Analysis of Boolean Functions

Transcription

1 Analysis of Boolean Functions Kavish Gandhi and Noah Golowich Mentor: Yufei Zhao 5th Annual MIT-PRIMES Conference Analysis of Boolean Functions, Ryan O Donnell May 16, Kavish Gandhi and Noah Golowich Boolean functions

2 Boolean Functions f : { 1, 1} n { 1, 1}. 2 Kavish Gandhi and Noah Golowich Boolean functions

3 Boolean Functions f : { 1, 1} n { 1, 1}. Applications of Boolean functions: Circuit design. Learning theory. 2 Kavish Gandhi and Noah Golowich Boolean functions

4 Boolean Functions f : { 1, 1} n { 1, 1}. Applications of Boolean functions: Circuit design. Learning theory. Voting rule for election with n voters and 2 candidates { 1, 1}; social choice theory. 2 Kavish Gandhi and Noah Golowich Boolean functions

5 Majority, Linear Threshold Functions Convention: x { 1, 1} n ; x 1, x 2,..., x n are coordinates of x. 3 Kavish Gandhi and Noah Golowich Boolean functions

6 Majority, Linear Threshold Functions Convention: x { 1, 1} n ; x 1, x 2,..., x n are coordinates of x. Majority function: Maj n (x) = sgn(x 1 + x x n ). (1, 1, 1) (-1, 1, -1) (-1, -1, -1) (1, -1, -1) 3 Kavish Gandhi and Noah Golowich Boolean functions

7 Majority, Linear Threshold Functions Convention: x { 1, 1} n ; x 1, x 2,..., x n are coordinates of x. Majority function: Maj n (x) = sgn(x 1 + x x n ). (1, 1, 1) (-1, 1, -1) (-1, -1, -1) (1, -1, -1) f is linear threshold function (weighted majority) if f (x) = sgn(a 0 + a 1 x a n x n ). 3 Kavish Gandhi and Noah Golowich Boolean functions

8 AND, OR, Tribes 1 True, 1 False. AND n (x) = x 1 x 2 x n. OR n (x) = x 1 x 2 x n. 4 Kavish Gandhi and Noah Golowich Boolean functions

9 AND, OR, Tribes 1 True, 1 False. AND n (x) = x 1 x 2 x n. OR n (x) = x 1 x 2 x n. Tribes w,s (x 1,..., x sw ) = (x 1 x w ) (x (s 1)w x sw ). n = ws is number of voters. s tribes, w people per tribe. x 1 x w x (s 1)w x sw 4 Kavish Gandhi and Noah Golowich Boolean functions

10 Influence Definition Impartial culture assumption: n votes independent, uniformly random: x { 1, 1} n. 5 Kavish Gandhi and Noah Golowich Boolean functions

11 Influence Definition Impartial culture assumption: n votes independent, uniformly random: x { 1, 1} n. Influence at coordinate i, Inf i : prob. that voter i changes outcome. Influence of f : I[f ] = n i=1 Inf i[f ]. Example: I[Maj 3 (x)] = 3/2. (1, 1, 1) (-1, -1, -1) 5 Kavish Gandhi and Noah Golowich Boolean functions

12 Oops Nassau County (NY) voting system: f (x) = sgn( x x x x 4 + 2x 5 + 2x 6 ). 6 Kavish Gandhi and Noah Golowich Boolean functions

13 Oops Nassau County (NY) voting system: f (x) = sgn( x x x x 4 + 2x 5 + 2x 6 ). Some towns have 0 influence! 6 Kavish Gandhi and Noah Golowich Boolean functions

14 Oops Nassau County (NY) voting system: f (x) = sgn( x x x x 4 + 2x 5 + 2x 6 ). Some towns have 0 influence! Lawyer Banzhaf sued Nassau County board (1965). 6 Kavish Gandhi and Noah Golowich Boolean functions

15 Influences of Tribes, Majority f monotone: x y coordinate-wise f (x) f (y). Theorem I[f ] I[Maj n ] = 2/π n + O(n 1/2 ) for all monotone f. 7 Kavish Gandhi and Noah Golowich Boolean functions

16 Influences of Tribes, Majority f monotone: x y coordinate-wise f (x) f (y). Theorem I[f ] I[Maj n ] = 2/π n + O(n 1/2 ) for all monotone f. For n = ws, define Tribes n = Tribes w,s with w, s such that Tribes w,s is essentially unbiased. Inf i [Tribes n ] = ln n n (1 + o(1)). 7 Kavish Gandhi and Noah Golowich Boolean functions

17 Influences of Tribes, Majority f monotone: x y coordinate-wise f (x) f (y). Theorem I[f ] I[Maj n ] = 2/π n + O(n 1/2 ) for all monotone f. For n = ws, define Tribes n = Tribes w,s with w, s such that Tribes w,s is essentially unbiased. Inf i [Tribes n ] = ln n n (1 + o(1)). Theorem (Kahn, Kalai, Linial) MaxInf[f ] Var[f ] Ω( log n n ). 7 Kavish Gandhi and Noah Golowich Boolean functions

18 Influences of Tribes, Majority f monotone: x y coordinate-wise f (x) f (y). Theorem I[f ] I[Maj n ] = 2/π n + O(n 1/2 ) for all monotone f. For n = ws, define Tribes n = Tribes w,s with w, s such that Tribes w,s is essentially unbiased. Inf i [Tribes n ] = ln n n (1 + o(1)). Theorem (Kahn, Kalai, Linial) MaxInf[f ] Var[f ] Ω( log n n ). Application: bribing voters. 7 Kavish Gandhi and Noah Golowich Boolean functions

19 Noise Stability Another important property of Boolean functions: noise stability. 8 Kavish Gandhi and Noah Golowich Boolean functions

20 Noise Stability Another important property of Boolean functions: noise stability. Definition For a fixed x { 1, 1} n and ρ [0, 1], y is ρ-correlated with x if, for each coordinate i, { x i with probability ρ y i = randomly chosen with probability 1 ρ. 8 Kavish Gandhi and Noah Golowich Boolean functions

21 Noise Stability Another important property of Boolean functions: noise stability. Definition For a fixed x { 1, 1} n and ρ [0, 1], y is ρ-correlated with x if, for each coordinate i, { x i with probability ρ y i = randomly chosen with probability 1 ρ. Definition For a Boolean function f and ρ [0, 1], the noise stability of f at ρ is Stab ρ [f ] = E[f (x)f (y)]. for x uniformly random and y ρ-correlated with x. 8 Kavish Gandhi and Noah Golowich Boolean functions

22 Noise Stability: Voting Example Imagine: 9 Kavish Gandhi and Noah Golowich Boolean functions

23 Noise Stability: Voting Example Imagine: Voting system represented by f. 9 Kavish Gandhi and Noah Golowich Boolean functions

24 Noise Stability: Voting Example Imagine: Voting system represented by f. Some chance 1 ρ 2 that the vote is misrecorded. 9 Kavish Gandhi and Noah Golowich Boolean functions

25 Noise Stability: Voting Example Imagine: Voting system represented by f. Some chance 1 ρ 2 that the vote is misrecorded. Noise stability: measure of how much f is resistant to misrecorded votes. 9 Kavish Gandhi and Noah Golowich Boolean functions

26 The Noise Stability of Majority Natural question: what is the noise stability of Majority? 10 Kavish Gandhi and Noah Golowich Boolean functions

27 The Noise Stability of Majority Natural question: what is the noise stability of Majority? Theorem For any ρ [0, 1], lim n Stab ρ [Maj n ] = 2 π arcsin ρ. 10 Kavish Gandhi and Noah Golowich Boolean functions

28 The Noise Stability of Majority Natural question: what is the noise stability of Majority? Theorem For any ρ [0, 1], lim n Stab ρ [Maj n ] = 2 π arcsin ρ. General idea of proof: use the multidimensional central limit theorem. 10 Kavish Gandhi and Noah Golowich Boolean functions

29 The Noise Stability of Majority Natural question: what is the noise stability of Majority? Theorem For any ρ [0, 1], lim n Stab ρ [Maj n ] = 2 π arcsin ρ. General idea of proof: use the multidimensional central limit theorem. Theorem (Majority is Stablest) Among Boolean functions that are unbiased and have only small influences, the Majority function has approximately the largest noise stability. 10 Kavish Gandhi and Noah Golowich Boolean functions

30 Arrow s Theorem: The Idea Noise stability also key in proving Arrow s Theorem. In particular, consider: 11 Kavish Gandhi and Noah Golowich Boolean functions

31 Arrow s Theorem: The Idea Noise stability also key in proving Arrow s Theorem. In particular, consider: Two candidate elections: most fair voting rule is Majority. 11 Kavish Gandhi and Noah Golowich Boolean functions

32 Arrow s Theorem: The Idea Noise stability also key in proving Arrow s Theorem. In particular, consider: Two candidate elections: most fair voting rule is Majority. Three candidate elections: not clear how to conduct the election. 11 Kavish Gandhi and Noah Golowich Boolean functions

33 Arrow s Theorem: The Idea Noise stability also key in proving Arrow s Theorem. In particular, consider: Two candidate elections: most fair voting rule is Majority. Three candidate elections: not clear how to conduct the election. One possibility: conduct pairwise Condorcet elections, each of which is evaluated by some voting rule f. 11 Kavish Gandhi and Noah Golowich Boolean functions

34 Arrow s Theorem: The Idea Noise stability also key in proving Arrow s Theorem. In particular, consider: Two candidate elections: most fair voting rule is Majority. Three candidate elections: not clear how to conduct the election. One possibility: conduct pairwise Condorcet elections, each of which is evaluated by some voting rule f. The Condorcet winner is the candidate that wins all his/her elections. 11 Kavish Gandhi and Noah Golowich Boolean functions

35 Arrow s Theorem: The Idea Noise stability also key in proving Arrow s Theorem. In particular, consider: Two candidate elections: most fair voting rule is Majority. Three candidate elections: not clear how to conduct the election. One possibility: conduct pairwise Condorcet elections, each of which is evaluated by some voting rule f. The Condorcet winner is the candidate that wins all his/her elections. May not always occur: might be some situations in which each candidate loses a pairwise election. 11 Kavish Gandhi and Noah Golowich Boolean functions

36 Arrow s Theorem: The Idea Noise stability also key in proving Arrow s Theorem. In particular, consider: Two candidate elections: most fair voting rule is Majority. Three candidate elections: not clear how to conduct the election. One possibility: conduct pairwise Condorcet elections, each of which is evaluated by some voting rule f. The Condorcet winner is the candidate that wins all his/her elections. May not always occur: might be some situations in which each candidate loses a pairwise election. Goal: find a function in which this contradiction never occurs. 11 Kavish Gandhi and Noah Golowich Boolean functions

37 Example: Contradiction with f Majority Example of contradiction: candidates A, B, C; voters x 1, x 2, and x 3 ; voting rule is Majority function. 12 Kavish Gandhi and Noah Golowich Boolean functions

38 Example: Contradiction with f Majority Example of contradiction: candidates A, B, C; voters x 1, x 2, and x 3 ; voting rule is Majority function. x 1 x 2 x 3 A vs. B A B A A vs. C C C A B vs. C C B B 12 Kavish Gandhi and Noah Golowich Boolean functions

39 Example: Contradiction with f Majority Example of contradiction: candidates A, B, C; voters x 1, x 2, and x 3 ; voting rule is Majority function. x 1 x 2 x 3 A vs. B A B A A vs. C C C A B vs. C C B B A wins the pairwise election with B. C wins the pairwise election with A. B wins the pairwise election with C. 12 Kavish Gandhi and Noah Golowich Boolean functions

40 Example: Contradiction with f Majority Example of contradiction: candidates A, B, C; voters x 1, x 2, and x 3 ; voting rule is Majority function. x 1 x 2 x 3 A vs. B A B A A vs. C C C A B vs. C C B B A wins the pairwise election with B. C wins the pairwise election with A. B wins the pairwise election with C. There is no Condorcet winner! 12 Kavish Gandhi and Noah Golowich Boolean functions

41 Arrow s Theorem: The Statement Theorem (Arrow s Theorem) In an n-candidate Condorcet election, if there is always a Condorcet winner, then f (x) = ±x i for some i (dictatorship). 13 Kavish Gandhi and Noah Golowich Boolean functions

42 Arrow s Theorem: The Statement Theorem (Arrow s Theorem) In an n-candidate Condorcet election, if there is always a Condorcet winner, then f (x) = ±x i for some i (dictatorship). The case n = 3 follows from the below result: connects it to stability. 13 Kavish Gandhi and Noah Golowich Boolean functions

43 Arrow s Theorem: The Statement Theorem (Arrow s Theorem) In an n-candidate Condorcet election, if there is always a Condorcet winner, then f (x) = ±x i for some i (dictatorship). The case n = 3 follows from the below result: connects it to stability. Theorem In a 3-candidate Condorcet election, the probability of a Condorcet winner is exactly 3/4(1 Stab 1/3 [f ]). 13 Kavish Gandhi and Noah Golowich Boolean functions

44 Arrow s Theorem: The Statement Theorem (Arrow s Theorem) In an n-candidate Condorcet election, if there is always a Condorcet winner, then f (x) = ±x i for some i (dictatorship). The case n = 3 follows from the below result: connects it to stability. Theorem In a 3-candidate Condorcet election, the probability of a Condorcet winner is exactly 3/4(1 Stab 1/3 [f ]). Dictator: only function for which Stab 1/3 [f ] = 1/3 3 4 (1 Stab 1/3[f ]) = Kavish Gandhi and Noah Golowich Boolean functions

45 Peres s Theorem Noise sensitivity of f at δ is probability that misrecorded votes change outcome: NS δ [f ] = Stab 1 2δ[f ]. Theorem (Peres, 1999) For any LTF f, NS δ [f ] O( δ). 14 Kavish Gandhi and Noah Golowich Boolean functions

46 Peres s Theorem Noise sensitivity of f at δ is probability that misrecorded votes change outcome: NS δ [f ] = Stab 1 2δ[f ]. Theorem (Peres, 1999) For any LTF f, NS δ [f ] O( δ). lim n NS δ [Maj n ] = 2 π δ + O(δ 3/2 ). 14 Kavish Gandhi and Noah Golowich Boolean functions

47 Applications of Peres s theorem Application: learning theory.???? 15 Kavish Gandhi and Noah Golowich Boolean functions

48 Applications of Peres s theorem Application: learning theory.???? Corollary An AND of 2 LTFs is learnable with error ɛ in time n O(1/ɛ2). Open problem: extend Peres s theorem to polynomial threshold functions: sgn(p(x)). 15 Kavish Gandhi and Noah Golowich Boolean functions

49 Fourier Analysis of Boolean Functions How to prove many of theorems: Fourier expansions, a representation of the function as a real, multilinear polynomial. 16 Kavish Gandhi and Noah Golowich Boolean functions

50 Fourier Analysis of Boolean Functions How to prove many of theorems: Fourier expansions, a representation of the function as a real, multilinear polynomial. For example, max 2 (x 1, x 2 ), outputs the maximum of x 1 and x 2 : max 2 (x 1, x 2 ) = x x x 1x Kavish Gandhi and Noah Golowich Boolean functions

51 Uniqueness of the Fourier Expansion For a given f : always exists a Fourier expansion. In particular: Theorem Every Boolean function can be uniquely expressed as a multilinear polynomial, called its Fourier expansion, f (x) = ˆf (S)x S, where x S = i S x i. S [n] 17 Kavish Gandhi and Noah Golowich Boolean functions

52 Uniqueness of the Fourier Expansion For a given f : always exists a Fourier expansion. In particular: Theorem Every Boolean function can be uniquely expressed as a multilinear polynomial, called its Fourier expansion, f (x) = ˆf (S)x S, where x S = i S x i. S [n] Coefficients ˆf (S): Fourier spectrum of f. 17 Kavish Gandhi and Noah Golowich Boolean functions

53 Parseval s and Plancherel s Theorems Theorem (Plancherel) For any Boolean functions f and g, E[f (x)g(x)] = ˆf (S)ĝ(S). S [n] Applies equally well to real-valued functions. Also yields corollary: 18 Kavish Gandhi and Noah Golowich Boolean functions

54 Parseval s and Plancherel s Theorems Theorem (Plancherel) For any Boolean functions f and g, E[f (x)g(x)] = S [n] ˆf (S)ĝ(S). Applies equally well to real-valued functions. Also yields corollary: Theorem (Parseval) For any Boolean function f, S [n] ˆf (S) 2 = E[f (x) 2 ] = Kavish Gandhi and Noah Golowich Boolean functions

55 Fourier Expansions for Stability, Influence Theorem For any Boolean function f and i [n], Inf i [f ] = S i ˆf (S) Kavish Gandhi and Noah Golowich Boolean functions

56 Fourier Expansions for Stability, Influence Theorem For any Boolean function f and i [n], Inf i [f ] = S i ˆf (S) 2. Theorem For any Boolean function f, Stab ρ [f ] = S [n] ρ S ˆf (S) Kavish Gandhi and Noah Golowich Boolean functions

57 Summary and Conclusion In summary: 20 Kavish Gandhi and Noah Golowich Boolean functions

58 Summary and Conclusion In summary: Looked at Boolean functions in the context of social choice theory and voting. 20 Kavish Gandhi and Noah Golowich Boolean functions

59 Summary and Conclusion In summary: Looked at Boolean functions in the context of social choice theory and voting. Fourier expansions of these functions along with noise stability and influence: allowed us to prove Arrow s Theorem and Peres s Theorem. 20 Kavish Gandhi and Noah Golowich Boolean functions

60 Summary and Conclusion In summary: Looked at Boolean functions in the context of social choice theory and voting. Fourier expansions of these functions along with noise stability and influence: allowed us to prove Arrow s Theorem and Peres s Theorem. Not just limited to voting theory: 20 Kavish Gandhi and Noah Golowich Boolean functions

61 Summary and Conclusion In summary: Looked at Boolean functions in the context of social choice theory and voting. Fourier expansions of these functions along with noise stability and influence: allowed us to prove Arrow s Theorem and Peres s Theorem. Not just limited to voting theory: Learning theory. Circuit design. 20 Kavish Gandhi and Noah Golowich Boolean functions

62 Acknowledgements We would like to thank: Yufei Zhao MIT-PRIMES Our parents 21 Kavish Gandhi and Noah Golowich Boolean functions