Quantized Average Consensus on Gossip Digraphs Hideaki Ishii Tokyo - - PowerPoint PPT Presentation

Quantized Average Consensus

• n Gossip Digraphs

Hideaki Ishii

Tokyo Institute of Technology Joint work with Kai Cai

Workshop on Uncertain Dynamical Systems Udine, Italy August 25th, 2011

Multi-Agent Consensus

Fl k f fi h/bi d F ti f t b t / Flocks of fish/birds Formation of autonomous robots/ mobile sensor networks

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Distributed randomized PageRank algorithm for ranking webpages

Ishii & Tempo (2010)

Multi-Agent Consensus

Some basic questions:  What are the necessary network connectivity for achieving consensus? achieving consensus?  Is it possible to enhance performance/capabilities of the

• verall system by introducing extra dynamics in agents?

 E g Acceleration of convergence in consensus  E.g. Acceleration of convergence in consensus

Liu, Anderson, Cao, & Morse (2009)

Focus of this talk: Average consensus on directed graphs

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with communication constraints

Average Consensus: Introduction

Edge Agent i

 Network of n agents on a directed graph (digraph)  Each agent updates its state based on neighbors’ info All t t t t th f th i i iti l l  All states must converge to the average of their initial values  Motivation: Sensor networks

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Known Conditions on Digraphs

 When the states are real valued  Update law:

L: Graph Laplacian

 Update law:  Average Consensus:

L: Graph Laplacian

Graph is strongly connected and balanced

The matrix I-L becomes doubly stochastic

Can this condition b l d? be relaxed?

Not balanced Balanced

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Olfati-Saber & Murray (2004)

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

B it Bl d l Thi T it ikli & V tt li (2010) Benezit, Blondel, Thiran, Tsitsiklis, & Vetterli (2010)

Our approach:  Conventional consensus based  Uses local variables that record changes in states

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Communication Constraint 1

Edge Edge Agent i

Quantized states: Integer valued  Model of finite data in communication and computation  Model of finite data in communication and computation  The average value may not be an integer nor unique:

• r

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Kashap, Basar, & Srikant (2007), Carli, Fagnani, Frasca, & Zampieri (2010)

Communication Constraint 2

Agent i Agent j

Gossip Algorithm  At each time instant, one edge is chosen randomly  Asynchronous protocol for distributed systems

B d Gh h P bh k & Sh h (2006)

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Boyd, Ghosh, Prabhakar, & Shah (2006)

Simpler Case: Quantized Consensus

Agent i Agent j

 Only agreement in the states (no averaging) Distributed algorithm  If then  If , then  If , then

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 If , then

Quantized Consensus

Theorem: For each initial state, there exists a finite such that with prob 1 with prob. 1. The underlying graph has a globally reachable node.  A d f hi h th i di t d th t th  A node from which there is a directed path to every other node in the graph

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Discussion

 Randomization is crucial for quantized states case.  With this algorithm, average consensus is not possible because the state sum can vary over time: 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.

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Towards Obtaining the Average

Additional elements for each agent i Surplus  Locally keeps track of state changes  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

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Quantized Average Consensus

Agent i Agent j

Distributed algorithm  Surplus:

Surplus of agent j is transferred to i

 Surplus:

Ch i th t t t ti k Surplus of agent j is transferred to i

 State:  If , then

Change in the state at time k

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If , then  If , then

Quantized Average Consensus

Agent i Agent j

 If , then there are three cases:  If and local max then  If and local max, then  If and local min, then

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 Otherwise,

Numerical Example

 Network of 50 agents on a random digraph  Initial values: Uniformly distributed in [ 5 5]  Initial values: Uniformly distributed in [-5,5]

Consensus but below the average Quantized Average Large surplus Average g p

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Surplus changes even after consensus

The Role of Surplus

 Sum of states and surpluses remains constant: f  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 surplus will eventually be collected at an agent i as This means too much surplus in the system.

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Quantized Average Consensus: Result

Theorem: For each initial state, there exists a finite such that

• r
• r

with prob. 1. with prob. 1. The underlying graph is strongly connected.

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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  The use of surplus variables is essential.  Condition on graphs: Balanced structure is no longer needed.  Proof is based on finite Markov chain arguments. g

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Discussion

 Scalability: Exact (quantized) average is obtained for any b f t number of agents. Tradeoffs  More communication and local computation are required. p q Convergence time may be slow.  M d t d d ft th t i t  More updates are needed even after the agents arrive at consensus (not at the average).

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Threshold Range

 may not be realistic in an uncertain environment. H iti i th l ith t th h i f ?  How sensitive is the algorithm to the choice of ? Theorem: The algorithm achieves quantized average The algorithm achieves quantized average Threshold satisfies

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Threshold vs Consensus Values

 The values that the agents potentially agree on.

Quantized Average

Threshold

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Threshold vs Convergence Time

 Convergence is faster for smaller .  This is because the decision to distribute surpluses can  This is because the decision to distribute surpluses can be made earlier.

For a complete digraph with 50 agents

Convergence Time

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Threshold

Convergence Time Analysis

 How does the convergence time scale with the number n of agents? 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

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p y p

Convergence Time Analysis: Result

 Simple case: Complete digraph Th Theorem:  Proof is based on the Lyapunov function:

“Good” “Bad” Conventional one

The problem is then reduced to hitting time analysis of

surplus surplus Conventional one

a Markov chain.

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Convergence Time: Comparison

Directed & Undirected Directed & Balanced Directed Complete Cyclic G l General

Zhu & Martinez Nedic, Olshevsky,

This work

Zhu & Martinez (2008) Nedic, Olshevsky, Ozdaglar, & Tsitsiklis (2009)

This work

Asynchronous Asynchronous Synchronous sy c

• ous

sy c

• ous

Sy c

• ous

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Numerical Example

Convergence Time R d G t i Time Random Geometric Digraphs Complete Digraphs Number of Agents

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Further Studies: Real-Valued Case

Agent i Agent j

Distributed algorithm  Surplus: Same as quantized case  Surplus: Same as quantized case  State:

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State:

Usual consensus Surplus

Further Studies: Real-Valued Case

 Average consensus on general strongly connected digraphs can be achieved for sufficiently small .  Surplus variables play similar roles  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. y p y

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Franceschelli, Giua, & Seatzu (2009)

Conclusion

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

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