Asymptotics, asynchrony, and asymmetry in distributed consensus

Transcription

1 DANCES Seminar 1 / Asymptotics, asynchrony, and asymmetry in distributed consensus Anand D. Information Theory and Applications Center University of California, San Diego 9 March 011 Joint work with Alex G. Dimakis, Tuncer Can Aysal, Mehmet Ercan Yildiz, Martin Wainwright, and Anna Scaglione, and Tara Javidi

2 DANCES Seminar > Introduction / Rapprochement, consensus, accord

3 DANCES Seminar > Introduction / Rapprochement, consensus, accord

4 DANCES Seminar > Introduction / Rapprochement, consensus, accord

5 DANCES Seminar > Introduction / Rapprochement, consensus, accord

6 DANCES Seminar > Introduction / Consensus is an important task Calibration Dissemination Coordination

7 DANCES Seminar > Introduction / Abstracting the task

8 DANCES Seminar > Introduction / Abstracting the task Network of agents, each with an observation

9 DANCES Seminar > Introduction / Abstracting the task Network of agents, each with an observation Communicate locally exchange messages about observations

10 DANCES Seminar > Introduction / Abstracting the task Network of agents, each with an observation Communicate locally exchange messages about observations Compute locally estimate a function of all values

11 DANCES Seminar > Introduction / There are many aspects to consider

12 DANCES Seminar > Introduction / There are many aspects to consider What are observations? continuous or discrete? scalar or vector?

13 DANCES Seminar > Introduction / There are many aspects to consider What are observations? continuous or discrete? scalar or vector? How can we communicate? point-to-point or broadcast? low resolution or high resolution?

14 DANCES Seminar > Introduction / There are many aspects to consider What are observations? continuous or discrete? scalar or vector? How can we communicate? point-to-point or broadcast? low resolution or high resolution? What do we compute? averages medians, quantiles convex optimization

15 DANCES Seminar > Introduction / The goal(s) for today x1(t) W1,i x(t) W,i x(t) xi(t) W,i x(t) W,i W,i x(t) 1 The basic mathematical model for consensus Routing and mobility can speed up convergence Broadcasting can trade off accuracy for speed The discreet charm of discrete messages

16 DANCES Seminar > A simple mathematical model / Building a mathematical model

17 DANCES Seminar > A simple mathematical model / The data model

18 DANCES Seminar > A simple mathematical model / The data model Set of n agents

19 DANCES Seminar > A simple mathematical model / The data model Set of n agents Agent i observes initial value x i (0) R for i = 1,... n

20 DANCES Seminar > A simple mathematical model / The data model Set of n agents Agent i observes initial value x i (0) R for i = 1,... n Assume data is bounded : x i (0) [0, 10], for example

21 DANCES Seminar > A simple mathematical model 9 / The communication graph

22 DANCES Seminar > A simple mathematical model 9 / The communication graph Agents are arranged in a graph G = (V, E).

23 DANCES Seminar > A simple mathematical model 9 / The communication graph Agents are arranged in a graph G = (V, E). Agents i can communicate with j if there is an edge (i, j) (e.g. j N i ).

24 DANCES Seminar > A simple mathematical model 9 / The communication graph Agents are arranged in a graph G = (V, E). Agents i can communicate with j if there is an edge (i, j) (e.g. j N i ). Bidirectional communication : agents exchange messages.

25 DANCES Seminar > A simple mathematical model 10 / Constraints on the communication

26 DANCES Seminar > A simple mathematical model 10 / Constraints on the communication Time is slotted : only one transmission per slot.

27 DANCES Seminar > A simple mathematical model 10 / Constraints on the communication Time is slotted : only one transmission per slot. Synchronous : use many edges, then update.

28 DANCES Seminar > A simple mathematical model 10 / Constraints on the communication Time is slotted : only one transmission per slot. Synchronous : use many edges, then update. Asynchronous : edges chosen randomly in each slot.

29 DANCES Seminar > A simple mathematical model 11 / Measuring performance The goal is to pass messages between agents such that they can estimate the average of the initial observations: ( ) x(t) x i (0) 1 i Averaging time T ave (n, ɛ) is time when x(t) is within ɛ of the average: { ( ) } x(t) xave 1 T ave (n, ɛ) = sup inf P Alg ɛ ɛ x(0) t x(0)

30 DANCES Seminar > A simple mathematical model 1 / A centralized solution 1 9 Simple centralized algorithm:

31 DANCES Seminar > A simple mathematical model 1 / A centralized solution 1 9 Simple centralized algorithm: 1 Build a spanning tree

32 DANCES Seminar > A simple mathematical model 1 / A centralized solution 1 9 Simple centralized algorithm: 1 Build a spanning tree Gather all the values at root

33 DANCES Seminar > A simple mathematical model 1 / A centralized solution Simple centralized algorithm: 1 Build a spanning tree Gather all the values at root Compute and disseminate average

34 DANCES Seminar > A simple mathematical model 1 / A centralized solution Simple centralized algorithm: 1 Build a spanning tree Gather all the values at root Compute and disseminate average Pro: requires Θ(n) messages Con: completely centralized

35 DANCES Seminar > A simple mathematical model 1 / Distributed synchronous consensus Suppose each agent linearly combines itself and its neighbors: x i (t + 1) = W ii x i (t) + W ij x j (t) j N i W ij = 1 i j W ij = W ji Synchronous algorithm where the update after each slot is given by: x(t + 1) = W x(t) where W is a doubly stochastic matrix.

36 DANCES Seminar > A simple mathematical model 1 / A simple result Theorem For synchronous consensus with update matrix W, T ave (n, ɛ) = Θ ( E T relax (W ) log ɛ 1) where T relax (W ) is the relaxation time of the matrix W : T relax (W ) = 1 1 λ (W ). Proof : W is the transition matrix of a Markov chain consensus is the convergence of the chain to its stationary distribution.

37 DANCES Seminar > A simple mathematical model 1 / A theme with variations Survey article by Dimakis et al. in Proc. IEEE. Synchronous DeGroot (19), Tsitsiklis (19) Time-varying topologies Chatterjee-Seneta (19), Tsitsiklis et al. (19), Jadbabaie et al. (00), Ren-Beard (00), Gao-Cheng (00), Fagnani-Zampieri (00) Asynchronous Boyd et al. (00) Quantization Kashyap et al. (00), Nedic et al. (009), Yildiz-Scaglione (00), Aysal et. al (009), Kar-Moura (010), Carli et al. (010), Lavaie-Murray (010) Discrete values Benezit et al. (010) Many others!

38 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip

39 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random.

40 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average.

41 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average. Set x i (t + 1) = x j (t + 1) = 1 (x i(t) + x j (t)).

42 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average. Set x i (t + 1) = x j (t + 1) = 1 (x i(t) + x j (t)).

43 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average. Set x i (t + 1) = x j (t + 1) = 1 (x i(t) + x j (t)).

44 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average. Set x i (t + 1) = x j (t + 1) = 1 (x i(t) + x j (t)).

45 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average. Set x i (t + 1) = x j (t + 1) = 1 (x i(t) + x j (t)).

46 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average. Set x i (t + 1) = x j (t + 1) = 1 (x i(t) + x j (t)).

47 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average. Set x i (t + 1) = x j (t + 1) = 1 (x i(t) + x j (t)).

48 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average. Set x i (t + 1) = x j (t + 1) = 1 (x i(t) + x j (t)).

49 DANCES Seminar > A simple mathematical model 1 / Asynchronous updates = gossip Node i wakes up at random, chooses neighbor j at random. Nodes i and j exchange x i (t) and x j (t) and compute average. Set x i (t + 1) = x j (t + 1) = 1 (x i(t) + x j (t)).

50 DANCES Seminar > A simple mathematical model 1 / Gossip uses random linear updates At each time a random pair (i, j) E averages: x i (t + 1) = x j (t + 1) = x i(t) + x j (t). Each update is linear : x(t + 1) = W (i,j) (t)x(t). Theorem Let W = E[W (i,j) ] over the edge selection process. Then T ave (n, ɛ) = Θ ( T relax ( W ) log ɛ 1)

51 DANCES Seminar > A simple mathematical model 1 / The implication for big graphs For the grid with uniform selection, gossip takes Θ(n ) transmissions! Selecting edges at random is inefficient! Local exchange is inefficient!

52 DANCES Seminar > Shrinking the graph 19 / Network properties can accelerate convergence Joint work with Alex Dimakis and Martin Wainwright

53 DANCES Seminar > Shrinking the graph 0 / Geographic gossip with routing

54 DANCES Seminar > Shrinking the graph 0 / Geographic gossip with routing Assume that packets can be routed between any two nodes.

55 DANCES Seminar > Shrinking the graph 0 / Geographic gossip with routing Assume that packets can be routed between any two nodes. Now select neighbor uniformly from all nodes and route message.

56 DANCES Seminar > Shrinking the graph 0 / Geographic gossip with routing Assume that packets can be routed between any two nodes. Now select neighbor uniformly from all nodes and route message. Effective graph is now the complete graph.

57 DANCES Seminar > Shrinking the graph 1 / Example : the grid algorithm T relax ( W )

58 DANCES Seminar > Shrinking the graph 1 / Example : the grid algorithm T relax ( W ) Local Θ(n )

59 DANCES Seminar > Shrinking the graph 1 / Example : the grid algorithm T relax ( W ) Local Θ(n ) With routing Θ(n)

60 DANCES Seminar > Shrinking the graph 1 / Example : the grid algorithm T relax ( W ) Local Θ(n ) With routing Θ(n) This is unfair, since routing costs in number of hops.

61 DANCES Seminar > Shrinking the graph / One-hop transmissions to reach consensus Count number of hops (power) to get within ɛ of the average: algorithm one-hop transmission

62 DANCES Seminar > Shrinking the graph / One-hop transmissions to reach consensus Count number of hops (power) to get within ɛ of the average: algorithm one-hop transmission Local Θ(n ) Boyd et al.

63 DANCES Seminar > Shrinking the graph / One-hop transmissions to reach consensus Count number of hops (power) to get within ɛ of the average: algorithm one-hop transmission Local Θ(n ) Boyd et al. With routing Θ(n / ) Dimakis,, Wainwright

64 DANCES Seminar > Shrinking the graph / One-hop transmissions to reach consensus Count number of hops (power) to get within ɛ of the average: algorithm one-hop transmission Local Θ(n ) Boyd et al. With routing Θ(n / ) Dimakis,, Wainwright Average on the way Θ(n) Benezit et al.

65 DANCES Seminar > Shrinking the graph / Gossip with mobility

66 DANCES Seminar > Shrinking the graph / Gossip with mobility Start with a grid of static nodes.

67 DANCES Seminar > Shrinking the graph / Gossip with mobility Start with a grid of static nodes. Add m fully mobile nodes.

68 DANCES Seminar > Shrinking the graph / Gossip with mobility Start with a grid of static nodes. Add m fully mobile nodes.

69 DANCES Seminar > Shrinking the graph / Gossip with mobility Start with a grid of static nodes. Add m fully mobile nodes. At each time, m mobile nodes choose new locations uniformly at random.

70 DANCES Seminar > Shrinking the graph / Gossip with mobility Start with a grid of static nodes. Add m fully mobile nodes. At each time, m mobile nodes choose new locations uniformly at random.

71 DANCES Seminar > Shrinking the graph / Gossip with mobility Start with a grid of static nodes. Add m fully mobile nodes. At each time, m mobile nodes choose new locations uniformly at random.

72 DANCES Seminar > Shrinking the graph / Gossip with mobility

73 DANCES Seminar > Shrinking the graph / Gossip with mobility Same local transmission model.

74 DANCES Seminar > Shrinking the graph / Gossip with mobility Same local transmission model. Mobile nodes reduce effective diameter to.

75 DANCES Seminar > Shrinking the graph / Gossip with mobility Same local transmission model. Mobile nodes reduce effective diameter to. Mobile nodes are accessed rarely.

76 DANCES Seminar > Shrinking the graph / Lower bounds on T relax ( W )

77 DANCES Seminar > Shrinking the graph / Lower bounds on T relax ( W ) Merge all mobile nodes into a super node.

78 DANCES Seminar > Shrinking the graph / Lower bounds on T relax ( W ) Merge all mobile nodes into a super node. T relax for induced chain T relax for original chain.

79 DANCES Seminar > Shrinking the graph / Lower bounds on T relax ( W ) Merge all mobile nodes into a super node. T relax for induced chain T relax for original chain. At most a m-factor improvement.

80 DANCES Seminar > Shrinking the graph / Upper bounds on T relax ( W ) π(i) π(i)wik π(j) Use a flow argument and the Poincaré inequality:

81 DANCES Seminar > Shrinking the graph / Upper bounds on T relax ( W ) π(i) π(i)wik π(j) Use a flow argument and the Poincaré inequality: Demands D ij = π(i)π(j) = n between each pair of nodes.

82 DANCES Seminar > Shrinking the graph / Upper bounds on T relax ( W ) π(i) π(i)wik π(j) Use a flow argument and the Poincaré inequality: Demands D ij = π(i)π(j) = n between each pair of nodes. Capacity C ik = π(i) W ik = n 1 Wik between each edge.

83 DANCES Seminar > Shrinking the graph / Upper bounds on T relax ( W ) D ij C ik D ij Use a flow argument and the Poincaré inequality: Demands D ij = π(i)π(j) = n between each pair of nodes. Capacity C ik = π(i) W ik = n 1 Wik between each edge.

84 DANCES Seminar > Shrinking the graph / Upper bounds on T relax ( W ) D ij C ik D ij Use a flow argument and the Poincaré inequality: Demands D ij = π(i)π(j) = n between each pair of nodes. Capacity C ik = π(i) W ik = n 1 Wik between each edge. Route flows i j to minimize overload on each edge.

85 DANCES Seminar > Shrinking the graph / Upper bounds on T relax ( W ) D ij C ik D ij Use a flow argument and the Poincaré inequality: Demands D ij = π(i)π(j) = n between each pair of nodes. Capacity C ik = π(i) W ik = n 1 Wik between each edge. Route flows i j to minimize overload on each edge.

86 DANCES Seminar > Shrinking the graph / Network effects on convergence algorithm transmissions

87 DANCES Seminar > Shrinking the graph / Network effects on convergence algorithm transmissions Local Θ(n ) Boyd et al.

88 DANCES Seminar > Shrinking the graph / Network effects on convergence algorithm transmissions Local Θ(n ) Boyd et al. With routing Θ(n / ) Dimakis--Wainwright

89 DANCES Seminar > Shrinking the graph / Network effects on convergence algorithm transmissions Local Θ(n ) Boyd et al. With routing Θ(n / ) Dimakis--Wainwright Average on the way Θ(n) Benezit et al.

90 DANCES Seminar > Shrinking the graph / Network effects on convergence algorithm transmissions Local Θ(n ) Boyd et al. With routing Θ(n / ) Dimakis--Wainwright Average on the way Θ(n) ) Benezit et al. Add m mobile Θ -Dimakis ( n m

91 DANCES Seminar > Shrinking the graph / Network effects on convergence algorithm transmissions Local Θ(n ) Boyd et al. With routing Θ(n / ) Dimakis--Wainwright Average on the way Θ(n) Benezit et al. ( ) Add m mobile Θ n -Dimakis k-local O ( m ) n k -Dimakis

92 DANCES Seminar > Trading accuracy for speed / Asymmetric gossip using broadcasting Joint work with T.C. Aysal, M.E. Yildiz and A. Scaglione

93 DANCES Seminar > Trading accuracy for speed 9 / Wireless is inherently broadcast

94 DANCES Seminar > Trading accuracy for speed 9 / Wireless is inherently broadcast In a wireless network, all neighbors can hear a transmission.

95 DANCES Seminar > Trading accuracy for speed 9 / Wireless is inherently broadcast In a wireless network, all neighbors can hear a transmission. Can perform multiple computations per slot.

96 DANCES Seminar > Trading accuracy for speed 9 / Wireless is inherently broadcast In a wireless network, all neighbors can hear a transmission. Can perform multiple computations per slot. When graph is well-connected, can get performance gains.

97 DANCES Seminar > Trading accuracy for speed 0 / Gossip in one direction

98 DANCES Seminar > Trading accuracy for speed 0 / Gossip in one direction All neighbors j N i of node i can hear transmission.

99 DANCES Seminar > Trading accuracy for speed 0 / Gossip in one direction All neighbors j N i of node i can hear transmission. Can do a simultaneous update x j (t + 1) = γx j (t) + (1 γ)x i (t).

100 DANCES Seminar > Trading accuracy for speed 0 / Gossip in one direction All neighbors j N i of node i can hear transmission. Can do a simultaneous update x j (t + 1) = γx j (t) + (1 γ)x i (t). 1 9

101 DANCES Seminar > Trading accuracy for speed 0 / Gossip in one direction All neighbors j N i of node i can hear transmission. Can do a simultaneous update x j (t + 1) = γx j (t) + (1 γ)x i (t). 1 9

102 DANCES Seminar > Trading accuracy for speed 0 / Gossip in one direction All neighbors j N i of node i can hear transmission. Can do a simultaneous update x j (t + 1) = γx j (t) + (1 γ)x i (t). No information exchange can get consensus (agreement) but not the true average.

103 DANCES Seminar > Trading accuracy for speed 1 / Analyzing the broadcast gossip algorithm Again, update given by a matrix multiplication: ( T x(t ) = t=1 W (it) ) x(0) For all t we have W (it) 1 = 1, so consensus is stable.

104 DANCES Seminar > Trading accuracy for speed / Benefits and challenges of broadcast No coordination to exchange data. Exploits potential long-range connections from shadowing/fading. No convergence to true average, but to consensus. Important to control the MSE of the consensus.

105 DANCES Seminar > Trading accuracy for speed / Main results Algorithm reaches consensus almost surely: ( ) P lim x(t) = c1 = 1. t The expected consensus value is the true average: E[c] = x Moreover, there is a closed form for the limiting MSE.

106 DANCES Seminar > Trading accuracy for speed / Simulations : MSE Per Node MSE Standard N=0 Geographic N=0 Broadcast N= Number of Radio Transmissions

107 DANCES Seminar > Trading accuracy for speed / Simulations : MSE 10 0 Randomized N=100 Geographic N=100 Broadcast N=100 Per Node MSE Number of Radio Transmissions

108 DANCES Seminar > Trading accuracy for speed / Extensions

109 DANCES Seminar > Trading accuracy for speed / Extensions Can look at effect of the wireless medium as well.

110 DANCES Seminar > Trading accuracy for speed / Extensions Can look at effect of the wireless medium as well. Fading allows long-distance connections.

111 DANCES Seminar > Trading accuracy for speed / Extensions Can look at effect of the wireless medium as well. Fading allows long-distance connections. Initial results suggest significant improvement when path loss is not too severe.

112 DANCES Seminar > Trading accuracy for speed / Implications

113 DANCES Seminar > Trading accuracy for speed / Implications Broadcasting is simpler than standard gossip no exchange.

114 DANCES Seminar > Trading accuracy for speed / Implications Broadcasting is simpler than standard gossip no exchange. More robust to packet drops which may occur in wireless.

115 DANCES Seminar > Trading accuracy for speed / Implications Broadcasting is simpler than standard gossip no exchange. More robust to packet drops which may occur in wireless. Faster convergence in small-to-medium networks.

116 DANCES Seminar > Discrete consensus / k k +1 R R Reaching consensus discretely Joint work with Tara Javidi

117 DANCES Seminar > Discrete consensus / Typical assumptions are unrealistic Existing work doesn t look practical :

118 DANCES Seminar > Discrete consensus / Typical assumptions are unrealistic Existing work doesn t look practical : Transmit and receive real numbers

119 DANCES Seminar > Discrete consensus / Typical assumptions are unrealistic Existing work doesn t look practical : Transmit and receive real numbers Consensus is the only goal of the network

120 DANCES Seminar > Discrete consensus / Typical assumptions are unrealistic Existing work doesn t look practical : Transmit and receive real numbers Consensus is the only goal of the network Asymptotics and universality

121 DANCES Seminar > Discrete consensus 9 / Synchronous quantized communication R R R R R R R R R R R R R R R R 1 0 9

122 DANCES Seminar > Discrete consensus 9 / Synchronous quantized communication R R R R R R R R R R R R R R R R At each time t all neighbors (i, j) exchange quantized values ˆx j (t).

123 DANCES Seminar > Discrete consensus 9 / Synchronous quantized communication R R R R R R R R R R R R R R R R At each time t all neighbors (i, j) exchange quantized values ˆx j (t). Messages i j and j i must take no more than R bits.

124 DANCES Seminar > Discrete consensus 9 / Synchronous quantized communication R R R R R R R R R R R R R R R R At each time t all neighbors (i, j) exchange quantized values ˆx j (t). Messages i j and j i must take no more than R bits. Update x i (t + 1) as a function of x i (t) and messages {ˆx j (t)}.

125 DANCES Seminar > Discrete consensus 0 / A simple protocol x 1(t) W 1,i x (t) W,i x (t) x (t) W,i x i(t) W,i W,i x (t) x i (t + 1) = (x i (t) ˆx i (t)) + j N i {i} W ij ˆx j (t). Quantization error plus weighted sum of messages Iterations preserve sum i x i(t)

126 DANCES Seminar > Discrete consensus 1 / So how well does it work?

127 DANCES Seminar > Discrete consensus 1 / So how well does it work? Random topology, 9 nodes, good connectivity 10 Log error Iterations

128 DANCES Seminar > Discrete consensus 1 / So how well does it work? Random topology, 9 nodes, poor connectivity 10 Log error Iterations

129 DANCES Seminar > Discrete consensus 1 / So how well does it work? 10 Grid, 100 nodes Log error Iterations

130 DANCES Seminar > Discrete consensus / Observations t t + 1 Quantization is important for practical applications. Average consensus to within reasonable resolution can be fast. Overhead can be reduced by piggybacking on existing traffic.

131 DANCES Seminar > Conclusions / Conclusions x1(t) W1,i x(t) W,i x(t) xi(t) W,i x(t) W,i W,i x(t) Algorithm can use network resources to accelerate convergence. Reaching consensus may be faster than computing averages. Lower-resolution averages can be fast and require less overhead.

132 DANCES Seminar > Conclusions / Some challenges for the future IEEE 0.xyz.pdq Implementing consensus in protocols for applications. Extending to other distributed computation problems. Quantifying robustness in rate, connectivity, etc.

133 DANCES Seminar > Conclusions / Thank you!