Quantized Average Consensus on Gossip Digraphs

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1 Quantized Average Consensus on Gossip Digraphs Hideaki Ishii Tokyo Institute of Technology Joint work with Kai Cai Workshop on Uncertain Dynamical Systems Udine, Italy August 25th, 2011

2 Multi-Agent Consensus Flocks of fish/birds Formation of autonomous robots/ mobile sensor networks Distributed randomized PageRank algorithm for ranking webpages Ishii & Tempo (2010) 2

3 Some basic questions: Multi-Agent Consensus What are the necessary network connectivity for achieving consensus? Is it possible to enhance performance/capabilities of the overall system by introducing extra dynamics in agents? E.g. Acceleration of convergence in consensus Liu, Anderson, Cao, & Morse (2009) Focus of this talk: Average consensus on directed graphs with communication constraints 3

4 Average Consensus: Introduction Agent i Edge Network of n agents on a directed graph (digraph) Each agent updates its state based on neighbors info All states t must converge to the average of their initial iti values Motivation: Sensor networks 4

5 Known Conditions on Digraphs When the states are real valued Update law: L: Graph Laplacian Average Consensus: Graph is strongly connected and balanced The matrix I-L becomes doubly stochastic Can this condition be relaxed? Not balanced Balanced Olfati-Saber & Murray (2004) 5

6 Recent Approaches for General Digraphs 1. Cooperative algorithm to make doubly stochastic 2. Use of variables in addition to states in agents Gharesifard & Cortes (2011) Computation of stationary distributions of Markov chains Benezit, Blondel, l Thiran, Tsitsiklis, ikli & Vetterli (2010) Our approach: Conventional consensus based Uses local variables that record changes in states 6

7 Communication Constraint 1 Agent i Edge Quantized states: Integer valued Model of finite data in communication and computation The average value may not be an integer nor unique: or Kashap, Basar, & Srikant (2007), Carli, Fagnani, Frasca, & Zampieri (2010) 7

8 Communication Constraint 2 Agent j Agent i Gossip Algorithm At each time instant, one edge is chosen randomly Asynchronous protocol for distributed systems Boyd, Ghosh, h Prabhakar, & Shah h (2006) 8

9 Simpler Case: Quantized Consensus Agent j Agent i Only agreement in the states (no averaging) Distributed algorithm If, then If, then If, then 9

10 Quantized Consensus Theorem: For each initial state, there exists a finite such that with prob. 1. The underlying graph has a globally reachable node. A node from which h there is a directed d path to every other node in the graph 10

11 Discussion Randomization is crucial for quantized states case. With this algorithm, average consensus is not possible because the state sum can vary over time: Hence, the true average is lost from the system. Key Idea: The agents must be aware of how much state change was made in the past. 11

12 Towards Obtaining the Average Additional elements for each agent i Surplus Locally keeps track of state changes Initial value Threshold Determines when to use surplus in state updates Simple choice: Local minimum & maximum: Keep the state bounded 12

13 Quantized Average Consensus Agent j Agent i Distributed algorithm Surplus: Surplus of agent j is transferred to i State: If, then If, then Change in the state t at time k 13

14 Quantized Average Consensus Agent j Agent i If, then there are three cases: If and local max, then If and local min, then Otherwise, 14

15 Numerical Example Network of 50 agents on a random digraph Initial values: Uniformly distributed in [-5,5] 5] Consensus but below the average Quantized Average Large surplus Surplus changes even after consensus 15

16 The Role of Surplus Sum of states and surpluses remains constant: Even after average consensus, nonzero surplus may be passed around. If states are in consensus but below average, then surplus will eventually be collected at an agent i as This means too much surplus in the system. 16

17 Quantized Average Consensus: Result Theorem: For each initial state, there exists a finite such that or with prob. 1. The underlying graph is strongly connected. 17

18 Quantized Average Consensus: Result Average consensus is possible for general directed graphs, where state sum can be varying. The use of surplus variables is essential. Condition on graphs: Balanced structure is no longer needed. Proof is based on finite Markov chain arguments. 18

19 Discussion Scalability: Exact (quantized) average is obtained for any number of agents. Tradeoffs More communication and local computation are required. Convergence time may be slow. More updates are needed d even after the agents arrive at consensus (not at the average). 19

20 Threshold Range may not be realistic in an uncertain environment. How sensitive is the algorithm to the choice of? Theorem: The algorithm achieves quantized average Threshold satisfies 20

21 Threshold vs Consensus Values The values that the agents potentially agree on. Quantized Average Threshold 21

22 Threshold vs Convergence Time Convergence is faster for smaller. This is because the decision to distribute surpluses can be made earlier. For a complete digraph with 50 agents Convergence Time Threshold 22

23 Convergence Time Analysis How does the convergence time scale with the number n of agents? Given initial states : Time to reach quantized average consensus Random variable Find a bound on the mean convergence time: Difficulty: Complicated dynamics of states and surpluses 23

24 Convergence Time Analysis: Result Simple case: Complete digraph Theorem: Proof is based on the Lyapunov function: Conventional one Good surplus Bad surplus The problem is then reduced to hitting time analysis of a Markov chain. 24

25 Convergence Time: Comparison Undirected Directed & Balanced Directed Complete Cyclic General Zhu & Martinez (2008) Nedic, Olshevsky, Ozdaglar, & Tsitsiklis (2009) This work Asynchronous ous Synchronous ous Asynchronous ous 25

26 Numerical Example Convergence Time Random Geometric Digraphs Complete Digraphs Number of Agents 26

27 Further Studies: Real-Valued Case Agent j Agent i Distributed algorithm Surplus: Same as quantized case State: Usual consensus Surplus 27

28 Further Studies: Real-Valued Case Average consensus on general strongly connected digraphs can be achieved for sufficiently small. Surplus variables play similar roles. Linear update laws for the state and surplus, but the system matrix is not stochastic. Analysis based on matrix perturbation theory. Franceschelli, Giua, & Seatzu (2009) 28

29 Conclusion Multi-agent average consensus with quantized states Distributed ib t randomized d algorithm via gossiping i Necessary and sufficient condition on graph structure Main message: The overall system capability can be enhanced by adding more dynamics to agents. 29