Analogical Transfer: a Form of Similarity-Based Inference? Fadi - PowerPoint PPT Presentation

Analogical Transfer: a Form of Similarity-Based Inference? Fadi Badra LIMICS, Paris, France DIG seminar, March 5, 2018 1

Agenda Analogical Transfer : definition, examples Analogy and the Qualitative Measurement of Similarity Analogical Transfer : a Similarity-Based Inference? Some more ideas I would like to share 2

Analogy [Gust et al., 08] Analogy is a cognitive process in which a structural pattern identified in a source conceptualization is transferred to a target domain (possibly the same) in order to learn a target conceptualization. 3 main steps : Retrieval, Mapping, Transfer [Holyoak and Thagard, 97] Analogy is guided by constraints on similarity, structure ( isomorphism between some elements of source and target), purpose (the reasoner’s goal) + consistency (solution viable in the reasoner’s model of the world) + simplicity [Cornuéjols, 96] 3

Analogical Transfer Assumption that if two situations are alike in some respect, they may be alike in others Logical characterization [Davis and Russell, 87] P p s q P p t q Q p s q (AJ) Q p t q What sufficient condition might justify “Analogical jump” (AJ)? weaker than generalization rule @ x , P p x q ñ Q p x q but stronger than single instance induction depends on the amount of similarity between sources and targets functional dependencies are good candidates 4

Let us take a few examples. . . 5

Example 1 : John’s Car P p s q P p t q Q p s q Q p t q s = Bob’s car t = John’s car P =“being a 1982 Mustang GLX V6 hatchbacks” Q =“having a price of 3 500 $” One can make the hypothesis that the price of t is « 3 500 $. 6

Example 2 : Tversky’s Faces R 1 p a , a 1 q R 2 p a , a 1 q R 1 p d , d 1 q R 2 p d , d 1 q R 1 =” differs in profile shape from round to sharp ” R 2 =” has the same eyebrow as ” Assuming that d 1 has a sharp profile (i.e., R 1 p d , d 1 q holds), one can make the hypothesis that d 1 has the same eyebrow as d (curved). 7

Example 3 : Analogical Classification For a “ p a 1 , . . . , a n q P B n , a : b :: c : d cls p a q : cls p b q :: cls p c q : cls p d q rewrites [Bounhas et al., 17] P k p a , b q P k p c , d q Q i p a , b q Q i p c , d q with P k p a , b q “ p a ´ b “ k q , with k P t´ 1 , 0 , 1 u n Q 1 p a , b q “ p cls p a q “ cls p b qq Q 2 p a , b q “ p cls p a q “ 0 ^ cls p b q “ 1 q Q 3 p a , b q “ p cls p a q “ 1 ^ cls p b q “ 0 q 8

Example 4 : the k -Nearest Neighbors x P N p x q cls p x q “ c x 0 P N p x q cls p x 0 q “ c N p x q = neighborhood of x cls p x q = class of x 9

Research Questions How to formalize Analogical Transfer? What should be transferred, and when? Transfer may require adaptation (not only copy) Adaptation relies on comparison, an assessement of differences between source and target Adaptation knowledge include qualitative proportionalities (“ All things equal, the more an apartment has rooms, the more expensive it is ”) or transformation rules (“ chocolate can be replaced by cocoa, but then sugar should be added ”) What is the role of similarity in Analogical Transfer? Different ways to measure similarity : geometric, feature-based, alignment-based, transformational [Goldstone, 13] + visual 10

A Qualitative Measurement of Similarity 11

Qualitative Similarity Relations U : a finite, non-empty set, the Universe We work on pairs p a , b q (or simply ab ) of the square product U ˆ U Ordinal similarity relations [Yao, 00] Strict Inequality “ a and b are more similar than c and d ” ab ą cd Equality ab „ cd ô � p ab ą cd q ^ � p cd ą ab q “ a and b are as similar as c and d ” Non-Strict Inequality ab ľ cd iff ab ą cd or ab „ cd 12

Scaling A variation υ is a scale on U ˆ U Scale = a map that preserves ľ , i.e. , ab ľ cd ô υ p ab q ľ υ p cd q Variations are used to discretize U ˆ U Id : xy ÞÑ xy 1 U : xy ÞÑ 1 # 1 if x “ y “ : xy ÞÑ 0 if x ‰ y # 1 if x “ y “ 1 1 : xy ÞÑ 0 otherwise 13

Scaling A variation υ is a scale on U ˆ U d p : xy ÞÑ } y ´ x } p X , Y : sets AP : XY ÞÑ p X z Y , Y z X q ∆ : XY ÞÑ | X z Y | ` | Y z X | 14

Scaling υ ϕ o p ab q “ o p ϕ p a q , ϕ p b qq variation between values taken by ϕ first, apply a feature ϕ : U Ñ X (e.g., age, gender) then, apply a scale o : X ˆ X Ý Ñ V on the values of ϕ # 1 if eyebrow p a q “ eyebrow p b q υ eyebrow p ab q “ “ 0 if eyebrow p a q ‰ eyebrow p b q # 1 if both P p a q and P p b q hold 1 P p ab q “ υ P 1 p ab q “ 0 otherwise υ age d 1 p ab q “ | age p b q ´ age p a q| 15

Scaling υ ϕ o p ab q “ o p ϕ p a q , ϕ p b qq variation between values taken by ϕ Difference between two vectors v p a q “ p ϕ 1 p a q , ϕ 2 p a q , . . . , ϕ n p a qq P R n for a P U Ý Ñ ab “ υ v ´ p ab q “ v p b q ´ v p a q Ranking difference (from a given object a) ϕ p p q “ size pÓ ap q with Ó ap “ t ab P t a u ˆ U | ap ľ ab uq υ a Ñ d 1 p bc q “ | size pÓ ac q ´ size pÓ ab q| Nearest neighbor (b=a) υ a Ñ d 1 p ac q “ | size pÓ ac q ´ 1 | “ k iff c is the k-NN of a 16

Relation to Analogy? 17

Intuition Analogical Equalities υ i p ab q „ υ i p a 1 b 1 q being in a same contour line Analogical Inequalities υ i p ab q ĺ υ i p cd q defines contour scales Analogical Dissimilarity ad p ab , cd q “ ad p α, β q distance between contour lines 18

Analogical Equalities υ p ab q „ υ p a 1 b 1 q Ñ “Contour lines” = sets of pairs equivalent for υ 19

Analogical Equalities υ p ab q „ υ p a 1 b 1 q Ñ “Contour lines” = sets of pairs equivalent for υ p aa 1 q “ υ eyebrow p bb 1 q “ 1 υ eyebrow “ “ aa ’ and bb ’ both share a commonality : having the same eyebrow. υ profile p aa 1 q “ υ profile p bb 1 q “ p round , sharp q Id Id aa ’ and bb ’ both share a difference : from round to sharp profile. 20

Analogical Equalities υ p ab q „ υ p a 1 b 1 q Ñ “Contour lines” = sets of pairs equivalent for υ X , Y : sets AP p XY q “ p X z Y , Y z X q AP p XY q “ AP p ZT q ô def X : Y :: Z : T X is to Y what Z is to T. 21

Analogical Inequalities υ p ab q ĺ υ p cd q Ñ “Contour scales” = ordering on values of υ 22

Analogical Inequalities υ p ab q ĺ υ p cd q Ñ “Contour scales” = ordering on values of υ 1 ľ 0 ô p υ eyebrow : 1 q ľ p υ eyebrow : 0 q “ “ Two faces having the same eyebrow are more similar than two faces having different eyebrows. n ĺ m ô p υ age d 1 : n q ľ p υ age d 1 : m q The lower the age difference, the more similar. ℓ Ď k ô p AP : ℓ q ľ p AP : k q All things equal, the less properties are lost or gained when going from a set X to a set Y, the more similar X and Y are. 23

Analogical Dissimilarity ad p ab , cd q Ñ measures the distance between two contour lines 24

Analogical Dissimilarity ad p ab , cd q Ñ measures the distance between two contour lines Dissimilarity between Nearest Neighbors υ a Ñ d 1 p bc q “ | size pÓ ac q ´ size pÓ ab q| υ a Ñ d 1 ad p ab , cd q “ υ p ab , cd q for a , b , c , d P U d 1 i.e. , for n , m P N ad p n , m q “ | n ´ m | AD is a ranking difference. 25

Analogical Dissimilarity ad p ab , cd q Ñ measures the distance between two contour lines Dissimilarity between Vectors v p a q “ p ϕ 1 p a q , ϕ 2 p a q , . . . , ϕ n p a qq P R n for a P U Ý Ñ ab “ υ v ´ p ab q “ v p b q ´ v p a q υ v d p p ab , cd q “ }Ý cd ´ Ý Ñ Ñ ad p ab , cd q “ υ ab } p ´ ad pÝ Ñ u , Ý Ñ v q “ }Ý Ñ v ´ Ý Ñ i.e. , u } p Consistent with AD defined in [Miclet et al., 2008]. 26

Analogical Dissimilarity ad p ab , cd q Ñ measures the distance between two contour lines Dissimilarity between Analogical Proportions υ ϕ AP with AP p XY q “ p X z Y , Y z X q υ ϕ ad p ab , cd q “ υ ∆ p ab , cd q AP ( ∆ : symmetric difference) “ | U z W | ` | W z U | ` | V z Q | ` | Q z V | with U “ X z Y , V “ Y z X , W “ Z z T , and Q “ T z Z where X “ ϕ p a q , Y “ ϕ p b q , Z “ ϕ p c q , and T “ ϕ p d q Consistent with AD defined in [Miclet et al., 2008]. 27

Analogical Transfer 28

Intuition ab Co-variation υ i ñ υ j inclusion of contour lines Analogical Transfer ab ad p ab , cd q ĺ k ñ υ j , υ i cd ñ υ j υ i similarity-based reasoning on contour lines 29

Co-Variations ab ñ υ j Ñ inclusion between two contour lines υ i ( υ i co-varies with υ j in ab) 30

Co-Variations ab ñ υ j Ñ inclusion between two contour lines υ i Bayesian semantics : r ab s υ i Ď r ab s υ j inclusion is verified on the whole equivalence class of ab 31

Co-Variations ab ñ υ j Ñ inclusion between two contour lines υ i Semi-Bayesian semantics : Ş k Pt i uY D r ab s υ k Ď r ab s υ j inclusion is verified around ab , but as long as we stay in the same contour line for some υ k ’s 32

Co-Variations ab ñ υ j Ñ inclusion between two contour lines υ i Ceteris paribus semantics : Ş k ‰ j r ab s υ k Ď r ab s υ j inclusion is verified ceteris paribus around ab , as long as we stay in the same contour line for all other υ k ’s 33