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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