Team-optimal distributed MMSE estimation in general and tree networks

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Contents lists available atScienceDirect

Digital

Signal

Processing

www.elsevier.com/locate/dsp

Team-optimal

distributed

MMSE

estimation

in

general

and

tree

networks

Muhammed

O. Sayin

a

,

∗

, Suleyman

S. Kozat

b

, Tamer Ba ¸sar

a

a_The_Department_of_Electrical_and_Computer_Engineering,_University_of_Illinois_at_{Urbana-Champaign,}_Champaign,_IL_61801,_USA b_The_Department_of_Electrical_and_Electronics_Engineering,_Bilkent_University,_Bilkent,_Ankara_06800,_Turkey

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline22February2017

Keywords: Distributednetworks DistributedKalmanﬁlter Teamproblem Finite-horizon

DistributedMMSEestimation

Weconstructteam-optimalestimationalgorithmsoverdistributednetworksforstateestimationinthe finite-horizon mean-square error (MSE)sense. Here, we have a distributed collection of agents with processingandcooperationcapabilities.Theseagentsobservenoisysamplesofadesiredstatethrough alinearmodelandseek tolearnthisstatebyinteractingwitheachother.Althoughthisproblemhas attracted significantattention and been studied extensively in fields includingmachine learningand signal processing, allthe well-knownstrategiesdo not achieveteam-optimal learningperformance in the finite-horizonMSEsense. To thisend, weformulatethe finite-horizondistributedminimum MSE (MMSE)whenthereisnorestriction onthesizeofthedisclosedinformation,i.e.,oracleperformance, overanarbitrarynetworktopology.Subsequently,weshowthatexchangeoflocalestimatesissufficient to achievethe oracle performance onlyovercertainnetwork topologies.By inspecting thesenetwork structures,weproposerecursivealgorithmsachievingtheoracleperformancethroughthedisclosureof localestimates.Forpracticalimplementationswealsoprovideapproachestoreducethecomplexityof thealgorithmsthroughthetime-windowingoftheobservations.Finally,inthenumericalexamples,we demonstratethesuperiorperformanceoftheintroducedalgorithmsinthefinite-horizonMSEsensedue tooptimalestimation.

1. Introduction

Overa distributednetwork ofagents withmeasurement, pro-cessing and communication capabilities, we can have enhanced processing performance, e.g., fast response time, relative to the centralizednetworksbydistributingtheprocessingpoweroverthe networks[1–4]. Mainly, distributedagents observethe true state ofthe system through noisy measurements from different view-points,processtheobservationdatainordertoestimatethestate, andcommunicatewitheachothertoalleviatetheestimation pro-cessinafullydistributedmanner.Notably,theagentscanrespond tostreaming data inan onlinemanner by disclosinginformation among each other at certain instances. This framework is con-veniently used to modelhighly complex structuresfrom defense applicationstosocialandeconomicalnetworks[5–8].Asan exam-ple,saythat wehave radarsystemsdistributedover an areaand seekingto locate hostile missiles, i.e., the location ofthe missile isthe underlyingstate of thesystem. In that respect,distributed

*

Correspondingauthor.

E-mailaddresses:sayin2@illinois.edu(M.O. Sayin),kozat@ee.bilkent.edu.tr (S.S. Kozat),basar1@illinois.edu(T. Ba ¸sar).

processing approach has vital importance in terms of detecting the missiles and reacting as fast as possible. In particular, even iftheviewpointsofafew radarsystemsareblockeddueto envi-ronmentalobstacles,throughthecommunicationamongtheradar systems,eachsystemshouldstillbeabletolocatethemissiles. Ad-ditionally,sinceeachradarsystemnotonlycollectsmeasurements butalsoprocessthemtolocatethemissiles,theoverallsystemcan respondtothemissilesfasterthanacentralizedapproachinwhich measurements ofallthe radarsystemsare collected ata central-izedunitandprocessedtogether.

Althoughthereis anextensive literatureon thistopic, e.g., [2, 6–11]andreferencestherein,westill havesigniﬁcantandyet un-explored problems for disclosure and utilization of information amongagents.Priorworkhasfocusedonthecomputationally sim-plealgorithmsthataimtominimizecertaincostfunctionsthrough theexchangeoflocalestimates,e.g., diffusionorconsensusbased estimation algorithms [2,9,3,12–14],due to processing power re-latedpracticalconcerns. However,thereis atrade-off intermsof computationalcomplexityandestimationperformance.

Formulatingtheoptimaldistributedestimationalgorithmswith respect to certain performance criteria isa signiﬁcant and unex-plored challenge. To this end, we consider here the distributed

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estimation problem asa team problem for distributed agents in whichagentstakeactions,e.g.,whichinformationtodiscloseand howtoconstructthelocalestimate.Thisdiffersfromtheexisting approaches in which agents exchange their local estimates. Fur-thermore,weaddresstheoptimalityofexchanginglocalestimates with respect to the team problemover arbitrary network struc-tures.

We examine the optimal usage of the exchanged information based on its content rather than a blind approach in which ex-changed information ishandled irrespective of the content asin thediffusionorconsensus basedapproaches. Insuch approaches, the agents utilize the exchanged information generally through certain static combination rules, e.g., the uniform rule [15], the Laplacianrule[16]ortheMetropolisrule[17].However,ifthe sta-tisticalproﬁle ofthe measurement data variesover thenetwork, i.e., each agent observes diverse signal-to-noise ratios, by ignor-ingthe variationinnoise, theserulesyield severedegradation in theestimationperformance[2].Insuchcasestheagentscan per-formbetter even without cooperation[2].Therefore, the optimal usageoftheexchangedinformationplaysanessentialrolein per-formanceimprovementintheteamproblem.

Consider distributed networks of agents that observe noisy samplesof an underlying state (possibly multi-dimensional) over a finite horizon. The agentscan exchange information withonly certainother agentsateach timeinstant.Inparticular,agents co-operate with each other as a team according to a certain team cost depending on the agents’ actions. To this end, each agent constructsa localestimateofthe underlyingstate andconstructs messages to disclose to the neighboring agents at each time in-stant.We particularlyconsidera quadraticcostfunction andthat theunderlyingstateandmeasurementnoisesarejointlyGaussian. We note that restrictions on the sent messages, e.g., on the size of the disclosed information,have significant impact on the optimalteamactions.Weintroducetheconceptoftheoracle per-formance,inwhich thereis norestrictionon thedisclosed infor-mation. In that case, a time-stamped information disclosure can beteam-optimal andweintroduce theoptimaldistributedonline learning (ODOL) algorithm using the time-stamped information disclosure. Through a counter example, we show that the oracle performancecannotbeachievedthroughtheexchangeoflocal es-timates in general. Then, we analytically show that over certain networks, e.g., treenetworks, agents can achieve the oracle per-formance through the exchange of local estimates. We propose the optimaland efficient distributedonline learning (OEDOL) al-gorithm, whichis practical forreal life applicationsand achieves theoracle performance over treenetworksthrough theexchange of local estimates. Finally, we introduce the time windowing of the measurements in the team cost andpropose a recursive al-gorithm,sub-optimaldistributedonlinelearning(SDOL)algorithm, combiningthe received messages linearlythrough time-invariant combinationweights.

We can list our main contributions as follows: 1) We intro-duce a team-problem to minimize ﬁnite horizon mean square error cost function in a distributed manner. 2) We derive the ODOL algorithm achieving the oracle performance over arbitrary networksthrough time-stampedinformationexchange.3) We ad-dresswhetheragentscanachievetheoracleperformancethrough the disclosure of local estimates. 4) We propose a recursive al-gorithm, the OEDOL algorithm, achieving the oracle performance overcertain networktopologieswithtremendously reduced com-municationload.5)Wealsoformulatesub-optimalversionsofthe algorithmswithreducedcomplexity.6) Weprovidenumerical ex-amplesdemonstratingthesigniﬁcantgainsdueto theintroduced algorithms.

Theremainder ofthepaperis organizedasfollows.We intro-duce the team problem fordistributed-MMSE estimation in

Sec-Fig. 1. The neighborhoods of ith agent over the distributed network. tion 2.We studythetreenetworks, exploitthenetwork topology to formulate the OEDOL algorithm that reducesthe communica-tion load andintroduce cell structures, which is relatively more connectedthan treenetworks,in Section3.We proposethe sub-optimalversionsoftheODOLalgorithmforpractical implementa-tions in Section 4. In Section 5,we provide numerical examples demonstratingsigniﬁcant gainsduetothe introducedalgorithms. WeconcludethepaperinSection6withseveralremarks.

Notation: We work with real data for notational simplicity.

N(

0

,

.)

denotes the multivariate Gaussian distribution with zero mean and designated covariance. For a vector a (or matrix A), a (or A) is its ordinary transpose. We denotethe vector whose terms are all 1s (or all 0s) by 1 (and 0). We denote random variablesbyboldlowercaseletters,e.g.,x.Theoperatorcol

{·}

pro-ducesacolumnvectororamatrixinwhichtheargumentsofcol

{·}

arestackedone undertheother.Foramatrix A,diag

{

A

}

operator constructs a diagonalmatrix with the diagonal entries of A. For agivenset

N

,diag

{

N }

createsadiagonalmatrixwhosediagonal block entriesareelementsoftheset.Theoperator

⊗

denotesthe Kroneckerproduct.

2. Teamproblemfordistributed-MMSEestimation

Consideradistributednetworkofm agentswithprocessingand communication capabilities. In Fig. 1, we illustrate this network through an undirected graph, where the vertices and the edges correspond totheagentsandthecommunicationlinksacrossthe network,respectively.Foreachagenti,wedenotethesetofagents whose information could be received at least after k hops, i.e.,

k-hopneighbors,by

N

_i(k),and

π

(k) i

:=

N

(k)

i

is thecardinalityof

N

(k)

i (see Fig. 1b)1.Weassume that

N

(0)

i

= {i}

and

N

(k)

i

= ∅

for

k

<

0.Notethatthesequenceofthesets

{

N

_i(0)

,

N

(1)

i

,

. . .

}

isa non-decreasingsequencesuchthat

N

_i(k)

⊆

N

_i(l)ifk

<

l.

1 _For_notational_simplicity,_we_deﬁne_N

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Here,atcertain time instants, theagentsobserve anoisy ver-sionof a time-invariantandunknown state vector x

∈ R

p which isa realizationofaGaussianrandomvariablex with mean

¯

x and

auto-covariance matrix

x. In particular, at time instant t, each agenti observesanoisyversionofthestateasfollows:

yi,t

=

Hix

+

ni,t

,

t

=

1

, . . . ,

T

,

whereHi

∈ R

q×p isamatrixcommonlyknownbyallagents,and

ni,t

∈ R

q is a realization of a zero-mean white Gaussian vector process

{

ni,t

}

with auto-covariance

ni. Correspondingly, the

ob-servation yi,t

∈ R

q is a realization of the random process

{

yi,t

}

, where yi,t

=

Hix

+

ni,t almosteverywhere (a.e.).Thenoise ni,t is also independent of the state x and the other noise parameters

nj,τ , j

=

i and

τ

≤

t.Weassumethatthestatisticalproﬁlesofthe

noiseprocesses are commonknowledge of theagents since they canreadilybeestimatedfromthedata[18].

Theagentshavecommunicationcapabilitiesandatcertaintime instants,i.e.,aftereachmeasurement,they canexchange informa-tionwiththeneighboringagentsasseeninFig. 1a.Letzi,j,t

∈ R

r denotetheinformationdisclosedbyi to j attime t,andr

≥

0 is thesizeofthedisclosedinformation.Weassumethatthereexists aperfectchannel betweenthe agentssuch thatthe disclosed in-formationcanbetransmittedwithinﬁniteprecision.Therefore,we denotetheinformationavailabletoagent-i attimet by

δ

i,t

=

yi,t

,

yi,τ

,

zj,i,τ

,

for j

∈

N

i

,

τ

=

1

, . . . ,

t

−

1

andlet

σ

i,t denotethesigma-algebrageneratedbytheinformation set

δ

i,t.Furthermore,wedeﬁnethesetofall

σ

i,t-measurable func-tionsfrom

R

qt

× R

rπi(t−1) _to

R

r _by

_i_,_t_._Importantly, _here,_which

information to disclose is not determined a priori in the prob-lemformulation. Let

γ

i,j,t bethe decisionstrategy forzi,j,t,then agent-i chooses

γ

i,j,t, j

∈

N

i,fromtheset

i,t,i.e.,

γ

i,j,t

∈

i,t and

γ

i,j,t

(δ

i,t

)

=

zi,j,t,basedonhis/herobjective.

In addition to the disclosed information zi,j,t, j

∈

N

i, agent-i takes action ui,t

∈ R

q, where the corresponding decision strat-egy

η

i,t is chosen from the set

i,t, which is the set of all

σ

i,t-measurablefunctionsfrom

R

qt

×R

rπi(t−1)to

R

p,i.e.,

η

i,t

∈

i,t and

η

i,t

(δ

i,t

)

=

ui,t.Here,weconsiderthattheagentshavea com-moncostfunction:

T

t=1 m

j=1

x

−

uj,t

2

,

where all actions ui,t, i

=

1

,

. . . ,

m and t

=

1

,

. . . ,

T are costly, andagent-i should take actions ui,t and zi,j,t, j

=

1

,

. . . , π

i and

t

=

1

,

. . . ,

T , accordingly. Therefore, this corresponds to a team-problem,inwhichagent-i facesthefollowingminimization prob-lem: min γi,j,t∈i,t,ηi,t∈i,t, j∈Ni,t=1,...,T T

t=1 m

j=1 E

x

−

η

j,t

(δ

j,t

)

2

.

(1)

Wepointout thatboth

i,t and

i,t areinﬁnitedimensional,i.e.,

(1)isafunctionaloptimizationproblemandtheoptimalstrategies canbe anonlinear functionoftheavailable information. Further-more,the agentsshould also constructthe disclosedinformation accordinglysince other agents’ decisions uj,t directly depend on thedisclosedinformation.

2.1.Alowerbound

Inordertocomputetheteamoptimalstrategies,weﬁrst con-structalowerboundontheperformance oftheagentsby remov-ing the limitation on the size of the disclosed information, i.e.,

r

→ ∞

. In that case, the following propositionprovides an opti-malinformationdisclosurestrategy.

Fig. 2. Timestampedinformationdisclosureover anetworkof6 agentsattime instantt=4.

Proposition2.1.Whenr

≥

mq,atimestampedinformationdisclosure strategy,inwhichagentstransmitthemostcurrentversionofthe avail-ableinformation(e.g.,seeFig. 2),canleadtotheteam-optimalsolution.

Proof. Through the time stamped information disclosure, each agentcanobtainthemeasurements oftheotheragentsseparately ina connectednetwork.However, themeasurements ofthe non-neighboringagentscould only bereceived aftercertain hopsdue to the partially connected structure, i.e., certain agents are not directly connected. As an example, the disclosed information of

j

∈

N

_i(2)reachestoi bypassingthroughtwocommunicationlinks asseeninFig. 1b.Inparticular,thiscaseassumesthateachagent has accessto full information fromthe other agents,albeit with certainhops,andcorrespondstothedirectaggregationofall mea-surementsacrossthenetworkateachagent.

Correspondingly,attimet,alltheinformationaggregatedatith

agentisgivenby

δ

_io_,_t

:=

yi,τ

τ≤t

,

yj,τ

τ_j_∈N≤t−1(1) i

, . . . ,

yj,τ

τ≤t−κi j∈N_i(κi)

,

(2)

wheretheinformationfromthefurthestagent isreceivedatleast after

κ

i hops. Therefore, a time-stamped information disclosure strategycanleadtotheteam-optimalsolution.

2

Let

σ

o

i,t denotethesigma-algebrageneratedbytheinformation set

δ

o_i_,_t and

o_i_,_t be theset of all

σ

o

i,t-measurable functionsfrom

R

qt

_×R

qπi(t−1)

×. . .×R

qπ

κi

i |t−κi|+ _(where

|

_t

−

_κ

i

|

+

=

0 ift

−

κ

i

<

0) to

R

p.Then,agent-i facesthefollowingminimizationproblem:

min ηi,t∈oi,t, t=1,...,T T

t=1 m

j=1 E

x

−

η

j,t

(δ

oj,t

)

2

,

(3) whichisequivalentto T

t=1 min ηi,t∈oi,t E

x

−

η

i,t

(δ

oi,t

)

2 (4) since

γ

i,j,t, j

∈

N

i,issettothetime-stampedstrategyand

η

i,t has impactonlyonthetermE

x

−

η

i,t

(δ

io,t

)

2.Let

δ

o i,t bedeﬁnedby

δ

o_i_,_t

:=

y_i_,_τ

τ≤t

,

y_j_,_τ

τ≤t−1 j∈N_i(1)

, . . . ,

y_j_,_τ

τ≤t−κi j∈N_i(κi)

(4)

Then,teamoptimalstrategy inthelowerbound,i.e., oracle strat-egy,

η

o

i,t andthecorrespondingactionu o

i,t aregivenby

η

o_i_,_t

(δ

o_i_,_t

)

=

uo_i_,_t

=

E

[

x

|δ

o_i_,_t

= δ

o_i_,_t

]

(5)

andwedeﬁneuo_i_,_t

:=

E

[

x

|δ

o_i_,_t

]

.

2.2. ODOLalgorithm

Since the state and the observation noise are jointly Gaus-sian randomparameters, we cancompute (5)through a Kalman-like recursion [19]. Therefore, we provide the following ODOL algorithm. We introduce a difference set

i,t

:= δ

oi,t

\δ

oi,t−1

=

yi,t

,

yj,t−1

j∈N_i(1)

,

. . . ,

yj,t−κi

j∈N(κi ) i

and a vector wi,t

=

col

{

i,t

}

.Then,fort

≥

1 theiterations oftheODOLalgorithmare givenby uo_i_,_t

=

I

−

Ki,tH

¯

i

uo_i_,_t₋₁

+

Ki,twi,t

,

Ki,t

= ˆ

i,t−1H

¯

i

¯

Hi

ˆ

i,t−1H

¯

i

+ ¯

ni

₋₁

,

ˆ

i,t

=

I

−

Ki,tH

¯

i

ˆ

i,t−1

,

where2uo_i_,₀

= ¯

x,

ˆ

i,0

=

x,H

¯

i

:=

Pi

⊗

Iq

H , H

:=

col

{

H1

, . . . ,

Hm

} ,

¯

ni

:=

Pi

⊗

Ip

n

Pi

⊗

Ip

,

n

:=

diag

n1

, . . . ,

nm

,andPi is thecorrespondingpermutationmatrix.

We point out that thisis a lower bound on the original cost function(1),i.e., T

t=1 m

i=1 E

x

−

uo_i_,_t

2

≤

min γi,j,t∈i,t,ηi,t∈i,t, i=1,...,m; j∈Ni;t=1,...,T T

t=1 m

j=1 E

x

−

η

j,t

(δ

j,t

)

2

,

(6)

wherewesubstituteteam-optimalaction(whenr

→ ∞

)uo_i_,_t back into(4)andsumovert

=

1

,

. . . ,

T andi

=

1

,

. . . ,

m.However,the lower boundis not necessarilytight depending onr. By Proposi-tion 2.1, time-stamped information disclosure strategy, in which the size of the disclosed information is q

×

m, yields the ora-cle solution.This implies that when r

≥

qm, the lower bound is tight.Furthermore,teamoptimalsolutions arelinearinthe avail-able information and can be constructed through the recursive algorithm ODOL. However, qm islinear inthe numberof agents,

m, andinlarge networksthiscancauseexcessivecommunication loadyetcommunicationloadiscrucialfortheapplicabilityofthe distributed learning algorithms [14,13]. Therefore, in the follow-ing section, we provide a suﬃcient condition on the size of the disclosed information, which depends on the network structure (ratherthanitssize),inordertoachievethelowerbound(6). 3. Distributed-MMSEestimationwithdisclosureoflocalestimate

Intheconventionaldistributedestimationalgorithms,e.g., con-sensus anddiffusion approaches, agentsdisclose their local esti-mates,whichhavesize p (notethat thisdoesnot dependonthe networksize). The followingexample addresseswhetherthe dis-closureoflocalestimatescanachievethelowerbound(6)ornot.

2 _If_the_inverse_fails_to_exist,_a_{pseudo-inverse}_can_replace_the_inverse_[19]_.

Fig. 3. A cycle network of 4 agents. 3.1. Acounterexample

Consider a cyclenetworkof 4 agents asseenin Fig. 3,where

p

=

q

=

1,Hi

=

1,

ni

=

σ

2

n,fori

=

1

,

. . . ,

4,and

x

=

σ

x2.Weaim toshowthatagent-1’soracleactionattimet

=

3,i.e.,uo₁_,₃,cannot beconstructedthroughtheexchangeoflocalestimates.

At time t

=

2, agent-2 and agent-3 have the following oracle actions: uo₂_,₂

=

E

[

x

|

y₂_,₂

=

y2,2

,

y2,1

=

y2,1

,

y1,1

=

y1,1

,

y4,1

=

y4,1

]

=

σ

x2 4

(

σ

2 x

+

σ

n2

)

(

y2,2

+

y2,1

+

y1,1

+

y4,1

),

(7) uo₃_,₂

=

E

[

x

|

y₃_,₂

=

y3,2

,

y3,1

=

y3,1

,

y1,1

=

y1,1

,

y4,1

=

y4,1

]

=

σ

x2 4

(

σ

2 x

+

σ

n2

)

(

y3,2

+

y3,1

+

y1,1

+

y4,1

).

(8) Note that since there are two hops between agents 2 and 3, at

t

=

2,agentsdonothaveaccesstoeachother’sanymeasurement yet.Attimet

=

3,agent-1’soracleactionisgivenby

uo₁_,₃

=

E

[

x

|

y1,3

=

y1,3

,

y1,2

=

y1,2

,

y1,1

=

y1,1

,

y2,2

=

y2,2

,

y2,1

=

y2,1

,

y3,2

=

y3,2

,

y3,1

=

y3,1

,

y₄_,₁

=

y4,1

],

=

σ

x2 8

(

σ

2 x

+

σ

n2

)

(

y1,3

+

y1,2

+

y1,1

+

y2,2

+

y2,1

+

y3,2

+

y3,1

+

y4,1

).

Assume that uo₁_,₃ canbe obtainedthrough the exchange oflocal estimates:

ˆ

u1,3

:=

E

[

x

|

y1,3

=

y1,3

,

y1,2

=

y1,2

,

y1,1

=

y1,1

,

uo₂_,₂

=

uo₂_,₂

,

uo₂_,₁

=

uo₂_,₁

,

uo₃_,₂

=

uo₃_,₂

,

uo₃_,₁

=

uo₃_,₁

].

(9)

Since all parameters are jointlyGaussian, the local estimates are alsojointlyGaussian,u

ˆ

1,3,islinearinuo₂_,₂ anduo₃_,₂.Furthermore, the measurements y2,2, y3,2, and y4,1 are only included in uo2,2 anduo₃_,₂.Therefore,weobtain

ˆ

u1,3

= · · · +

α

u2o,2

+ β

uo3,2

= · · · +

α

σ

2 x 4

(

σ

2 x

+

σ

n2

)

(

y2,2

+

y4,1

+ · · · )

+ β

σ

x2 4

(

σ

2 x

+

σ

n2

)

(

y3,2

+

y4,1

+ · · · ),

where

· · ·

referstotheotherterms.However,theequalityofu

ˆ

1,3 anduo₁_,₃ implies

α

= β =

1

/

2 due tothecombinationweights of

y2,2 and y3,2,respectively,and

α

+ β =

1

/

2 duetothe combina-tion weightof y4,1,whichleads toa contradiction.Hence, which information to discloseover arbitrary networksforteam-optimal

(5)

Fig. 4. Anexampletreenetwork.NoticetheeliminatedlinksfromFig. 1toavoid multi-pathinformationdiffusion.

solutions should be considered elaborately. In the following, we analyticallyshowthatlowerboundcouldbeachievedthroughthe disclosureoflocalestimatesover“treenetworks”.

3.2.Treenetworks

A network has a “tree structure” if its corresponding graph is a tree, i.e., connected andundirected without any cycles [20]. As an example, the conventional star or line networks have tree structures. We remark that for an arbitrary network topol-ogy we can also construct the spanning tree of the network andeliminate thecycles. Inthe literature, thereexists numerous distributed algorithms for minimum spanning tree construction

[21–25].

Importantly, the following theoremshows that over tree net-works we can achieve the performance of the oracle algorithm throughthedisclosureofthelocalestimatesonly.

Theorem3.1.Considertheteam-problemoveratreenetwork,inwhich r

=

p.Then,exchangeoflocalestimatescanleadtotheteam-optimal solution,i.e.,agentscanachievetheoracleperformance.

Proof. Initially,agent-i hasaccessto yi,1 onlyandthe oracle ac-tion is uo_i_,₁

=

E

[

x

|

yi,1

=

yi,1

]

. At time t

=

2, the oracle action is givenby uo_i_,₂

=

E

x

|

y_i_,_τ

=

yi,τ

_τ₌₁_,₂

,

yj,1

=

yj,1

j∈Ni

,

(10)

whichcanbewrittenas uo_i_,₂

=

E

x

|

y_i_,_τ

=

yi,τ

τ=1,2

,

E

[

x

|

y_j_,₁

=

yj,1

]

j∈Ni

=

E

x

|

y_i_,_τ

=

yi,τ

τ=1,2

,

uo_j_,₁

=

uo_j_,₁

j∈Ni

.

(11)

Thisimpliesthatfort

=

1 andt

=

2,the oracleperformance can beachievedthroughthedisclosureoflocalestimate.Therefore,we can consider the oracle action (10) even though agents disclose their local estimate instead of time-stamped information disclo-sure.

As seen in Fig. 4, over a tree network, fork

∈ {

1

,

. . . , κ

i

}

we have

N

(k) i

=

j∈Ni

N

(k) i

∩

N

(k−1) j

.

(12)

Notethatthesetsin(12)aredisjointas

N

(k) i

∩

N

(k−1) j1

∩

N

(k) i

∩

N

(k−1) j2

= ∅

(13)

forall j1

,

j2

∈

N

i and j1

=

j2. Notably, over a tree network, by

(13),we can partitionthe collectionsetofthemeasurements re-ceivedafteratleastk-hopsasfollows

yj,τ

j∈N(k) i

=

yj,τ

j∈N(k) i ∩N (k−1) j1

,

. . . ,

yj,τ

j∈N_i(k)∩N(_j_πk−1) i

.

(14)

In the time-stamped information disclosure, at time t

=

3, agent-i hasaccessto

δ

o

i,3,deﬁnedin(2).Wedenotethesetofnew measurementsreceivedbyi over j attimet

=

2 by

o_j_,_i_,₂

:=

=

yj,2

yk,2

k∈N(1) i ∩N (0) j

,

yk,1

k∈N(2) i ∩N (1) j

,

whichcanalsobewrittenas

o_j_,_i_,₂

= δ

o_j_,₂

\

=δoj,1

yj,1

,

yi,1

=o i,j,1=δoi,1

,

(15)

where we exclude the informationsent by i to j at time t

=

1, i.e., yi,1.Then,wecanwritetheaccessedinformationastheunion of new measurement yi,3, new measurements received over the neighboringagentsandtheaccessed informationattimet

=

2 as follows:

δ

_io_,₃

=

yi,3

,

oj1,i,2

, . . . ,

o jπ_i,i,2

, δ

oi,2

.

(16)

Notethat thesetsonthe righthandside of(16)aredisjointdue totreestructure.Furthermore,by(15)and(16),thesigma-algebra generatedby

δ

o

i,3 isequivalent to thesigma-algebrageneratedby theset

{

yi,τ}τ≤3

,

{δ

o_j_,_τ

}

τ_j_∈≤_N2_i

.Since uo_j_,_t

=

E

[

x

|δ

o_j_,_t

= δ

o_j_,_t

]

,we ob-tain uo_i_,₃

=

E

x

|

y_i_,_τ

=

yi,τ

τ≤3

,

uo_j_,_τ

=

uo_j_,_τ

τ≤2 j∈Ni

.

(17) By(15),wehave

o_j_,_i_,_t

= δ

o_j_,_t

\

δ

o_j_,_t₋₁

∪

o_i_,_j_,_t₋₁

,

(18)

which implies that for t

≥

2,

o_j_,_i_,_t is constructible from

δ

_io_,_τ

and

δ

o_j_,_{τ for}

τ

≤

t. Hence, by induction, we conclude that the lower bound can be achieved through the exchange of local es-timates.

2

Remark3.1. When theexpectationofthe state isconditioned on inﬁnitenumberofobservationsover evenaconstructedspanning tree,onlyaﬁnitenumberoftheobservationsismissingcompared tothecaseoverafullyconnectednetwork.Hence,evenifwe con-struct the spanning tree of that network, we would still achieve the lower bound over a fullyconnected (or centralized)network asymptotically. As an illustrative example,in Fig. 10, we observe thattheMMSEperformanceoverthefullyconnected,starandline networks are asymptotically the same. Similarly, in [26–28], the authors show that the performance of the diffusion based algo-rithmscould approachthe performance ofafully connected net-workundercertainregularityconditions.

In thesequel, we propose the OEDOL algorithm that achieves thelowerboundovertreenetworksiteratively.

3.3. OEDOLalgorithm

ByTheorem 3.1,overatreenetwork,oracleactioncanbe con-structedby

(6)

uo_i_,_t

=

E

x

|

y_i_,_τ

=

yi,τ

τ≤t

,

uo_j_,_τ

=

uo_j_,_τ

τ≤t−1 j∈Ni

(19)

through the disclosure of oracle actions, i.e., local estimates. We remarkthat uo

i,t is linear inthe previous actions uoi,τ ,

τ

≤

t

−

1.

Inordertoextractnewinformation,i.e.,innovationpart,weneed to eliminate the previously received information at each instant ontheneighboringagents.Thisbringsinadditionalcomputational complexity.On thecontrary, agentscan justsend thenew infor-mationcomparedtothepreviouslysentinformation,e.g.,si,t.Note that here agents disclose the same informationto the neighbor-ing agents.Sincewe are conditioningon thelinear combinations oftheconditionedvariableswithouteffectingtheirspannedspace, i.e., si,t is computable from uoi,τ for

τ

≤

t andvice versa, agents

canstillachievetheoracleperformancebyreducedcomputational load,yet.

Attimet,agent-i receiveslocalmeasurement yi,t andsent in-formationfromtheneighboringagents,

ri,t

:=

col

{

sj1,t−1

, . . . ,

sjπi,t−1

}.

Weaimto determinethecontent ofthereceived informationri,t toextract the innovationwithin them andutilizethisinnovation intheupdateoftheoracleaction.

Initially,attimet

=

1,agent-i hasonlyaccesstothelocal mea-surementyi,1.Then,theoracleactionisgivenby

uo_i_,₁

= (

I

−

xHi

(

Hi

xHi

+

ni

)

− 1_H i

)

¯

x

+

xHi

(

Hi

xHi

+

ni

)

− 1_y i,1

.

Letuo_i_,₀

= ¯

x and

ˆ

i,0

=

x,andsetBi,1

= ˆ

i,0H_i

(

Hi

ˆ

i,0H_i

+

ni

)

−

1 andAi,1

=

I

−

Bi,1Hi.Then,weobtain

uo_i_,₁

=

Ai,1uoi,0

+

Bi,1yi,1

,

ˆ

i,1

=

Ai,1

ˆ

i,0

.

Next, instead of sending uo

i,1, agent-i sends to the neighboring agents, j

∈

N

i,

si,1

=

uoi,1

−

Ai,1uoi,0

=

Bi,1yi,1

.

Correspondingly, attime t

=

2, agent-i receives yi,2 andri,2. Let

ri,2 be the corresponding random vector. Then, conditioning the stateandthereceivedinformationonthepreviously available in-formation yi,1

=

yi,1,wehave

x yi,2 ri,2

yi,1

=

yi,1

∼ N

⎛

⎝

⎡

⎣

u o i,1 Hiuoi,1 ¯ Hi,1uoi,1

⎤

⎦ ,

⎡

⎣

î,1 î,1H i î,1H¯i,1 Hiî,1 Hiî,1Hi+ni Hiî,1H¯i,1 ¯ Hi,1î,1 H¯i,1î,1Hi H¯i,1î,1H¯i,1+ ¯Gi,1

⎤

⎦

⎞

⎠ ,

where H

¯

i,1

:=

col

Bj1,1Hj1

, . . . ,

Bjπi,1Hjπi

andG

¯

i,1

=

diag

{

Gi,1

}

, whereGi,1

:=

col

Bj1,1

n_j1Bj1,1

, . . . ,

Bjπi,1

njπiB jπi,1

. LetH

˜

i,1

:=

col

{

Hi

,

H

¯

i,1

}

andG

˜

i,1

:=

diag

{

ni

,

G

¯

i,1

}

andset

!

Bi,2 Ci,2

"

= ˆ

i,1H

˜

i,1

˜

Hi,1

ˆ

i,1H

˜

i,1

+ ˜

Gi,1

₋1

,

Ai,2

=

I

−

Bi,2Hi

−

Ci,2H

¯

i,1

.

Then,weobtain uo_i_,₂

=

Ai,2uoi,1

+

Bi,2yi,2

+

Ci,2ri,2

,

(20)

ˆ

i,2

=

Ai,2

ˆ

i,1 andagent-i sends

si,2

=

uoi,2

−

Ai,2uoi,1

=

Bi,2yi,2

+

j∈Ni C_i(_,j₂)Bj,1yj,1

=sj,1

,

whereC(_i_,j₂)denotesthecorresponding jthblock ofCi,2.Therefore, attimet

=

3,agent-i receivesfrom j

∈

N

i:

sj,2

=

Bj,2yj,2

+

k∈Nj\i

(

C(_jk_,₂)Bj,1yj,1

)

+

C(ji,)2Bi,1yi,1

.

(21)

Since thelast termontherighthandside of(21)isknownby i,

wehave E

[

ri,2

|δ

io,2

= δ

oi,2

] =

= ¯Hi,2

⎡

⎢

⎣

B_j1,2+$k∈N_j1\iC(_j1k),2Bk,2

..

.

Bjπ_i,2+$k∈Njπ_i\iC (k) jπ_i,2Bk,2

⎤

⎥

⎦

uoi,2

+

Di,2si,1

,

(22) where Di,2

:=

col

{

C(ji1),2

,

. . . ,

C (i) jπi,2

}

,and Gi,2

=

⎡

⎢

⎣

B_j1,2n j₁B_j1,2+ $ k∈N_j1\iC( k) j1,2Bk,1nkBk,1(C (k) j1,2)

..

.

Bjπ_i,2_{n j}_π iB jπ_i,2+ $ k∈Njπ_i\iC (k) jπ_i,2Bk,1_nkBk,1(C (k) jπ_i,2)

⎤

⎥

⎦

(23)

By(20),(22),and(23),thenextoracleactionuo_i_,₃ isgivenby uo_i_,₃

=

Ai,3uoi,2

+

Bi,3yi,3

+

Ci,3

(

ri,3

−

Di,2si,1

).

Subsequently, agent-i sends si,3

=

uo_i_,₃

−

Ai,3uo_i_,₂ andthe received informationfrom j

∈

N

i yields

sj,3

=

Bj,3yj,3

+

k∈Nj C(_jk_,₃)

⎛

⎝

Bk,2yk,2

+

l∈Nk\j C_k(l_,)₂Bl,1yl,1

⎞

⎠

=

Bj,3yj,3

+

k∈Nj\i C(_jk_,₃)

⎛

⎝

Bk,2yk,2

+

l∈Nk\j C_k(l_,)₂Bl,1yl,1

⎞

⎠

+

C(_ji_,)₃

si,2

−

C_i(_,j₂)sj,1

.

Then, H

¯

i,3 isgivenby

¯

Hi,3

=

⎡

⎢

⎣

B_j1,3Hj1+Cj1,3H¯j1,2

..

.

Bjπ_i,3Hjπ_i+Cjπ_i,3H¯jπ_i,2

⎤

⎥

⎦ −

⎡

⎢

⎣

C(i) j1,3H¯ (i) j1,2

..

.

C(_ji_π) i,3H¯ (i) jπ_i,2

⎤

⎥

⎦ .

(24) Correspondingly,wehave Gi,3

=

⎡

⎢

⎣

B_j1,3n j₁B_j1,3+Cj1,3G¯j1,2C_j1,3

..

.

Bjπ_i,3n jπ_iBjπ_i,3+Cjπ_i,3G¯jπ_i,2Cjπ_i,3

⎤

⎥

⎦

−

⎡

⎢

⎣

C(_j1i)_,₃G(_j1i)_,₂(C(_j1i)_,₃)

..

.

C(_ji) π_i,3G (i) jπ_i,2(C (i) jπ_i,3)

⎤

⎥

⎦ .

(25)

Therefore,theoracleactioncanbewrittenas

uo_i_,₄

=

Ai,4uoi,3

+

Bi,4yi,4

+

Ci,4

(

ri,4

−

Di,3si,2

+

Ti,3ri,2

),

where Ai,4

,

Bi,4

,

Ci,4

,

Di,3 aredeﬁnedaccordinglyand

(7)

Ti,3

:=

⎡

⎢

⎣

C(_j1i)_,₃C(_i_,j1₂)

. .

_.

C(_ji_π) i,3C (jπ_i) i,2

⎤

⎥

⎦ .

Following identical steps, for t

≥

1, the OEDOL algorithm is givenby

uo_i_,_t

=

Ai,tuoi,t−1

+

Bi,tyi,t

+

Ci,twi,t

,

(26)

ˆ

i,t

=

Ai,t

ˆ

i,t−1

,

(27)

wherewi,t istheinnovationtermextractedfromthereceived in-formation,whichevolvesaccordingto

wi,t

=

ri,t

−

Di,t−1si,t−2

+

Ti,t−1wi,t−2

.

(28) Theweightingmatrices Ai,t, Bi,t,Ci,t,Di,t,andTi,t aredeﬁnedby

!

Bi,t Ci,t

"

= ˆ

i,t−1H

˜

i,t−1

×

˜

Hi,t−1

ˆ

i,t−1H

˜

i,t−1

+ ˜

Gi,t−1

−1

,

(29) Ai,t

=

I

−

Bi,tHi

−

Ci,tH

¯

i,t−1

,

(30) Di,t

=

col

C(_ji) 1,t

, . . . ,

C (i) jπ_i,t

(31) Ti,t

=

⎡

⎢

⎣

C(i) j1,tC (j1) i,t−1 ··· 0

..

.

. .

.

..

.

0 ··· C(_ji_π) i,tC (jπ_i) i,t−1

⎤

⎥

⎦ ,

(32)

where H

˜

i,t

=

col

{

Hi

,

H

¯

i,t

}

, G

˜

i,t

=

diag

{

ni

,

G

¯

i,t

}

and G

¯

i,t

=

diag

{

Gi,t

}

.By(24)and(25),theintermediateparameters H

¯

i,t and

Gi,t evolveaccordingto

¯

Hi,t

=

⎡

⎢

⎣

B_j1,tHj1+Cj1,tH¯j1,t−1

..

.

Bjπ_i,tHjπ_i+Cjπ_i,tH¯jπ_i,t−1

⎤

⎥

⎦ −

⎡

⎢

⎣

C(i) j1,tH¯ (i) j1,t−1

..

.

C(_ji_π) i,tH¯ (i) jπ_i,t−1

⎤

⎥

⎦ ,

(33) Gi,t

=

⎡

⎢

⎣

B_j1,t_{n j} 1B j1,t+Cj1,tG¯j1,t−1C_j1,t

..

.

Bjπ_i,tn jπ_iBjπ_i,t+Cjπ_i,tG¯jπ_i,t−1Cjπ_i,t

⎤

⎥

⎦

−

⎡

⎢

⎣

C(i) j1,tG (i) j1,t−1 C(i) j1,t

..

.

C(_ji_π) i,tG (i) jπ_i,t−1 C(_ji_π) i,t

⎤

⎥

⎦

(34)

and we initialize the parameters as H

¯

j,τ

=

0 and Gj,τ

=

0 for

τ

<

1.Then,agent-i sends si,t

=

uoi,t

−

Ai,tuoi,t−1

.

ThedetaileddescriptionofthealgorithmisprovidedinTable 1.

3.4.Computationalcomplexity

In (26), the combination matrices Ai,t

,

Bi,t, Ci,t, Di,t−1, and Ti,t−1 are independent of the streaming data although they are time-varying. Hence they can be computed before-hand. In that case,thecomputationalcomplexityoftheiterationsforeachagent is dominated by the term Ci,twi,t. Therefore, the average com-putational complexity is on the order of p2

_π

2_, _where

_π

2

_:=

1

/

m

$

m_i₌₁

π

2

i, i.e., O

(

p

2

_π

2

₎

_. _Otherwise, _the _{computational} com-plexity of the algorithm is mainly dominated by the matrix in-version in (29), note that G

¯

i,t

∈ R

pπi×pπi, unless the network is sparsely connected, i.e.,

π

i

m for i

=

1

,

. . . ,

m. Therefore, over

Table 1

ThedescriptionoftheOEDOLalgorithm. Algorithm: The OEDOL Algorithm. Initialization: For i=1 to m do uo i,0= ¯x, ˆi,0= x, ¯ Hi,τ=0, Gi,τ=0, and wi,τ=0 forτ<1 End for Iterations: Do for t≥1 For i=1 to m do Construction of Weights: For j=1 to m do

CalculateH¯j,tand Gj,tby(33)and(34).

Determine combination matrices via(29)–(32). End for

Construct ri,tthrough received sj,t−1for j∈ Ni

Extraction of Innovation: wi,t=ri,t−Di,t−1si,t−2+Ti,t−1wi,t−2 Update: uo i,t=Ai,tuoi,t−1+Bi,tyi,t+Ci,twi,t ˆi,t=Ai,tˆi,t−1

Disclose si,t=uoi,t−Ai,tuoi,t−1to the neighbors.

End for

Table 2

Acomparisonofthecomputationalcomplexitiesoftheproposedalgorithms. Algorithm Without weights Pre-computed weights ODOL O(qm)3 O(qm)2

OEDOL Omp3_π3 _O_p2_π2

a non-sparse network, theaverage complexity is on theorder of

mp3

π

3 ₍

_π

3 _is _deﬁned _accordingly) _since _each _agent _{i computes} Bj,t andCj,t for j

=

1

. . . ,

m.Inparticular,thecomplexityisgiven by O

mp3

_π

3

_,_while_it_is _O

₍

_qm

₎

3

_for_the_ODOL_algorithm._Note that over tree networks, we have m

−

1 edges and correspond-inglyaverageneighborhoodsizeissmall.Hence,disclosureoflocal estimatesovertreenetworksalsoreducesthecomputational com-plexitycomparedto thetime-stampeddisclosurestrategy in gen-eral(inadditiontothesubstantialreductionincommunication).In

Table 2,we tabulatethe computationalcomplexities ofthe intro-ducedalgorithms.

Wepointoutthatdiffusionorconsensusbasedalgorithmshave relatively low complexity, i.e., on the order of p2 _in _the least-mean-square based algorithms andon the order of p3 in quasi-Newtonbasedalgorithms,sinceexchangedinformationishandled irrespective ofthe content. Such algorithms also present appeal-ing performance for certain applications in addition to the low computational complexity. However, they do not achieve the or-acleperformance.

In the following, we analyze whether the agents can still achievetheoracleperformancethroughtheexchangeoflocal esti-matesoverthenetworksnotintreestructure.

3.5. Treenetworksinvolvingcellstructures

While constructing thespanning tree, we cancelcertain com-municationlinksin ordertoavoidmulti-pathinformation propa-gation.However,wealsoobservethatinafullyconnectednetwork agents can achieve the oracle performance through the disclo-sureoflocalobservations.Inparticular,sincealloftheagentsare connected,eachagentcanreceivetheobservationsacrossthe net-work directly. Correspondingly, in a fully connectednetwork, we can achieve identicalperformance withthe ODOLalgorithm only throughthe disclosureofthelocalestimatesasstatedinthe fol-lowingcorollaryformally.

(8)

Fig. 5. An example tree network involving cell structures.

Corollary3.1.Considertheteam-problemoverafullyconnected net-work,inwhichr

=

p.Then,exchangeoflocalestimatescanleadtothe team-optimalsolution,i.e.,agentscanachievetheoracleperformance.

Proof. Overa fullyconnectednetwork,

κ

i

=

1 andthe oracle ac-tionisgivenby uo_i_,_t

=

E

&

x

|{

y_i_,_τ

=

yi,τ

}

τ≤t

,

{

yj,τ

=

yj,τ

}

τj∈N≤t−i1

'

(35)

andwecanalsoobtain(35)by(11)sincewehave

o_j_,_i_,_t

= δ

o_j_,_t

\ {δ

_io_,_t

\ {

yi,t

}},

which implies that

o_j_,_i_,_t is constructible from

δ

_io_,_{τ and}

δ

o_j_,_{τ for}

τ

≤

t.Theproofisconcluded.

2

We point out that the team-optimal strategies can fail if a link or node failure occurs. However, once a link failure is de-tected,team-optimalstrategiescan berecomputedbyeliminating thefailed linkin thenewnetwork conﬁguration.Hence, through suchstrategies,wecanincreasetherobustnessoftheteamagainst linkandnodefailures.

We deﬁne a “cell structure” as a sub-network in which all agentsare connectedto each other.Intuitively, consideringa cell structureasa“single”agent,thecell(i.e.,alltheagentsinthecell) canbe involvedinthe treesuch thatthe agentscanstill achieve the oracle performance through thedisclosure of local estimates (althoughtheremaybe loopsinthecell). Welist thefeatures of thecellstructures,e.g.,seeninFig. 5,asfollows:

•

Agentsoutofacellcanconnect toatmostoneoftheagents withinthatcell.

•

Acellstructureconsistsofatleast2agents.

•

Anagentcanbelongtomorethanonecell.

•

Twodifferentagentscannot belong to morethan one cell at thesametime.

•

Alloftheagentsbelongtoatleastacellina connected net-work.

•

Each agent has also the knowledge ofthe cells of the other agents.

•

Each agent labels its cells from its own and its ﬁrst order neighbor’spoint ofview.As anexample,foragent-i,

C

i,i1

de-notesthe cell involving both i and i1. Notethat ifthe same cellalsoincludesi2,

C

i,i1

=

C

i,i2.

Thefollowingtheoremshowsthatagentscanachievetheoracle performance over treenetworksinvolvingcell structuresthrough thedisclosureofthelocalestimates.

Theorem3.2.Considertheteamproblemovertreenetworksinvolving cellstructures.Then,exchangeoflocalestimatescanleadtothe team-optimalsolution,i.e.,agentscanachievetheoracleperformance.

Proof. Initially, we have

o_j_,_i_,₁

= δ

j,1

= {

yj,1

}

and the oracle action is also given by (11) over this network topology. Note that the information received by j at t

=

2 is given by

δ

o_j_,₂

=

yj,2

,

=ok, j,0

yk,1

k∈Nj

,

δ

o_j_,₁

,

whichyields

o_j_,_i_,₂

= δ

o_j_,₂

\

⎧

⎨

⎩

δ

oj,1

∪

k∈Ci,j

yk,1

⎫

⎬

⎭

,

= δ

o j,2

\

⎧

⎨

⎩

δ

oj,1

∪

k∈Ci,j

o_k_,_j_,₁

⎫

⎬

⎭

and

o

j,j,t

= ∅

bydeﬁnition.Duetothecellstructure,wehave

k∈Cj,i

_ko_,_j_,₁

=

o_i_,_j_,₁

∪

k∈Ci,j\j

o_k_,_i_,₁

,

= δ

o i,1

∪

k∈Ci,j\j

δ

_ko_,₁

.

Correspondingly,fort

>

0 wehave

o_j_,_i_,_t

= δ

o_j_,_t

\

⎧

⎨

⎩

δ

oj,t−1

∪

oi,j,t−1

∪

k∈Ci,j\j

o_k_,_i_,_t₋₁

⎫

⎬

⎭

(36) and

o

j,i,t isconstructible bythesets

δ

io,τ and

δ

oj,τ for j

∈

N

i and

τ

≤

t.Note that overtree networks,

C

i,j

\

j

= ∅

fori

=

1

,

. . . ,

m,

j

∈

N

i,and(36)leadsto(18).Hence,fort

>

0 weobtain(19)and theproofisconcluded.

2

Notethatthenetworkcanhaveloopswithinthecellstructures and agents can still achieve the oracle performance through the diffusionofthelocalestimates.Thisincreasestherobustnessofthe teamstrategiesagainstthelinkfailures.Inthesequel,weprovide thesub-optimalextensionsofthealgorithmsforpractical applica-tions.

4. Sub-optimalapproaches

Minimization of the cost function (4) optimally requires rel-atively excessive computations. We aim to mitigate the problem sub-optimally yet in a computationally eﬃcient approach while achieving comparable performance withthe optimal case. As an example,we canapproximatethe costmeasure (4)through To

≥

max

{

κ

i

}

i=1,...,msizetime-windowingasfollows

T

t=1 min ηi,t∈si,t E

x

−

η

i,t

(δ

is,t

)

2

,

(37) whereagent-i hastheinformationset

δ

_is_,_t

:=

yi,τ

t−To<τ≤t

_,

_y j,τ

t−To<τ≤t−1 j∈N_i(1)

, . . . ,

yj,τ

t−To<τ≤t−κi j∈N(κi) i

,

(38)

if t

≥

To and other cases are deﬁned accordingly,

σ

is,t is the sigma-algebra generated by

δ

s_i_,_t, and

o_i_,_t denotes the set of all