Complexity of Teaching by a Restricted Number of Examples Hayato Kobayashi and Ayumi Shinohara Tohoku University, Japan COLT2009. Montreal, Canada. 21 June. 1 / 18
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] COLT2009. Montreal, Canada. 21 June. 2 / 18
Illustrative problem: Censored Phone-a-friend lifeline “Who Wants to Be a Millionaire?” • Millionaire (=“Who Wants to Be a Millionaire?”) – Challenge multiple-choice questions – Win a cash award depending on the number of correct answers Censored – Get help from the three lifelines during the game • Lifelines – Phone-a-friend • Will give you advice from friends – 50:50 Censored • Removes two incorrect answers – Ask the Audience • Lets you see the answers of audience COLT2009. Montreal, Canada. 21 June. 3 / 18
Classical model (1/3) • Millionaire (one correct choice) When was the COLT conference first held? B: 1988 A: 1983 C: 1992 D: 1997 Concept class (available answers) C = {{A}, {B}, {C}, {D}} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 4 / 18
Classical model (1/3) • Millionaire (one correct choice) When was the COLT conference first held? B: 1988 B: 1988 A: 1983 C: 1992 D: 1997 Target concept (correct answer) Concept class (available answers) c = {B} C = {{A}, {B}, {C}, {D}} S = {(A, False), (B, True), (C, False), (D, False)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 5 / 18
Classical model (1/3) • Millionaire (one correct choice) When was the COLT conference first held? B: 1988 B: 1988 A: 1983 C: 1992 D: 1997 Target concept (correct answer) Concept class (available answers) c = {B} C = {{A}, {B}, {C}, {D}} Teaching set S = {(A, False), (B, True), (C, False), (D, False)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 6 / 18
Classical model (1/3) • Millionaire (one correct choice) When was the COLT conference first held? B: 1988 B: 1988 A: 1983 Teaching Dimension C: 1992 D: 1997 TD(c, C ) := |Minimum teaching set| In this case, TD(c, C )=1 Target concept (correct answer) Concept class (available answers) c = {B} C = {{A}, {B}, {C}, {D}} Teaching set S = {(A, False), (B, True), (C, False), (D, False)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 7 / 18
Classical model (2/3) • Millionaire 2.0 (two correct choices) Which are the two cities where the Olympic games were held in Canada? B: Calgary A: Montreal C: Ottawa D: Vancouver Concept class (available answers) C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 8 / 18
Classical model (2/3) • Millionaire 2.0 (two correct choices) Which are the two cities where the Olympic games were held in Canada? A: Montreal B: Calgary B: Calgary A: Montreal C: Ottawa D: Vancouver Target concept (correct answer) Concept class (available answers) c = {A, B} C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} S = {(A, True), (B, True)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 9 / 18
Classical model (2/3) • Millionaire 2.0 (two correct choices) Which are the two cities where the Olympic games were held in Canada? A: Montreal B: Calgary B: Calgary A: Montreal Teaching Dimension C: Ottawa D: Vancouver TD(c, C ) := |Minimum teaching set| In this case, TD(c, C )=2 Target concept (correct answer) Concept class (available answers) c = {A, B} C = {{A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}} Minimum teaching set S = {(A, True), (B, True)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 10 / 18
Classical model (3/3) • Generalized Millionaire (unknown # of correct choices) Which choices are correct specialties of Canada? B: Maple butter A: Maple tea C: Maple dressing D: Maple mustard Concept class (available answers) Target concept (correct answer) C = 2 {A,B,C,D} c = {A, B, C, D} S = {(A, True), (B, True), (C, True), (D, True)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 11 / 18
Classical model (3/3) • Generalized Millionaire (unknown # of correct choices) Which choices are correct specialties of Canada? B: Maple butter B: Maple butter A: Maple tea A: Maple tea C: Maple dressing C: Maple dressing D: Maple mustard D: Maple mustard Concept class (available answers) Target concept (correct answer) C = 2 {A,B,C,D} c = {A, B, C, D} S = {(A, True), (B, True), (C, True), (D, True)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 12 / 18
Classical model (3/3) • Generalized Millionaire (unknown # of correct choices) Which choices are correct specialties of Canada? B: Maple butter B: Maple butter A: Maple tea A: Maple tea Teaching Dimension TD(c, C ) := |Minimum teaching set| C: Maple dressing C: Maple dressing D: Maple mustard D: Maple mustard In this case, TD(c, C )=4 Concept class (available answers) Target concept (correct answer) C = 2 {A,B,C,D} c = {A, B, C, D} Minimum teaching set S = {(A, True), (B, True), (C, True), (D, True)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 13 / 18
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 optimally incremental teachabilities for concept classes of + : Monotone monomials – M n – M n ’ : Monomials without the empty concept – M n : Monomials COLT2009. Montreal, Canada. 21 June. 14 / 18
Our model • Restriction: # of examples ≦ k – Phone-a-friend lifeline: 30 sec. – This presentation: 25 min. – Lectures in our univ.: 90 min. Millionaire 2.0 (two correct choices) Concept class (available answers) Target concept (correct answer) C = {{A,B}, {A,C}, {A,D}, c = {A, B} {B,C}, {B,D}, {C,D}} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 15 / 18
Our model • Restriction: # of examples ≦ k – Phone-a-friend lifeline: 30 sec. – This presentation: 25 min. – Lectures in our univ.: 90 min. Millionaire 2.0 (two correct choices) Concept class (available answers) “The question is Target concept (correct answer) C = {{A,B}, {A,C}, {A,D}, which two are …“ c = {A, B} {B,C}, {B,D}, {C,D}} (He used 29 sec.) Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 16 / 18
Our model • Restriction: # of examples ≦ k – Phone-a-friend lifeline: 30 sec. – This presentation: 25 min. – Lectures in our univ.: 90 min. Millionaire 2.0 (two correct choices) Concept class (available answers) “The question is Target concept (correct answer) C = {{A,B}, {A,C}, {A,D}, which two are …“ c = {A, B} “ A is True . B is“ {B,C}, {B,D}, {C,D}} (He used 29 sec.) S = {(A, True)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 17 / 18
Our model • Restriction: # of examples ≦ k – Phone-a-friend lifeline: 30 sec. – This presentation: 25 min. – Lectures in our univ.: 90 min. (Complexity) Optimal Teaching Error Millionaire 2.0 (two correct choices) Concept class (available answers) “The question is Target concept (correct answer) C = {{A,B}, {A,C}, {A,D}, which two are …“ c = {A, B} “ A is True . B is“ {B,C}, {B,D}, {C,D}} (He used 29 sec.) S = {(A, True)} Teaching Learner (challenger) Teacher (friend) COLT2009. Montreal, Canada. 21 June. 18 / 18
Optimal Teaching Error Definition Worst case error ( , ) : min max ( , ) OptTErr c C Err c h c k :| | S S k ( , ) h CONS S C k-optimal teaching sets achieving the optimal teaching error ( , ) : arg min max ( , ) OptTSets c C Err c h k ( , ) h CONS S C :| | S S k h A B C D Err(c, h) Millionaire 2.0 (two correct choices) c = {A,B} T T F F 0/4 C = {{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}} c = {A, B} {A,C} T F T F 2/4 {A,D} T F F T 2/4 OptTSets 1 (c, C ) = { {(A, True)}, {(B, True)} {B,C} F T T F 2/4 {(C, False)}, {(D, False)} } {B,D} F T F T 2/4 OptTSets 2 (c, C ) = MinTSets(c, C ) {C,D} F F T T 4/4 = { {(A, True), (B, True)}, {(C, False), (D, False)} } | | | | c h c h c h ( , ) : Err c h COLT2009. Montreal, Canada. 21 June. 19 / 18 | | | { , , , } | X A B C D
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