Quantized Average Consensus

on Gossip Digraphs

Hideaki Ishii Tokyo Institute of Technology

Joint work with Kai Cai

Workshop on Uncertain Dynamical SystemsUdine, 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

22Distributed 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

overall system by introducing extra dynamics in agents?

E g Acceleration of convergence in consensus E.g. Acceleration of convergence in consensusLiu, Anderson, Cao, & Morse (2009)

Focus of this talk: Average consensus on directed graphs

33with communication constraints

Average Consensus: Introduction

EdgeAgent 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 networks44

Known Conditions on DigraphsWhen 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

55Olfati-Saber & Murray (2004)

Recent Approaches for General Digraphs

1. Cooperative algorithm to make doubly stochastic

2. Use of variables in addition to states in agentsGharesifard & Cortes (2011)

Computation of stationary distributions of Markov chainsB 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

66

Communication Constraint 1

EdgeEdgeAgent 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:

or

77Kashap, Basar, & Srikant (2007), Carli, Fagnani, Frasca, & Zampieri (2010)

Communication Constraint 2

Agent iAgent 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)88

Boyd, Ghosh, Prabhakar, & Shah (2006)

Simpler Case: Quantized Consensus

Agent iAgent j

Only agreement in the states (no averaging)

Distributed algorithm

If then If , then

If , then

99 If , then

Quantized Consensus

Theorem:

For each initial state, there exists a finite such that

with prob 1with 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

1010

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.

1111

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 iAgent 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

1313

If , then

If , then

Quantized Average Consensus

Agent iAgent j

If , then there are three cases:

If and local max then If and local max, then

If and local min, then

1414 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

QuantizedAverage

Large surplus

Average

g p

1515Surplus 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 assurplus 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

oror

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 tnumber 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).

1919

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 averageThe algorithm achieves quantized average

Threshold satisfies

2020

Threshold vs Consensus Values

The values that the agents potentially agree on.

Quantized Average

Threshold

2121

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

2222Threshold

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 surpluses2323

p y p

Convergence Time Analysis: Result Simple case: Complete digraph

ThTheorem:

Proof is based on the Lyapunov function:

“Good” “Bad”Conventional one

The problem is then reduced to hitting time analysis of

surplus surplusConventional one

a Markov chain.

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Convergence Time: Comparison

Directed &Undirected Directed &Balanced Directed

Complete

Cyclic

G lGeneral

Zhu & Martinez Nedic, Olshevsky, This workZhu & Martinez (2008)

Nedic, Olshevsky, Ozdaglar, & Tsitsiklis

(2009)

This work

Asynchronous AsynchronousSynchronoussy c o ous sy c o ousSy c o ous

25

Numerical Example

Convergence Time

R d G t i

Time

Random Geometric Digraphs

Complete Digraphs

Number of Agents

2626

Further Studies: Real-Valued Case

Agent iAgent j

Distributed algorithm

Surplus: Same as quantized case Surplus: Same as quantized case

State:2727

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

2828Franceschelli, 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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