ALMOST SURE CONVERGENCE OF RANDOM GOSSIP ALGORITHMS Giorgio Picci with T. Taylor, ASU Tempe AZ. Wofgang Runggaldier s Birthday, Brixen July 2007 1
CONSENSUS FOR RANDOM GOSSIP ALGORITHMS Consider a finite set of nodes representing say wireless sensors or distributed computing units, can they achieve a common goal by exchanging information only locally? exchanging information locally for the purpose of forming a common estimate of some physical variable x; Each node k forms his own estimate x k (t), t Z + and updates according to exchange of information with a neighbor. Neighboring pairs are chosen randomly Q: will all local estimates {x k (t), k = 1,...,n} converge to the same value as t?. 2
DYNAMICS OF RANDOM GOSSIP ALGORITHMS While two nodes v i and v j are in communication, they exchange information to refine their own estimate using the neighbor s estimate. Model this adjustament in discrete time by a simple symmetric linear relation x i (t + 1) = x i (t) + p(x j (t) x i (t)) x j (t + 1) = x j (t) + p(x i (t) x j (t)) where p is some positive gain parameter modeling the speed of adjustment. For stability need to impose that 1 2p 1 and hence 0 p 1. For p = 1 2 you take the average of the two measurements so that x i(t +1) = x j (t + 1). 3
DYNAMICS OF RANDOM GOSSIP ALGORITHMS The whole coordinate vector x(t) R n evolves according to x(t +1) = A(e)x(t), the matrix A(e) R n n depending on the edge e = v i v j selected at that particular time instant; A(e) = 1 0 0 0.......... 1 p p. 1....... 1. p 1 p... ) T = I n p (1 vi 1 v j )(1 vi 1 v j 1 4
EIGENSPACES The vector 1 vi has the i th entry equal to 1 and zero otherwise. A(e) is a symmetric doubly stochastic matrix. The value 1 2p is a simple eigenvalue associated to the eigenvector (1 vi 1 v j ), A(e)(1 vi 1 v j ) = (1 vi 1 v j ) p(1 vi 1 v j )2 = (1 2p)(1 vi 1 v j ) ( ) the orthogonal (codimension one) subspace 1 vi 1 v j is the eigenspace of the eigenvalue 1. Let 1 := [1,...,1]. Want x(t) to converge to the subspace {1} := {α1; α R}. This would be automatically true for a fixed irreducible d-stochastic matrix. 5
A CONTROLLABILITY LEMMA Lemma 1 Let G = (V,E) be a graph. Then iff G is connected. span{1 vi 1 v j : (v i v j ) E} = 1 ( 1 vi 1 v j ) span {1}; i.e. Corollary 1 Let G = (V,E ) with E E be a subgraph of G. Let {e i : 1 i m } be an ordering of E, and let π denote a permutation of {1,2,,m }. Let B(E,π) = m i=1 A(e πi), where the product is ordered from right to left. Then if and only if G is connected. B(E,π) 1 < 1 6
THE EDGE PROCESS Let Ω = E N, be the space of all semi-infinite sequences taking values in E, and let σ : Ω Ω denote the shift map: σ(e 0,e 1,e 2,,e n, ) = (e 1,e 2,,e n, ). Let ev k : Ω E denote the evaluation on the k th term. Let µ denote an ergodic shift invariant probability measure on Ω, so that the edge process e(k) : ω ev k (ω) is ergodic. Special cases: e(k) is iid, or an ergodic Markov chain. However, what we shall do works for general ergodic processes. Consider the function C : Ω Z R n n, C(ω,t) := A(ev i (ω)) = A(ev 0 (σ i ω)) t 1 i=0 which by stationarity of e obeys the composition rule C(ω,t +s) =C(σ t ω,s)c(ω,t) with C(ω,0) = I. Such a function is called a matrix cocycle. t 1 i=0 7
MULTIPLICATIVE ERGODIC THEOREM Theorem 1 [Oseledet s Multiplicative Ergodic Theorem] Let µ be a shift invariant probability measure on Ω and suppose that the shift map σ : Ω Ω is ergodic and that log + C(ω,t) is in L 1. Then the limit Λ = lim t ( C(ω,t) T C(ω,t)) 1 2t (1) exists with probability one, is symmetric and nonnegative definite, and is µ a.s. independent of ω. Let λ 1 < λ 2 < λ k for k n be the distinct eigenvalues of Λ, let U i denote the eigenspace of λ i, and let V i = i j=1 U j. Then for u V i V i 1, 1 lim t t log C(ω,t)u = log(λ i). (2) The numbers λ i are called the Lyapunov exponents of C. 8
MULTIPLICATIVE ERGODIC THEOREM The Lyapunov exponents control the exponential rate of convergence (or non-convergence) to consensus. The matrices A(e) are doubly stochastic as are any matrix products of them, C(ω,t). If follows that the constant functions on V, {1}, as well as the mean zero functions in {1} are invariant under the action of this cocycle and of its transpose. Thus these subspaces are also invariant under the limiting matrix Λ of the Oseledet s theorem. There is a Lyapunov exponent associated with the subspace {1} which, it is not difficult to see, is one. There are n 1 Lyapunov exponents associated with the subspace 1, so the key point is to characterize them. 9
CONVERGENCE TO CONSENSUS For x R n use the symbol x := 1 n ni=1 x i. The main convergence result follows. Theorem 2 Let G = (V,E) be a connected graph and let e(t) be an ergodic stochastic process taking values on E. Suppose that the support of the probability distribution induced by e(t) is all of E. Let the gossip algorithm be initialized at x(0) = x 0. Then there is a (deterministic) constant λ < 1 and a (random) constant K λ such that µ-almost surely. x(t) x 0 1 < K λ λ t x 0 x 0 1 10
OPEN QUESTIONS Rate of convergence (for L 2...) Multiple gossiping : more than one pair of communicating edges per time slot, Convergence is merely associated to the time T it takes the algorithm to visit a spanning tree with positive probability. Indeed, the actual rate of convergence of the algorithm is just determined by T. Much remains to be done!!! 11
REFERENCES W. Runggaldier (circa 1970): STILLE WASSER GRUNDEN TIEF, unpublished (although well known among specialists). 12