Csr2011 june18 11_30_remila

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Csr2011 june18 11_30_remila

  1. 1. The Optimal Strategy for the Average Long-Lived Consensus Eric R´mila e Universit´ de Lyon e Laboratoire de l’Informatique du Parall´lisme e (LIP, umr 5668 CNRS - Universit´ Lyon 1 - ENS de Lyon) eIXXI (Institut des Syst`mes Complexes - Complex System Institute, Lyon) e IUT Roanne, Universit´ de Saint-Etienne ePartially supported by Program Ecos C09E04, ANR Subtile and ANR Mint. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  2. 2. The consensus problem: informal approach What qualities are requested for a collective decision, called consensus? university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  3. 3. The consensus problem: informal approach What qualities are requested for a collective decision, called consensus? Representativity: the consensus corresponds to a sufficient number of individual opinions, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  4. 4. The consensus problem: informal approach What qualities are requested for a collective decision, called consensus? Representativity: the consensus corresponds to a sufficient number of individual opinions, Stability: the consensus is robust to individual opinion variations. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  5. 5. The consensus problem: informal approach What qualities are requested for a collective decision, called consensus? Representativity: the consensus corresponds to a sufficient number of individual opinions, Stability: the consensus is robust to individual opinion variations. Problem: the qualities above are often incompatible. What can we do in in the real life? university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  6. 6. An illustrative example: the wedding party problem A disc-jockey with two kinds of musics: traditional music and techno music, people going in and out. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  7. 7. An illustrative example: the wedding party problem A disc-jockey with two kinds of musics: traditional music and techno music, people going in and out. The music must appeal to at least some of the people in the room (representativity). university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  8. 8. An illustrative example: the wedding party problem A disc-jockey with two kinds of musics: traditional music and techno music, people going in and out. The music must appeal to at least some of the people in the room (representativity). changing style after every song is frowned upon (stability). university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  9. 9. An illustrative example: the wedding party problem A disc-jockey with two kinds of musics: traditional music and techno music, people going in and out. The music must appeal to at least some of the people in the room (representativity). changing style after every song is frowned upon (stability). The party has to last ’till the end of the night. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  10. 10. An illustrative example: the wedding party problem A disc-jockey with two kinds of musics: traditional music and techno music, people going in and out. The music must appeal to at least some of the people in the room (representativity). changing style after every song is frowned upon (stability). The party has to last ’till the end of the night. Another example: governments simultaneously need stability for consistence of politics and representativity to university-logo remain. Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  11. 11. The consensus problem: system formalization n sensors, each one is given an input component, chosen in Zm = {0, 1, ...., m − 1}, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  12. 12. The consensus problem: system formalization n sensors, each one is given an input component, chosen in Zm = {0, 1, ...., m − 1}, A global input: an n-uple of Vn = (Zm )n , university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  13. 13. The consensus problem: system formalization n sensors, each one is given an input component, chosen in Zm = {0, 1, ...., m − 1}, A global input: an n-uple of Vn = (Zm )n , For each pair (x, b) of Vn × Zm , #x (b) is the number of input components of x that are equal to b, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  14. 14. The consensus problem: system formalization n sensors, each one is given an input component, chosen in Zm = {0, 1, ...., m − 1}, A global input: an n-uple of Vn = (Zm )n , For each pair (x, b) of Vn × Zm , #x (b) is the number of input components of x that are equal to b, For each x of Vn , dom(x) is the integer b that maximizes #x (b) (the lowest one in case of ties) university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  15. 15. The consensus problem: system formalization n sensors, each one is given an input component, chosen in Zm = {0, 1, ...., m − 1}, A global input: an n-uple of Vn = (Zm )n , For each pair (x, b) of Vn × Zm , #x (b) is the number of input components of x that are equal to b, For each x of Vn , dom(x) is the integer b that maximizes #x (b) (the lowest one in case of ties) Example: x = (0, 3, 1, 3, 1). #x (0) = 1, #x (1) = 2, #x (2) = 0, #x (3) = 2. dom(x) = 1. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  16. 16. The consensus problem: evolution formalization The input graph (Vn , En ): the undirected graph whose edges are input pairs only differing in a unique sensor. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  17. 17. The consensus problem: evolution formalization The input graph (Vn , En ): the undirected graph whose edges are input pairs only differing in a unique sensor. A trajectory: a sequence x0 → x1 → x2 → .... with ∀k ∈ N, {xk+1 , xk } ∈ En . At each step: change of one input component. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  18. 18. The consensus problem: adding a memory a memory: a set Q of states, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  19. 19. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  20. 20. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, A consensus protocol: university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  21. 21. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, A consensus protocol: A memory evolution function τ : Vn × Q → Q, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  22. 22. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, A consensus protocol: A memory evolution function τ : Vn × Q → Q, A consensus function f : Vn × Q → Zm , university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  23. 23. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, A consensus protocol: A memory evolution function τ : Vn × Q → Q, A consensus function f : Vn × Q → Zm , An execution: sequence (x0 , s0 ) → (x1 , s1 ) → (x2 , s2 ) → .... with university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  24. 24. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, A consensus protocol: A memory evolution function τ : Vn × Q → Q, A consensus function f : Vn × Q → Zm , An execution: sequence (x0 , s0 ) → (x1 , s1 ) → (x2 , s2 ) → .... with x0 → x1 → x2 → .... is a trajectory, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  25. 25. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, A consensus protocol: A memory evolution function τ : Vn × Q → Q, A consensus function f : Vn × Q → Zm , An execution: sequence (x0 , s0 ) → (x1 , s1 ) → (x2 , s2 ) → .... with x0 → x1 → x2 → .... is a trajectory, sk+1 = τ (xk , sk ), university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  26. 26. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, A consensus protocol: A memory evolution function τ : Vn × Q → Q, A consensus function f : Vn × Q → Zm , An execution: sequence (x0 , s0 ) → (x1 , s1 ) → (x2 , s2 ) → .... with x0 → x1 → x2 → .... is a trajectory, sk+1 = τ (xk , sk ), s0 = ⊥, the empty memory state university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  27. 27. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, A consensus protocol: A memory evolution function τ : Vn × Q → Q, A consensus function f : Vn × Q → Zm , An execution: sequence (x0 , s0 ) → (x1 , s1 ) → (x2 , s2 ) → .... with x0 → x1 → x2 → .... is a trajectory, sk+1 = τ (xk , sk ), s0 = ⊥, the empty memory state Each trajectory induces a unique execution, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  28. 28. The consensus problem: adding a memory a memory: a set Q of states, Vn × Q: configuration set, A consensus protocol: A memory evolution function τ : Vn × Q → Q, A consensus function f : Vn × Q → Zm , An execution: sequence (x0 , s0 ) → (x1 , s1 ) → (x2 , s2 ) → .... with x0 → x1 → x2 → .... is a trajectory, sk+1 = τ (xk , sk ), s0 = ⊥, the empty memory state Each trajectory induces a unique execution, At each time step: change (or not) of the memory and of the consensus value f (xk , sk ). university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  29. 29. Representativity formalization We fix a threshold t. The consensus function f is representative when: f (x, s) = k =⇒ #x (k) > t. Remark: we need to have: n > k t, in order to be able to satisfy the threshold condition in any case. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  30. 30. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  31. 31. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with xp → xp+1 → ... → xp is a trajectory, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  32. 32. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with xp → xp+1 → ... → xp is a trajectory, sk+1 = τ (xk , sk ), university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  33. 33. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with xp → xp+1 → ... → xp is a trajectory, sk+1 = τ (xk , sk ), each input component is changed at most once. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  34. 34. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with xp → xp+1 → ... → xp is a trajectory, sk+1 = τ (xk , sk ), each input component is changed at most once. Remark: necessarily, p − p ≤ n university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  35. 35. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with xp → xp+1 → ... → xp is a trajectory, sk+1 = τ (xk , sk ), each input component is changed at most once. Remark: necessarily, p − p ≤ n instability: maximal number of consensus changes in a geodesic university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  36. 36. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with xp → xp+1 → ... → xp is a trajectory, sk+1 = τ (xk , sk ), each input component is changed at most once. Remark: necessarily, p − p ≤ n instability: maximal number of consensus changes in a geodesic A worst case criterion, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  37. 37. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with xp → xp+1 → ... → xp is a trajectory, sk+1 = τ (xk , sk ), each input component is changed at most once. Remark: necessarily, p − p ≤ n instability: maximal number of consensus changes in a geodesic A worst case criterion, which considers geodesics which may never appear (or with very low probability) university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  38. 38. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with xp → xp+1 → ... → xp is a trajectory, sk+1 = τ (xk , sk ), each input component is changed at most once. Remark: necessarily, p − p ≤ n instability: maximal number of consensus changes in a geodesic A worst case criterion, which considers geodesics which may never appear (or with very low probability) and leads to non intuitive results university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  39. 39. Stability formalization, geodesic criterion A geodoesic: a sequence (xp , sp ) → (xp+1 , sp+1 ) → ... → (xp , sp ), with xp → xp+1 → ... → xp is a trajectory, sk+1 = τ (xk , sk ), each input component is changed at most once. Remark: necessarily, p − p ≤ n instability: maximal number of consensus changes in a geodesic A worst case criterion, which considers geodesics which may never appear (or with very low probability) and leads to non intuitive results Not a good criterion. Forget it !! university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  40. 40. Stability formalization, average criterion A uniform random walk S0 , S1 , S2 .... on (Vn , En ). university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  41. 41. Stability formalization, average criterion A uniform random walk S0 , S1 , S2 .... on (Vn , En ). Np : the number of consensus changes during the first p execution steps. Np = card{k ∈ N | (0 ≤ k < p) ∧ (f (Sk , Xk ) = f (Sk+1 , Xk+1 ))} university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  42. 42. Stability formalization, average criterion A uniform random walk S0 , S1 , S2 .... on (Vn , En ). Np : the number of consensus changes during the first p execution steps. Np = card{k ∈ N | (0 ≤ k < p) ∧ (f (Sk , Xk ) = f (Sk+1 , Xk+1 ))} Definition: Np instability = lim (E ( )) p→∞ p university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  43. 43. Stability formalization, average criterion A uniform random walk S0 , S1 , S2 .... on (Vn , En ). Np : the number of consensus changes during the first p execution steps. Np = card{k ∈ N | (0 ≤ k < p) ∧ (f (Sk , Xk ) = f (Sk+1 , Xk+1 ))} Definition: Np instability = lim (E ( )) p→∞ p Proposition: For Q finite, instability is well defined, does not depend on the origin distribution of S0 , and we have: Np instability = E( lim ) p→∞ p Proof: Consequence of the Ergodic Theorem and the Lebesgue’s dominated convergence Theorem. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  44. 44. The result Problem. Find a protocol (a pair (f , τ )) university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  45. 45. The result Problem. Find a protocol (a pair (f , τ )) satisfying representativity for a fixed t, university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  46. 46. The result Problem. Find a protocol (a pair (f , τ )) satisfying representativity for a fixed t, which minimizes the instability. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  47. 47. The result Problem. Find a protocol (a pair (f , τ )) satisfying representativity for a fixed t, which minimizes the instability. Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by: - f (x, s) = s if s > t, - f (x, s) = dom(x) otherwise. The pair (f , f ) is a solution the problem university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  48. 48. The result Problem. Find a protocol (a pair (f , τ )) satisfying representativity for a fixed t, which minimizes the instability. Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by: - f (x, s) = s if s > t, - f (x, s) = dom(x) otherwise. The pair (f , f ) is a solution the problem Interpretation. Optimal protocol: university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  49. 49. The result Problem. Find a protocol (a pair (f , τ )) satisfying representativity for a fixed t, which minimizes the instability. Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by: - f (x, s) = s if s > t, - f (x, s) = dom(x) otherwise. The pair (f , f ) is a solution the problem Interpretation. Optimal protocol: when it is possible, do not change the consensus value (forced change rule) university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  50. 50. The result Problem. Find a protocol (a pair (f , τ )) satisfying representativity for a fixed t, which minimizes the instability. Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by: - f (x, s) = s if s > t, - f (x, s) = dom(x) otherwise. The pair (f , f ) is a solution the problem Interpretation. Optimal protocol: when it is possible, do not change the consensus value (forced change rule) when a change is forced, change to the dominating value dom(x) of the input x (dominating value rule). university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  51. 51. The result Problem. Find a protocol (a pair (f , τ )) satisfying representativity for a fixed t, which minimizes the instability. Theorem. [CSR’ 11 !!] Let f : Vn × Zm → Zm defined by: - f (x, s) = s if s > t, - f (x, s) = dom(x) otherwise. The pair (f , f ) is a solution the problem Interpretation. Optimal protocol: when it is possible, do not change the consensus value (forced change rule) when a change is forced, change to the dominating value dom(x) of the input x (dominating value rule). only store the current consensus value. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  52. 52. Proof. Main ideas There exists an optimal protocol with forced changes, by postponing unforced changes method. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  53. 53. Proof. Main ideas There exists an optimal protocol with forced changes, by postponing unforced changes method. What happens after a forced change in (xk , sk )? Consider the part of the execution until the next forced change. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  54. 54. Proof. Main ideas There exists an optimal protocol with forced changes, by postponing unforced changes method. What happens after a forced change in (xk , sk )? Consider the part of the execution until the next forced change. Let b be the chosen consensus value in (xk , sk ). We have: #xk (b) ≤ #xk (dom(xk ))xk university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  55. 55. Proof. Main ideas There exists an optimal protocol with forced changes, by postponing unforced changes method. What happens after a forced change in (xk , sk )? Consider the part of the execution until the next forced change. Let b be the chosen consensus value in (xk , sk ). We have: #xk (b) ≤ #xk (dom(xk ))xk if, at any time, #(b) ≤ #(dom(xk )), then b can be replaced by dom(xk ), with no loss of stability. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  56. 56. Proof. Main ideas There exists an optimal protocol with forced changes, by postponing unforced changes method. What happens after a forced change in (xk , sk )? Consider the part of the execution until the next forced change. Let b be the chosen consensus value in (xk , sk ). We have: #xk (b) ≤ #xk (dom(xk ))xk if, at any time, #(b) ≤ #(dom(xk )), then b can be replaced by dom(xk ), with no loss of stability. if it happens that #(b) > #(dom(xk )), then it happens before that #(b) = #(dom(xk )) (symmetry). b ≡ dom(xk ) for the future stability. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  57. 57. Proof. Main ideas There exists an optimal protocol with forced changes, by postponing unforced changes method. What happens after a forced change in (xk , sk )? Consider the part of the execution until the next forced change. Let b be the chosen consensus value in (xk , sk ). We have: #xk (b) ≤ #xk (dom(xk ))xk if, at any time, #(b) ≤ #(dom(xk )), then b can be replaced by dom(xk ), with no loss of stability. if it happens that #(b) > #(dom(xk )), then it happens before that #(b) = #(dom(xk )) (symmetry). b ≡ dom(xk ) for the future stability. it is never worse to choose dom(xk ) than b. QED university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  58. 58. But .... we have cheated. If it happens that #(b) > #(dom(xk )), then it happens that #(b) = #(dom(xk )) ± 1 university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  59. 59. But .... we have cheated. If it happens that #(b) > #(dom(xk )), then it happens that #(b) = #(dom(xk )) ± 1 The patch: put an artificial intermediary state on each edge. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  60. 60. But .... we have cheated. If it happens that #(b) > #(dom(xk )), then it happens that #(b) = #(dom(xk )) ± 1 The patch: put an artificial intermediary state on each edge. The given protocol is optimal in the modified process (even for a finite fixed time). university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  61. 61. But .... we have cheated. If it happens that #(b) > #(dom(xk )), then it happens that #(b) = #(dom(xk )) ± 1 The patch: put an artificial intermediary state on each edge. The given protocol is optimal in the modified process (even for a finite fixed time). It can be deduced that the protocol is asymptotically optimal for the original process. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  62. 62. Conclusion We have exhibited an optimal protocol for the average consensus stability. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  63. 63. Conclusion We have exhibited an optimal protocol for the average consensus stability. This protocol only uses a small memory: Zm . university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)
  64. 64. The end Thanks. university-logo Eric R´mila e CSR, 2011 June, St-Petersburg (Russia)

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