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Average consensus by gossip algorithms with quantized communication

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Average consensus by gossip algorithms with quantized communication Powered By Docstoc
					Average consensus by gossip algorithms
    with quantized communication
                             Paolo Frasca
                       paolo.frasca@polito.it


   joint work with Ruggero Carli, Fabio Fagnani, and Sandro Zampieri

                         Politecnico di Torino




                                            Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 1/2
Consensus over networks

Several agents in a network have to communicate in order to
achieve an agreement about their states.




                                     Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 2/2
Problem Statement

Given a network, or graph G(V, E)

    nodes are agents
    edges are communication links.

Every agent i ∈ V has a state xi (t), evolving in time                        t ∈ N.

                  xi (t + 1) = τ (xi (t); xj (t) s.t. j ∈ Ni ).


τ depends on the topology of the communication graph:
Goal: to design τ such that

                                        N
                                 1
                    lim xi (t) =             xj (0) ∀ i ∈ V
                   t→+∞          N     j=1

This is average consensus!

                                                Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 3/2
Applications & motivations

In distributed control and information theory.

    Data fusion in sensor networks
    Coordination and rendezvous of robots or biological swarms
    Load balancing between processors




                                           Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 4/2
Deterministic and randomized algorithms

Basic algorithms are deterministic
    consist in repeated averaging with (all) neighbors
    with fixed or time varying topology

Recently, the gossip communication schedule has been proposed
    randomized
    communication is in pairs.




                                          Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 5/2
Gossip algorithm

At each time step,

 1. One edge {i, j} is randomly selected
   P(e(t) = {i, j}) = P{i,j} , such that   {i,j} ∈ E   P{i,j} = 1. Let P ∈ RN ×N be
                                    
                                        P{i,j}
                                    
                                          2
                                                   if           {i, j} ∈ E
                      Pij = Pji =
                                     0      otherwise


 2. The two agents insisting on the selected edge average their states

                        xi (t + 1) = (1 − α)xi (t) + αxj (t)
                        xj (t + 1) = (1 − α)xj (t) + αxi (t).

        α ∈ (0, 1).



                                                       Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 6/2
Example

              Edges are randomly selected.
             0.9

             0.8

             0.7

             0.6

             0.5

             0.4

             0.3

             0.2

             0.1

              0
                   0   0.2   0.4   0.6        0.8             1




  This is a typical proximity graph for a sensor network.




                                         Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 7/2
Example

              Edges are randomly selected.
              0.9

              0.8

              0.7

              0.6

              0.5

              0.4

              0.3

              0.2

              0.1

               0
                    0   0.2   0.4   0.6         0.8            1




  This is a typical proximity graph for a sensor network.




                                          Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 7/2
Example

              Edges are randomly selected.
              0.9

              0.8

              0.7

              0.6

              0.5

              0.4

              0.3

              0.2

              0.1

               0
                    0   0.2   0.4   0.6          0.8            1




  This is a typical proximity graph for a sensor network.




                                          Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 7/2
Performance

As performance we mean how near and how fast
the algorithm approaches the average consensus.

The gossip algorithm converges exactly and exponentially fast to the
average consensus.
Define 1 be the N -dimensional vector whose components are 1,
       1   N
xave = N j=1 xj (0)

                               ||x(t) − xave 1||2
           T = sup inf t : P                      ≥                    ≤
                x(0)                ||x(0)||2

   T = Θ(N ) for N → ∞, for the complete graph;
   T = Θ(N 3 ) for N → ∞, for the ring graph.




                                         Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 8/2
Edges communication


Real communication is not perfect between linked neighbors
   there can be errors in measurements;
   sent messages can be lost for some reason;
   in technology, the communication is digital, then with finite
   precision.
We suppose that the agents can exchange information through
digital channels. The simplest model is a uniform quantization,
rounding to the nearest integer.

                     q[x(t)] = round [x(t)].


                                      Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 9/2
Quantization effects

Then ⇒ exact consensus can not be reached.

How can we adapt the gossip algorithm to work with integer
communication?

How much its performance degrades?




                                        Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 10/2
Quantized Gossip algorithm


Let {i, j} be selected at time t. Then

                 xi (t + 1) = xi (t) − αq[xi (t)] + αq[xj (t)]
                                                                                                               (1)
                 xj (t + 1) = xj (t) − αq[xj (t)] + αq[xi (t)],


This algorithm
    preserves the initial average of states,
    approaches near-optimally the consensus.
For simplicity, we assume that edges are selected with equal
probability and that α = 1/2.




                                              Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 11/2
Simulations: Complete graph

Let
         1   N
xave =   N   j=1   xj (0)                                                       Complete graph




         1
V (t) = √ ||x(t) − xave ||2
         N                                      1
                                               10




                                       ave 2
                                 ||
                                 ||x−x
                                −1/2
                                 N
                                                0
                                               10




                                                    0     200            400           600            800           1000           1200
                                                                                        t



                              Quantized (solid lines) and non-quantized communication (dashed).
                              N = 5, 10, 20, 40, 80, average of 50 trials.
                                                        Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 12/2
Simulations: Ring graph


                                                                       Ring graph




                                   1
                           10




                           ave 2
                     ||
                     ||x−x
                    −1/2
                     N
                                   0
                           10




                                       0   2000     4000        6000       8000      10000      12000      14000       16000
                                                                             t



                Quantized (solid lines) and non-quantized communication (dashed).
                N = 5, 10, 20, 40, average of 50 trials.
                                            Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 13/2
Simulations: Random geometric graph

Given N uniform random
points in [0, 1]2 ,                                                      Random geometric graph
they are linked when they
are closer than some
threshold to each other.
                                              1
                                             10




                                     ave 2
                               ||
                               ||x−x
                              −1/2
                               N
                                              0
                                             10




                                                  0   1000     2000        3000       4000       5000       6000       7000       8000
                                                                                        t



                            Quantized (solid lines) and non-quantized communication (dashed).
                            N = 5, 10, 20, 40, average of 50 trials.
                                                       Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 14/2
An exponential bound

   Let P be the matrix of the probability of selecting edges,
   given a vector v, let diagv denote a diagonal matrix having v
   as diagonal.

Let λ be the smallest non zero eigenvalue of diag(P 1) − P .
Then,
                       2          t     2    1
               E[V (t) ] ≤ (1 − λ) V (0) +      .
                                           2λN




                                      Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 15/2
Remarks

Then, in expectation,
      for small t, V (t) decreases exponentially with rate 1 − λ
      (the same of the ideal case!),
                                               1
      eventually saturates to a constant,     2λN .
 1
2λN   depends on N and on the graph topology via λ. Namely,
            1
      λ=   N −1    for the complete graph
           2π 2
      λ=   N3
                  + o(1/N 3 ), for N → ∞, for the ring graph.
Then the saturation level can grow with N .
But this is not apparent in simulations!



                                            Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 16/2
Convergence

We have a stronger result about the convergence of the
algorithm.

Almost surely, there exists Tcon ∈ N such that

              |xi (t) − xj (t)| ≤ 1   ∀ i, j      ∀ t > Tcon ,                                              (2)

 and hence x(t) − xave     ∞   ≤ 1 and N −1/2 x(t) − xave                                 2   ≤ 1/2.




                                           Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 17/2
Symbolic dynamics I

Let ni (t) = 2xi (t) .
This quantity follows an integer dynamics.
                    ni (t)
Since q[xi (t)] =     2      , from (1)


              (ni (t + 1), nj (t + 1)) = g(ni (t), ni+1 (t))
where g : Z2 → Z2 is

                                h   k   k   h
              g(h, k) =           +   ,   +                                   .
                                2   2   2   2




                                           Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 18/2
Symbolic dynamics II

If we define m(t) = min1≤i≤N ni (t)
M (t) = max1≤i≤N ni (t),
D(t) = M (t) − m(t),
we can prove that

    D(t) is non increasing,
    if D(t0 ) ≥ 2, then there exists τ ∈ N such that
    P[D(t0 + τ ) < D(t)] > 0.

This implies that
almost surely there exists Tcon ∈ N such that D(t) < 2, for all t > Tcon .
Then the convergence statement follows.




                                           Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 19/2
Conclusions

   When quantization step is small compared to states
   ⇒ it’s ok to approximate the quantized system with the ideal
   version
   when states are smaller (big t)
   ⇒ granularity becomes important ⇒ we have finite time
   convergence.

We proved convergence to the best that the discrete nature of
the problem allows.
But we do not have tight estimates of Tcon .




                                     Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 20/2
References
[1] Stephen Boyd, Arpita Ghosh, Balaji Prabhakar, and Devavrat Shah. Randomized gossip algorithms. IEEE/ACM
    Trans. Netw., 14(SI):2508–2530, 2006.
[2] Ruggero Carli, Fabio Fagnani, Paolo Frasca, Thomas Taylor, and Sandro Zampieri. Average consensus on
    networks with transmission noise or quantization. In European Control Conference (ECC’07), Kos, 2007.
[3] Fabio Fagnani and Sandro Zampieri. Randomized consensus algorithms over large scale networks. Information
    Theory and Applications Workshop, 2007, pages 150–159, Jan. 29 2007-Feb. 2 2007.
[4] Paolo Frasca, Ruggero Carli, Fabio Fagnani, and Sandro Zampieri. Average consensus by gossip algorithms
    with quantized communication. 2008. submitted to CDC.
[5] Paolo Frasca, Ruggero Carli, Fabio Fagnani, and Sandro Zampieri. Average consensus on networks with
    quantized communication. 2008. submitted.
[6] P. Gupta and P.R. Kumar. The capacity of wireless networks. Information Theory, IEEE Transactions on,
    46(2):388–404, Mar 2000.
[7] Akshay Kashyap, Tamer Basar, and R. Srikant. Quantized consensus. Automatica, 43(7):1192–1203, July 2007.
[8] J. Tsitsiklis. Problems in decentralized decison making and computation. PhD thesis, MIT, 1984.




                                                                  Mathematics and Cognition Seminar, Arizona State University, April 15, 2008 – p. 21/2

				
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