 {B,D} F T F T 2/4 {C,D} F F T T 4/4

Millionaire 2.0 (two correct choices) C = {{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}} c = {A, B} OptTSets1(c, C) = { {(A, True)}, {(B, True)} {(C, False)}, {(D, False)} } OptTSets2(c, C) = MinTSets(c, C) = { {(A, True), (B, True)}, {(C, False), (D, False)} }

c =

| } , , , { | | | | | | | : ) , ( D C B A h c h c X h c h c Err      

c Definition

) , ( max min : ) , (

) , ( | :|

h c Err C c OptTErr

C S CONS h k S S k  



Worst case error

) , ( max min arg : ) , (

) , ( | :|

h c Err C c OptTSets

C S CONS h k S S k  



k-optimal teaching sets achieving the optimal teaching error

20

• COLT2009. Montreal, Canada. 21 June.

Err(c, {A,C}) = 2/4

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Optimal Teaching Error

h A B C D Err(c, h) {A,B} T T F F 0/4 {A,C} T F T F 2/4 {A,D} T F F T 2/4 {B,C} F T T F 2/4 {B,D} F T F T 2/4 {C,D} F F T T 4/4

Millionaire 2.0 (two correct choices) C = {{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}} c = {A, B} OptTSets1(c, C) = { {(A, True)}, {(B, True)} {(C, False)}, {(D, False)} } OptTSets2(c, C) = MinTSets(c, C) = { {(A, True), (B, True)}, {(C, False), (D, False)} }

c =

| } , , , { | | | | | | | : ) , ( D C B A h c h c X h c h c Err      

c Definition

) , ( max min : ) , (

) , ( | :|

h c Err C c OptTErr

C S CONS h k S S k  



Worst case error

) , ( max min arg : ) , (

) , ( | :|

h c Err C c OptTSets

C S CONS h k S S k  



k-optimal teaching sets achieving the optimal teaching error

21

• COLT2009. Montreal, Canada. 21 June.

Err(c, {C,D}) = 4/4

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Optimal Teaching Error

h A B C D Err(c, h) {A,B} T T F F 0/4 {A,C} T F T F 2/4 {A,D} T F F T 2/4 {B,C} F T T F 2/4 {B,D} F T F T 2/4 {C,D} F F T T 4/4

Millionaire 2.0 (two correct choices) C = {{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}} c = {A, B} OptTSets1(c, C) = { {(A, True)}, {(B, True)} {(C, False)}, {(D, False)} } OptTSets2(c, C) = MinTSets(c, C) = { {(A, True), (B, True)}, {(C, False), (D, False)} }

c =

| } , , , { | | | | | | | : ) , ( D C B A h c h c X h c h c Err      

c Definition

) , ( max min : ) , (

) , ( | :|

h c Err C c OptTErr

C S CONS h k S S k  



Worst case error

) , ( max min arg : ) , (

) , ( | :|

h c Err C c OptTSets

C S CONS h k S S k  



k-optimal teaching sets achieving the optimal teaching error

22

• COLT2009. Montreal, Canada. 21 June.

Worst case error = 2/4 if teaching (A, True) (It is optimal)

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Optimal Teaching Error

h A B C D Err(c, h) {A,B} T T F F 0/4 {A,C} T F T F 2/4 {A,D} T F F T 2/4 {B,C} F T T F 2/4 {B,D} F T F T 2/4 {C,D} F F T T 4/4

Millionaire 2.0 (two correct choices) C = {{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}} c = {A, B} OptTSets1(c, C) = { {(A, True)}, {(B, True)} {(C, False)}, {(D, False)} } OptTSets2(c, C) = MinTSets(c, C) = { {(A, True), (B, True)}, {(C, False), (D, False)} }

c =

| } , , , { | | | | | | | : ) , ( D C B A h c h c X h c h c Err      

c Definition

) , ( max min : ) , (

) , ( | :|

h c Err C c OptTErr

C S CONS h k S S k  



Worst case error

) , ( max min arg : ) , (

) , ( | :|

h c Err C c OptTSets

C S CONS h k S S k  



k-optimal teaching sets achieving the optimal teaching error

23

• COLT2009. Montreal, Canada. 21 June.

Worst case error = 4/4 if teaching (A, False) (It is NOT optimal)

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Features of our model (1/2)

– Special teaching strategies are necessary

• A subset of a minimum teaching set is not always optimal

h A B C Err(c, h) {A,B,C} T T T 0/3 {A,C} T F T 1/3 {A} T F F 2/3 {B,C} F T T 1/3 {B} F T F 2/3

C = {{A, B, C}, {A, C}, {A}, {B, C}, {B}} c = {A, B, C} k = 1 MinTSets(c, C) = { {(A, True), (B, True)} } OptTSets1(c, C) = { {(C, True)} }

c = Theorem

), , ( ' , ' ), , ( , , , C C C C c OptTSets S S S c MinTSets S k c

k

         

(Proof)

24

• COLT2009. Montreal, Canada. 21 June.

/ 18

Features of our model (1/2)

– Special teaching strategies are necessary

• A subset of a minimum teaching set is not always optimal

h A B C Err(c, h) {A,B,C} T T T 0/3 {A,C} T F T 1/3 {A} T F F 2/3 {B,C} F T T 1/3 {B} F T F 2/3

C = {{A, B, C}, {A, C}, {A}, {B, C}, {B}} c = {A, B, C} k = 1 MinTSets(c, C) = { {(A, True), (B, True)} } OptTSets1(c, C) = { {(C, True)} }

c = Theorem

), , ( ' , ' ), , ( , , , C C C C c OptTSets S S S c MinTSets S k c

k

         

(Proof)

25

• COLT2009. Montreal, Canada. 21 June.

Minimum set to teach c

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Features of our model (1/2)

– Special teaching strategies are necessary

• A subset of a minimum teaching set is not always optimal

h A B C Err(c, h) {A,B,C} T T T 0/3 {A,C} T F T 1/3 {A} T F F 2/3 {B,C} F T T 1/3 {B} F T F 2/3

C = {{A, B, C}, {A, C}, {A}, {B, C}, {B}} c = {A, B, C} k = 1 MinTSets(c, C) = { {(A, True), (B, True)} } OptTSets1(c, C) = { {(C, True)} }

c = Theorem

), , ( ' , ' ), , ( , , , C C C C c OptTSets S S S c MinTSets S k c

k

         

(Proof)

26

• COLT2009. Montreal, Canada. 21 June.

Worst case error = 2/3

/ 18

Features of our model (1/2)

– Special teaching strategies are necessary

• A subset of a minimum teaching set is not always optimal

h A B C Err(c, h) {A,B,C} T T T 0/3 {A,C} T F T 1/3 {A} T F F 2/3 {B,C} F T T 1/3 {B} F T F 2/3

C = {{A, B, C}, {A, C}, {A}, {B, C}, {B}} c = {A, B, C} k = 1 MinTSets(c, C) = { {(A, True), (B, True)} } OptTSets1(c, C) = { {(C, True)} }

c = Theorem

), , ( ' , ' ), , ( , , , C C C C c OptTSets S S S c MinTSets S k c

k

         

(Proof)

27

• COLT2009. Montreal, Canada. 21 June.

Worst case error = 2/3

/ 18

Features of our model (1/2)

– Special teaching strategies are necessary

• A subset of a minimum teaching set is not always optimal

h A B C Err(c, h) {A,B,C} T T T 0/3 {A,C} T F T 1/3 {A} T F F 2/3 {B,C} F T T 1/3 {B} F T F 2/3

C = {{A, B, C}, {A, C}, {A}, {B, C}, {B}} c = {A, B, C} k = 1 MinTSets(c, C) = { {(A, True), (B, True)} } OptTSets1(c, C) = { {(C, True)} }

c = Theorem

), , ( ' , ' ), , ( , , , C C C C c OptTSets S S S c MinTSets S k c

k

         

(Proof)

28

• COLT2009. Montreal, Canada. 21 June.

Worst case error = 1/3 (optimal)

/ 18

Features of our model (2/2)

– Smart teachers dare to tell a lie

• A k-optimal teaching set can be inconsistent with c

h A B C D E Err(c, h) {A,B,C,D,E} T T T T T 0/5 {B,C,D,E} F T T T T 1/5 {A,B} T T F F F 3/5 {A,C} T F T F F 3/5 {A,D} T F F T F 3/5 {A,E} T F F F T 3/5

C = {{A, B, C, D, E}, {B, C, D, E}, {A, B}, {A, C}, {A, D}, {A, E}} c = {A, B, C, D, E} k = 1 OptTSets1(c,C) = { {(A, False)} } CONS( {(A, False)}, C) = { {B,C,D,E} }

c = Theorem

) , ( ), , ( , , , C S CONS c c OptTSets S k c

k

        C C C

(Proof)

29

• COLT2009. Montreal, Canada. 21 June.

/ 18

Features of our model (2/2)

– Smart teachers dare to tell a lie

• A k-optimal teaching set can be inconsistent with c

h A B C D E Err(c, h) {A,B,C,D,E} T T T T T 0/5 {B,C,D,E} F T T T T 1/5 {A,B} T T F F F 3/5 {A,C} T F T F F 3/5 {A,D} T F F T F 3/5 {A,E} T F F F T 3/5

C = {{A, B, C, D, E}, {B, C, D, E}, {A, B}, {A, C}, {A, D}, {A, E}} c = {A, B, C, D, E} k = 1 OptTSets1(c,C) = { {(A, False)} } CONS( {(A, False)}, C) = { {B,C,D,E} }

c = Theorem

) , ( ), , ( , , , C S CONS c c OptTSets S k c

k

        C C C

(Proof)

30

• COLT2009. Montreal, Canada. 21 June.

Worst case error = 1/5 (optimal) athough (A, False) is a lie

/ 18

Features of our model (2/2)

– Smart teachers dare to tell a lie

• A k-optimal teaching set can be inconsistent with c

h A B C D E Err(c, h) {A,B,C,D,E} T T T T T 0/5 {B,C,D,E} F T T T T 1/5 {A,B} T T F F F 3/5 {A,C} T F T F F 3/5 {A,D} T F F T F 3/5 {A,E} T F F F T 3/5

C = {{A, B, C, D, E}, {B, C, D, E}, {A, B}, {A, C}, {A, D}, {A, E}} c = {A, B, C, D, E} k = 1 OptTSets1(c,C) = { {(A, False)} } CONS( {(A, False)}, C) = { {B,C,D,E} }

c = Theorem

) , ( ), , ( , , , C S CONS c c OptTSets S k c

k

        C C C

(Proof)

31

• COLT2009. Montreal, Canada. 21 June.

Worst case error = 3/5 If teaching the truth (A, True)

/ 18

Optimally Incremental Teachability

– Optimal teaching strategies independent of k

c Definition

c is optimally incrementally teachable w.r.t. C

) , ( } ,..., { )], , ( , 1 [ , ,...,

1 ) , ( 1

C C

C

c OptTSets z z c TD k z z

k k c TD

    def 

Millionaire 2.0 (two correct choices) C = {{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}} c = {A, B} OptTSets2(c, C) = MinTSets(c, C) = { {(A, True), (B, True)}, {(C, False), (D, False)} } OptTSets1(c, C) = { {(A, True)}, {(B, True)} {(C, False)}, {(D, False)} } c of Millionaire 2.0 is opt.

• inc. teachable w.r.t. C

Optimal order c 〈 (A, True), (B, True) 〉 Fact 1-opt. 2-opt.

32

• COLT2009. Montreal, Canada. 21 June.

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Concept classes Mn

+, Mn’, Mn

• Mn: Monomials

– A concept is a set of x∈{0,1}n satisfying a monomial

• Mn’: Monomials w/o the empty concept

– Mn’ := Mn-{φ}

• Mn

+: Monotone monomials

– Monomials consisting of only positive literals

Ex.) Concepts for monomials on 3 variables : {100, 101, 110, 111} ←Monotone monomial : {010} : φ (Empty concept)

1 1v

v

1

v

3 2 1 v

v v

33

• COLT2009. Montreal, Canada. 21 June.

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• Opt. inc. teachability of Mn

+

2 1 3

: v v M c





Ex.) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True) (Proof sketch) The condition for c is satisfied by 〈z1, z2,…〉 ℓ: # of variables of the target monomial

Minimum teaching set

Optimal order

h∈M3

+

Err(c, h) (All true) 6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23

Mn

+ is opt. inc. teachable

Theorem

     

  

) (i ) (i ) , 1 ( ) , 01 1 ( :

1

 

 

True False z

n i n i i

c →

34

• COLT2009. Montreal, Canada. 21 June.

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• Opt. inc. teachability of Mn

+

2 1 3

: v v M c





Ex.) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True) (Proof sketch) The condition for c is satisfied by 〈z1, z2,…〉 ℓ: # of variables of the target monomial

Minimum teaching set Exist

Optimal order

h∈M3

+

Err(c, h) (All true) 6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23 by z1 by z1 by z1 by z1

Mn

+ is opt. inc. teachable

Theorem

     

  

) (i ) (i ) , 1 ( ) , 01 1 ( :

1

 

 

True False z

n i n i i

c →

35

• COLT2009. Montreal, Canada. 21 June.

/ 18

• Opt. inc. teachability of Mn

+

2 1 3

: v v M c





Ex.) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True) (Proof sketch) The condition for c is satisfied by 〈z1, z2,…〉 ℓ: # of variables of the target monomial

Minimum teaching set Exist Exist

Optimal order

h∈M3

+

Err(c, h) (All true) 6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23 by z1 by z1 by z1 by z1 by z2 by z2

Mn

+ is opt. inc. teachable

Theorem

     

  

) (i ) (i ) , 1 ( ) , 01 1 ( :

1

 

 

True False z

n i n i i

c →

36

• COLT2009. Montreal, Canada. 21 June.

/ 18

• Opt. inc. teachability of Mn

+

2 1 3

: v v M c





Ex.) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True) (Proof sketch) The condition for c is satisfied by 〈z1, z2,…〉 ℓ: # of variables of the target monomial

Minimum teaching set Exist Exist Not exist

Optimal order

h∈M3

+

Err(c, h) (All true) 6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23 by z1 by z1 by z1 by z1 by z2 by z2 by z3

Mn

+ is opt. inc. teachable

Theorem

     

  

) (i ) (i ) , 1 ( ) , 01 1 ( :

1

 

 

True False z

n i n i i

c →

37

• COLT2009. Montreal, Canada. 21 June.

/ 18

h∈M3’ Err(c, h) v1 6/23 v2 6/23 v3 4/23 v1v2 4/23 v1v2 4/23 v1v2 4/23 v2v3 4/23 … …

• Opt. inc. teachability of Mn’

2 1 3

: ' v v M c

Ex.) z0 = (1 1 1, True) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True)

h∈M3’ Err(c, h)

(All true)

6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23

zopt = (0 0 0, False)

1-optimal teaching set

Not subset

Mn

’ is not opt. inc. teachable

Theorem

(Proof sketch) zopt := (0n, False) is a special teaching strategy when k=1

c → Not all negated

38

• COLT2009. Montreal, Canada. 21 June.

4/23 Minimum teaching set

/ 18

h∈M3’ Err(c, h) v1 6/23 v2 6/23 v3 4/23 v1v2 4/23 v1v2 4/23 v1v2 4/23 v2v3 4/23 … …

• Opt. inc. teachability of Mn’

2 1 3

: ' v v M c

Ex.) z0 = (1 1 1, True) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True)

h∈M3’ Err(c, h)

(All true)

6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23 Not negated

zopt = (0 0 0, False)

1-optimal teaching set

Not subset

Mn

’ is not opt. inc. teachable

Theorem

(Proof sketch) zopt := (0n, False) is a special teaching strategy when k=1

c →

39

• COLT2009. Montreal, Canada. 21 June.

4/23 For Mn

+

Minimum teaching set

/ 18

h∈M3’ Err(c, h) v1 6/23 v2 6/23 v3 4/23 v1v2 4/23 v1v2 4/23 v1v2 4/23 v2v3 4/23 … …

• Opt. inc. teachability of Mn’

2 1 3

: ' v v M c

Ex.) z0 = (1 1 1, True) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True)

h∈M3’ Err(c, h)

(All true)

6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23 Not negated

zopt = (0 0 0, False)

1-optimal teaching set

Not subset

Mn

’ is not opt. inc. teachable

Theorem

(Proof sketch) zopt := (0n, False) is a special teaching strategy when k=1

c →

40

• COLT2009. Montreal, Canada. 21 June.

4/23 For Mn

+

Minimum teaching set Can’t exclude (All true)

/ 18

h∈M3’ Err(c, h) v1 6/23 v2 6/23 v3 4/23 v1v2 4/23 v1v2 4/23 v1v2 4/23 v2v3 4/23 … …

• Opt. inc. teachability of Mn’

2 1 3

: ' v v M c

Ex.) z0 = (1 1 1, True) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True)

h∈M3’ Err(c, h)

(All true)

6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23 Not negated

zopt = (0 0 0, False)

1-optimal teaching set

Not subset

Mn

’ is not opt. inc. teachable

Theorem

(Proof sketch) zopt := (0n, False) is a special teaching strategy when k=1

c →

41

• COLT2009. Montreal, Canada. 21 June.

4/23 For Mn

+

Minimum teaching set Can’t exclude (All true) Can’t exclude v2

/ 18

h∈M3’ Err(c, h) v1 6/23 v2 6/23 v3 4/23 v1v2 4/23 v1v2 4/23 v1v2 4/23 v2v3 4/23 … …

• Opt. inc. teachability of Mn’

2 1 3

: ' v v M c

Ex.) z0 = (1 1 1, True) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True)

h∈M3’ Err(c, h)

(All true)

6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23 Not negated

zopt = (0 0 0, False)

1-optimal teaching set

Not subset

Mn

’ is not opt. inc. teachable

Theorem

(Proof sketch) zopt := (0n, False) is a special teaching strategy when k=1

c →

42

• COLT2009. Montreal, Canada. 21 June.

4/23 For Mn

+

Minimum teaching set Can’t exclude (All true) Can’t exclude v2 Can’t exclude v1

/ 18

h∈M3’ Err(c, h) v1 6/23 v2 6/23 v3 4/23 v1v2 4/23 v1v2 4/23 v1v2 4/23 v2v3 4/23 … …

• Opt. inc. teachability of Mn’

2 1 3

: ' v v M c

Ex.) z0 = (1 1 1, True) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True)

h∈M3’ Err(c, h)

(All true)

6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23 Not negated

zopt = (0 0 0, False)

1-optimal teaching set

Not subset

Mn

’ is not opt. inc. teachable

Theorem

(Proof sketch) zopt := (0n, False) is a special teaching strategy when k=1

c →

43

• COLT2009. Montreal, Canada. 21 June.

4/23 For Mn

+

Minimum teaching set Can’t exclude (All true) Can’t exclude v2 Can’t exclude v1 Can’t exclude (All true)

/ 18

h∈M3’ Err(c, h) v1 6/23 v2 6/23 v3 4/23 v1v2 4/23 v1v2 4/23 v1v2 4/23 v2v3 4/23 … …

• Opt. inc. teachability of Mn’

2 1 3

: ' v v M c

Ex.) z0 = (1 1 1, True) z1 = (0 1 1, False) z2 = (1 0 1, False) z3 = (1 1 0, True)

h∈M3’ Err(c, h)

(All true)

6/23 v1 2/23 v2 2/23 v3 4/23 v1v2 0/23 v2v3 2/23 v1v3 2/23 v1v2v3 1/23 Not negated

zopt = (0 0 0, False)

1-optimal teaching set

Not subset

Mn

’ is not opt. inc. teachable

Theorem

(Proof sketch) zopt := (0n, False) is a special teaching strategy when k=1

c →

44

• COLT2009. Montreal, Canada. 21 June.

4/23 For Mn

+

Minimum teaching set Can’t exclude (All true) Can’t exclude v2 Can’t exclude v1

Mn is not opt. inc. teachable

Theorem Can’t exclude (All true)

/ 18

Interesting result

– Teachers must tell a lie to optimally teach φ in Mn

(Proof sketch when k>n) TD(c’, Mn) = n+1 S∈MinTSets(c’, Mn) is k-optimal for c However, S is inconsistent with c

Theorem

) M CONS(S, ), , ( ], 1 2 , 4 [

n 1

     



 

n k n

M OptTSets S k

Theorem ([Goldman and Kearns 1995])

1} n 2, min{ ) , (    

n

M c TD

h∈Mn Err(c, h)

φ

0/2n … … v1v2…vn 1/2n … …

n n

v v v M c ... : '

2 1



c = c' =

45

• COLT2009. Montreal, Canada. 21 June.

/ 18

Interesting result

– Teachers must tell a lie to optimally teach φ in Mn

(Proof sketch when k>n) TD(c’, Mn) = n+1 S∈MinTSets(c’, Mn) is k-optimal for c However, S is inconsistent with c

Theorem

) M CONS(S, ), , ( ], 1 2 , 4 [

n 1

     



 

n k n

M OptTSets S k

Theorem ([Goldman and Kearns 1995])

1} n 2, min{ ) , (    

n

M c TD

h∈Mn Err(c, h)

φ

0/2n … … v1v2…vn 1/2n … …

n n

v v v M c ... : '

2 1



c = c' =

Theorem

) , ( ), , ( , , , C S CONS c c OptTSets S k c

k

        C C C

46

• COLT2009. Montreal, Canada. 21 June.

Natural example

/ 18

Interesting result

– Teachers must tell a lie to optimally teach φ in Mn

(Proof sketch when k>n) TD(c’, Mn) = n+1 S∈MinTSets(c’, Mn) is k-optimal for c However, S is inconsistent with c

Theorem

) M CONS(S, ), , ( ], 1 2 , 4 [

n 1

     



 

n k n

M OptTSets S k

Theorem ([Goldman and Kearns 1995])

1} n 2, min{ ) , (    

n

M c TD

h∈Mn Err(c, h)

φ

0/2n … … v1v2…vn 1/2n … …

n n

v v v M c ... : '

2 1



c = c' =

Theorem

) , ( ), , ( , , , C S CONS c c OptTSets S k c

k

        C C C

47

• COLT2009. Montreal, Canada. 21 June.

Natural example

/ 18

Summary

Mn

+

Mn’ Mn Teaching Dim., TD(C)

[Goldman and Kearns 1991]

n n+1 2n Teachability

[Shinohara and Miyano 1991]

True True False

• Opt. Teaching Error

OptTEk(C)

• Opt. Inc. Teachability

True False False

n k n

2 1 2 



n k n

2 1 2

1   

              

   

) 2 ( 2 1 ) 2 ( 2 2 ) 2 ( 2 1 2

1 1 n n n k n n k n

k n n k k Our results

Different boundary Quite small

48

• COLT2009. Montreal, Canada. 21 June.

/ 18

Thank you for your attention!

49

• COLT2009. Montreal, Canada. 21 June.