Modeling Preferences with Formal Concept Analysis

Transcription

1 Modeling Preferences with Formal Concept Analysis Sergei Obiedkov Higher School of Economics, Moscow, Russia

2 John likes strawberries more than apples. John likes raspberries more than pears. Can we generalize? John likes red berries more than tree fruit.

3 John likes strawberries more than apples. John likes raspberries more than pears. Can we generalize? John likes red berries more than tree fruit. How do we step from preferences over objects to preferences over concepts? What is the right level of generalization (berries vs. red berries)? What exactly do we mean by likes more than?

4 Two ways to specify concepts: intensionally, by describing what it takes to be an instance of the concept;

5 Two ways to specify concepts: intensionally, by describing what it takes to be an instance of the concept; extensionally, by enumerating instances of the concept.

6 Two ways to specify concepts: intensionally, by describing what it takes to be an instance of the concept; extensionally, by enumerating instances of the concept. Cf. a logical formula and the set of its models.

7 Two ways to specify concepts: intensionally, by describing what it takes to be an instance of the concept; extensionally, by enumerating instances of the concept. Cf. a logical formula and the set of its models. Thus, preferences over concepts are preferences over descriptions are preferences over sets of objects.

8 The approach of preference logics: Start with a preference relation over interpretations or possible worlds, in modal logics. Derive a preference relation over sets of interpretations in one of several reasonable ways. Lift the derived relation to the level of propositions by associating each proposition with the set of its models.

9 The approach of preference logics: Start with a preference relation over interpretations or possible worlds, in modal logics. Derive a preference relation over sets of interpretations in one of several reasonable ways. Lift the derived relation to the level of propositions by associating each proposition with the set of its models. We follow this approach, but: Interpretations are objects. Atomic propositions are attributes.

10 The approach of preference logics: Start with a preference relation over interpretations or possible worlds, in modal logics. Derive a preference relation over sets of interpretations in one of several reasonable ways. Lift the derived relation to the level of propositions by associating each proposition with the set of its models. We follow this approach, but: Interpretations are objects. Atomic propositions are attributes. We consider only conjunctive propositions and treat them as sets of attributes. In so doing, we obtain preferences over conjunctive concepts.

11 Formal Concept Analysis Formal context K = (G, M, I ) a set of objects G a set of attributes M objects are described with attributes: the binary relation I G M

12 Formal Concept Analysis Formal context K = (G, M, I ) a set of objects G a set of attributes M objects are described with attributes: the binary relation I G M Derivation operators For A G and B M: A I = {m M g A : gim} B I = {g G m B : gim}

13 Formal Concept Analysis Formal context K = (G, M, I ) a set of objects G a set of attributes M objects are described with attributes: the binary relation I G M Derivation operators For A G and B M: A = {m M g A : gim} B = {g G m B : gim}

14 Formal Concept Analysis Derivation operators For A G and B M: A = {m M g A : gim} B = {g G m B : gim} Formal concept (A, B) A G A = B B M B = A A is the concept extent and B is the concept intent. (A, B) (C, D) A C ( D B) The concept set of the context K forms a lattice B(K).

15 Canteens in Dresden Bergstraße Reichenbachstraße Klinikum Siedepunkt

16 Canteens in Dresden Bergstraße Reichenbachstraße Klinikum Siedepunkt Siedepunkt Preferences: Reichenbachstraße Klinikum Bergstraße

17 Preference context P = (G, M, I, ) (G, M, I ) is a formal context. preference relation is a preorder on G.

18 Preference context P = (G, M, I, ) (G, M, I ) is a formal context. preference relation is a preorder on G. P as a special case of a relational context family: (G, M, I ) with derivation operators ( ) ; (G, G, ) with derivation operators ( ) and ( ).

19 Menus (G, M, I ) Preferences (G, G, ) B R K S {B, K} = {, } {, } = {R} B R K S B R K S {B, K} = {S} {B, K} =

20 Implications B R K S Implication A B holds in the context (G, M, I ) if A B. The Duquenne-Guigues basis of attribute implications:,,, A set of implications a Horn formula

21 We define several kinds of preferences and provide context-based semantics for them. P = π: When does a preference π hold in a preference context P?

22 We define several kinds of preferences and provide context-based semantics for them. P = π: When does a preference π hold in a preference context P? Based on this: P = Π Π is sound for P π Π(P = π) Π = π π is a semantic consequence of Π if, for all P, P = Π = P = π. (Π is a set of preferences) Π is complete for P if, for all π, P = π = Π = π.

23 Universal preferences Universal preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y)

24 Universal preferences Universal preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y) or Y X.

25 Universal preferences Universal preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y) or Y X. might not be reflexive: {x, y} {x, y} if x y; irreflexive: {x} {x}; transitive: X and Y hold for all X and Y ;

26 Universal preferences Universal preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y) or Y X. might not be reflexive: {x, y} {x, y} if x y; irreflexive: {x} {x}; transitive: X and Y hold for all X and Y ; but can be easily transformed into a strict partial order: X Y X Y and Y X.

27 Universal preferences Universal preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y) or Y X. If Y is preferred to X, then the subsets of Y are preferred to the subsets of X. X and Y are maximal with respect to Y X if an only if (X, Y ) is a formal concept of (G, G, ).

28 Universal preferences Universal preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y) or Y X. If Y is preferred to X, then the subsets of Y are preferred to the subsets of X. X and Y are maximal with respect to Y X if an only if (X, Y ) is a formal concept of (G, G, ). The concepts of (G, G, ) provide a complete representation of universal preferences over object sets.

29 Universal preferences Lifting preferences to propositions/attribute sets In preference logics: φ is preferred to ψ models of φ are preferred to models of ψ

30 Universal preferences Lifting preferences to propositions/attribute sets In preference logics: φ is preferred to ψ models of φ are preferred to models of ψ For A M and B M: A is preferred to B A is preferred to B

31 Universal preferences Universal preferences von Wright s version Universal preferences over object sets A set Y G is preferred to a set X G if x X y Y (x y). Universal preferences over attribute sets A set B M is preferred to a set A M if P = A B x A y B (x y)

32 Universal preferences Universal preferences von Wright s version Universal preferences over object sets A set Y G is preferred to a set X G if x X y Y (x y). Universal preferences over attribute sets A set B M is preferred to a set A M if P = A B x A y B (x y) or B A.

33 Universal preferences Simple properties If B =, then P = {A B, B A} for any A M. If B = G then P = A B A are the least preferable objects in G; P = B A A are the most preferable objects in G.

34 Universal preferences Canteens in Dresden Bergstraße Reichenbachstraße Klinikum Siedepunkt,, Siedepunkt Reichenbachstraße Klinikum Bergstraße

35 Universal preferences Inference A system of one rule X Y X U Y V is sound and complete with respect to universal preferences.

36 Universal preferences Universal preferences as implications Universal translation of P K P = (G G, (M {1, 2}) { }, I ) (g 1, g 2 )I m 1 g 1 Im (g 1, g 2 )I m 2 g 2 Im (g 1, g 2 )I g 1 g 2 The derivation operators of K P are denoted by ( ).

37 Universal preferences Canteens in Dresden Bergstraße Reichenbachstraße Klinikum Siedepunkt Siedepunkt Reichenbachstraße Klinikum Bergstraße

38 Universal preferences Universal translation B, B B, R B, K B, S R, B R, R R, K R, S K, B K, R K, K K, S S, B S, R S, K S, S

39 Universal preferences Universal preferences as implications Translation of a universal preference A B T (A B) is the implication (A {1}) (B {2}) { } of the formal context K P. Example,, 1, 1, 1 2 1, 2

40 Universal preferences Universal preferences as implications Translation of a universal preference A B T (A B) is the implication (A {1}) (B {2}) { } of the formal context K P. A universal preference A B is valid in a preference context P = (G, M, I, ) if and only if its translation is valid in K P : P = A B K P = T (A B).

41 Universal preferences Basis For a preference context P {A B (A {1}) (B {2}) is minimal w.r.t. K P = (A {1}) (B {2}) { }} is the minimal (in the number of preferences) basis of the universal preferences valid in P.

42 Universal preferences Basis For a preference context P {A B (A {1}) (B {2}) is minimal w.r.t. K P = (A {1}) (B {2}) { }} is the minimal (in the number of preferences) basis of the universal preferences valid in P. Canteens in Dresden,,,,

43 Universal preferences Computing the basis If G is large, G 2 is even bigger and it may be hard to compute this basis from K P.

44 Universal preferences Computing the basis If G is large, G 2 is even bigger and it may be hard to compute this basis from K P. Query learning (Angluin 1988) Learn from a teacher rather than from fixed data. Example: attribute exploration.

45 Universal preferences Computing the basis If G is large, G 2 is even bigger and it may be hard to compute this basis from K P. Query learning (Angluin 1988) Learn from a teacher rather than from fixed data. Example: attribute exploration. Explore implications over the attributes of K P and simulate the teacher by testing implications in P.

46 Existential preferences Existential preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y).

47 Existential preferences Existential preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y). Example When X and Y are sets of game positions reachable in one turn, how do you decide which set you prefer?

48 Existential preferences Existential preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y). Example When X and Y are sets of game positions reachable in one turn, how do you decide which set you prefer? -preferences if it is your turn.

49 Existential preferences Existential preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y). Example When X and Y are sets of game positions reachable in one turn, how do you decide which set you prefer? -preferences if it is your turn. -preferences if it is your opponent s turn.

50 Existential preferences Existential preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y). Existential preferences over attribute sets A set B M is preferred to a set A M if P = A B x A y B (x y)

51 Existential preferences Existential preferences over object sets: X Y A set Y G is preferred to a set X G if x X y Y (x y). Existential preferences over attribute sets A set B M is preferred to a set A M if P = A B x A y B (x y) or A g B g.

52 Existential preferences Canteens in Dresden Bergstraße Reichenbachstraße Klinikum Siedepunkt,, Siedepunkt Reichenbachstraße Klinikum Bergstraße

53 Existential preferences In terms of description logics A B: A R.B

54 Existential preferences In terms of description logics A B: A R.B A B: A R. B

55 Existential preferences Existential preferences are generalized implications For a preference context P = (G, M, I, ) 1 If (G, M, I ) = A B, then P = A B. 2 If is the identity relation and P = A B, then (G, M, I ) = A B.

56 Existential preferences Inference A system of three rules X X, X Y U X V Y, X Y, Y Z. X Z is sound and complete with respect to existential preferences.

57 Existential preferences Existential preferences as implications Existential translation of P K P = (G, P(M), I ) gi A g A. The derivation operators of K P are denoted by ( ). { } { } {, } { } {, } {, } {,, } { } {, } {, } {,, } {, } {,, } {,, } {,,, } B R K S

58 Existential preferences Existential preferences as implications Translation of an existential preference A B T (A B) is the implication {A} {B} of the formal context K P. Example,, {{,, }} { } { } {{ }} {{ }} {{ }}

59 Existential preferences Existential preferences as implications Translation of an existential preference A B T (A B) is the implication {A} {B} of the formal context K P. An existential preference A B is valid in a preference context P = (G, M, I, ) if and only if its translation is valid in K P : P = A B K P = T (A B).

60 Existential preferences Complete set Let P be a preference context. The set Σ = {A B A is minimal and B is maximal w.r.t. K P = {A} {B}} is a sound and complete subset of existential preferences of P.

61 Existential preferences Complete set Let P be a preference context. The set Σ = {A B A is minimal and B is maximal w.r.t. K P = {A} {B}} is a sound and complete subset of existential preferences of P. Example,,,,,,,,,,,,,

62 Existential preferences are a relaxation of universal preferences (almost).

63 Existential preferences are a relaxation of universal preferences (almost). Both are global: propositions are compared w.r.t. all their models.

64 Existential preferences are a relaxation of universal preferences (almost). Both are global: propositions are compared w.r.t. all their models. Ceteris paribus preferences Preferences under normal conditions: John prefers red wine to white wine (but not when having fish).

65 Existential preferences are a relaxation of universal preferences (almost). Both are global: propositions are compared w.r.t. all their models. Ceteris paribus preferences Preferences under normal conditions: John prefers red wine to white wine (but not when having fish). Preferences assuming everything else being equal : John prefers an academic job to a job in industry (other things being equal).

66 Existential preferences are a relaxation of universal preferences (almost). Both are global: propositions are compared w.r.t. all their models. Ceteris paribus preferences Preferences under normal conditions: John prefers red wine to white wine (but not when having fish). Preferences assuming everything else being equal : John prefers an academic job to a job in industry (other things being equal).... or more specifically: John prefers an academic job to a job in industry (given the salary is the same).

67 In preference logics X Γ Y (van Benthem et al. 2009) Y is preferred to X ceteris paribus with respect to a set Γ of propositions if x X y Y ( ϕ Γ(x = ϕ y = ϕ) x y). Γ is a (true) relaxation of.

68 In preference logics X Γ Y (van Benthem et al. 2009) Y is preferred to X ceteris paribus with respect to a set Γ of propositions if x X y Y ( ϕ Γ(x = ϕ y = ϕ) x y). Γ is a (true) relaxation of. Adding the ceteris paribus condition to the definition of -preferences results in stronger preferences.

69 Ceteris paribus preferences B M is preferred ceteris paribus to A M with respect to C M in P = (G, M, I, ) if A C B, i.e., g A h B ({g} C = {h} C g h). A C B

70 Ceteris paribus preferences as implications Ceteris paribus translation of P K P = (G G, (M {1, 2, 3}) { }, I ) (g 1, g 2 )I (m, 1) g 1 Im, (g 1, g 2 )I (m, 2) g 2 Im, (g 1, g 2 )I (m, 3) {g 1 } {m} = {g 2 } {m}, (g 1, g 2 )I g 1 g 2. The derivation operators of K P are denoted by ( ). Example R, K...

71 Ceteris paribus preferences as implications Translation of a ceteris paribus preference A C B T (A C B) is the implication (A {1}) (B {2}) (C {3}) { } of the formal context K P. A ceteris paribus preference A C B is valid in a preference context P = (G, M, I, ) if and only if its translation is valid in K P : P = A C B K P = T (A C B).

72 A C B is in canonical form if A B = A C = B C. The canonical form of A C B: A (B C) C (A B) B (A C). Proposition The set Π = {A C B (A {1}) (B {2}) (C {3}) is minimal w.r.t. K P = T (A C B) and A B = A C = B C} is sound and complete for the preference context P.

73 Π Let Π be a set of ceteris paribus preferences. Then Π = {D F E A C B Π(A D, B E, C F )}. Proposition For any preference A C B in canonical form, we have Π = A C B if and only if Π contains all canonical-form preferences D F E such that A D, B E, C F, and M = D E F.

74 Ceteris Paribus Consequence(A C B, Π) Input: A ceteris paribus preference A C B and a set Π of ceteris paribus preferences (over a universal set M). Output: true, if Π = A C B; false, otherwise. S := [A (B C) C (A B) B (A C)] {stack} repeat D F E := pop(s) if D F E Π then X := M \ (D E F ) if X = then return false choose m X push(d {m} F E, S) push(d F E {m}, S) push(d F {m} E, S} until empty(s) return true

75 The algorithm is exponential in M. The theory implied by ceteris paribus preferences is generated by Horn formulas into which we translate preferences;

76 The algorithm is exponential in M. The theory implied by ceteris paribus preferences is generated by Horn formulas into which we translate preferences; m i m j m k for each m M and i j k {1, 2, 3};

77 The algorithm is exponential in M. The theory implied by ceteris paribus preferences is generated by Horn formulas into which we translate preferences; m i m j m k for each m M and i j k {1, 2, 3}; m 1 m 2 m 3 for each m M.

78 The algorithm is exponential in M. The theory implied by ceteris paribus preferences is generated by Horn formulas into which we translate preferences; m i m j m k for each m M and i j k {1, 2, 3}; m 1 m 2 m 3 for each m M. However, the algorithm is linear in Π.

79 Preferences may be biased All the canteens serve different otherwise?. Would the preferences be Observed data covers only some possibilities. Derived preferences hold in the data, but may have counterexamples in the entire domain.

80 Horn bias The Horn bias induced by a preference context P = (G, M, I, ) is the set of implications that hold in (G, M, I ). The canonical basis of (G, M, I ),,,

81 Biased preferences If H is the Horn bias induced by P = (G, M, I, ), Π is the set of all preferences a that hold in P, π Π is a preference, Π 1 Π \ {π}, such that Π 1 = π and H Π 1 = π then π is Horn-biased in P. a of a certain kind

82 Example {,,, } =,,,, but = is Horn-biased in P

83 Universal preferences Proposition A universal preference A B is Horn-biased if and only if A A or B B. Thus, unbiased preferences are preferences over formal concepts.

84 Universal preferences How do we compute unbiased universal preferences? Compute H, the canonical basis of (G, M, I ). Transform it into Background knowledge A {i} B {i} for A B H, i {1, 2}

85 Universal preferences How do we compute unbiased universal preferences? Compute H, the canonical basis of (G, M, I ). Transform it into Background knowledge A {i} B {i} for A B H, i {1, 2} A {i} for A B H, i {1, 2} and A =

86 Universal preferences How do we compute unbiased universal preferences? Compute H, the canonical basis of (G, M, I ). Transform it into Background knowledge A {i} B {i} for A B H, i {1, 2} A {i} for A B H, i {1, 2} and A = Compute the canonical basis of K P relative to this background knowledge with Next Closure, making the first attribute.

87 Universal preferences How do we compute unbiased universal preferences? Compute H, the canonical basis of (G, M, I ). Transform it into Background knowledge A {i} B {i} for A B H, i {1, 2} A {i} for A B H, i {1, 2} and A = Compute the canonical basis of K P relative to this background knowledge with Next Closure, making the first attribute. Equivalently, compute the minimal hypotheses of K P for.

88 Universal preferences Universal preferences Example The canonical basis of (G, M, I ),,, The relative basis of universal preferences,,,,,,,,,

89 Universal preferences Universal preferences and implications A hybrid system 1 Armstrong rules 2 The rule for universal preferences 3 X Y, X Y Z X Z X X, X X Y, Z X Y Z X

90 Existential preferences Proposition An existential preference A B is Horn-biased if and only if A A or B B. Thus, unbiased preferences are preferences over formal concepts.

91 Existential preferences How do we compute unbiased existential preferences? Can use Background knowledge {P} {P } for every pseudo-intent P in (G, M, I ) {A} {A \ {a} a A} for every A M { } But this is computationally unrealistic.

92 Existential preferences Existential preferences { } { } {, } { } {, } {, } {,, } { } {, } {, } {,, } {, } {,, } {,, } {,,, } B R K S Infeasible for all but very small M. Reduce the representation size by making use of the dependencies in the data.

93 Existential preferences Existential preferences as implications between concepts Conceptual existential translation of P C P = (G, B(G, M, I ), I ) gi (A, B) g A. The derivation operators of C P are denoted by ( ). Conceptual translation of an existential preference A B T C(A B) is the implication {(A, A )} {(B, B )} of the formal context C P.

94 Existential preferences Example (BRKS, ) (R, ) (BK, ) (B, ) (, ) Bergstraße Reichenbachstraße Klinikum Siedepunkt,, (B, ) (BRKS, ) (BRKS, ) (R, ) (B, ) (R, )

95 Existential preferences Existential preferences as implications between concepts Siedepunkt Reichenbachstraße Klinikum Bergstraße

96 Existential preferences Existential preferences as implications between concepts A complete set {A B C P = {(A, A)} {(B, B)} and B A}. to be considered relative to the implications of (G, M, I ). Example For our example, this gives just one preference:,,,.

97 Existential preferences Existential preferences and implications A hybrid system 1 Armstrong rules 2 The rules for existential preferences 3 A B A B

98 Ceteris paribus preferences For ceteris paribus preferences, bias can be reduced even further. A doubly conditional functional dependency [A, B]C D[E, F ] holds in (G, M, I ) if, for every g, h G A g, B h, C g = C h E g, F h, D g = D h. The induced bias includes the Horn bias.

99 Ceteris paribus preferences 2CFD bias The 2CFD bias induced by a preference context P = (G, M, I, ) is the set of doubly conditional functional dependencies that hold in the formal context (G, M, I ). Define 2CFD-biased preferences similarly to how Horn-biased preferences were defined. Doubly conditional functional dependencies are in one-to-one correspondence with implications of K P (without ). Unbiased preferences are obtained by backward translation from minimal hypotheses for in K P.

100 Limitation We derive only preferences over conjunctions of boolean variables. Clearly, less expressive than preference modal logics (van Benthem et al. 2009). Seems less expressive than state-of-the art approaches to preference handling: CP-nets (Boutilier et al 1999); TCP-nets (Brafman and Domshlak 2002); cp-theories (Wilson 2011).

101 cp-theories A conditional preference over a set V of variables u : x 1 > x 2 [W ] u is an assignment to U V ; x 1 and x 2 are different values of some X V ; Interpretation W V \ (U {X }). Given u, we prefer x 1 to x 2 provided that the values of variables outside W remain the same. Example When going to Dresden, I prefer train to plane provided that the other parameters with a possible exception of time are the same. Dresden: train > plane[{time}]

102 cp-theories To express weak conditional preferences in our framework: introduce a separate attribute for each variable value X = x; denote by Ŵ the set of all attributes for the variables in W ; associate a conditional preference u : x 1 > x 2 [W ] with a ceteris paribus preference u {X = x 2 } {M\ c W } u {X = x 1 }. Start with a strict preference relation over objects if you need to express strict conditional preferences.

103 cp-theories Conditional preferences in cp-theories are always over values of a single variable. Example But we can do more: u {X = x 2, Y = y 2 } {M\ c W } u {X = x 1, Y = y 1 }.

104 cp-theories... And, in principle, even more: For variables with ordinal values, use attributes of the form x 1 X x 2 instead of just X = x, see many-valued contexts and scaling in FCA.

105 cp-theories... And, in principle, even more: For variables with ordinal values, use attributes of the form x 1 X x 2 instead of just X = x, see many-valued contexts and scaling in FCA. Customize ceteris paribus conditions to specify relations other than equality. In so doing, we can express preferences like Between two ways of travel, I prefer a cheap one provided that it is at least as fast as the other.

106 cp-theories Universal preference exploration Two approaches 1 Explore preferences based on minimal generating sets of in K P. 2 Find the basis of (G, M, I ) and use standard attribute exploration for K P (making the first attribute). A counterexample to A B is a pair of objects (g, h) with A g, B h, and g h.

107 cp-theories Universal preference exploration Two approaches 1 Explore preferences based on minimal generating sets of in K P. 2 Find the basis of (G, M, I ) and use standard attribute exploration for K P (making the first attribute). A counterexample to A B is a pair of objects (g, h) with A g, B h, and g h. Would be interesting to design exploration, where the user is asked to compare two objects.

108 cp-theories Existential preference exploration Probably, trickier than for universal preferences: Adding objects requires adding attributes to the translated context.