Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication

0.89 MB·

by  Solmaz S. Kia

Distributedconvexoptimizationviacontinuous-time
⋆
coordinationalgorithmswithdiscrete-timecommunication
SolmazS. Kiaa JorgeCort´esb SoniaMart´ınezb
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1 aDepartment of Mechanical and Aerospace Engineering, University ofCalifornia,Irvine
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2 bDepartment ofMechanical andAerospace Engineering, University of California,San Diego

g
u
A
Abstract

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2 This paper proposes a novel class of distributed continuous-time coordination algorithms to solve network optimization
problemswhosecostfunctionisasumoflocalcostfunctionsassociatedtotheindividualagents.Weestablishtheexponential

] convergence of the proposed algorithm under (i) strongly connected and weight-balanced digraph topologies when the local
C
costs are strongly convex with globally Lipschitz gradients, and (ii) connected graph topologies when the local costs are
O strongly convex with locally Lipschitz gradients. When the local cost functions are convex and the global cost function is
strictly convex,we establish asymptotic convergence underconnected graph topologies. Wealso characterize thealgorithm’s
.
h correctness under time-varying interaction topologies and study its privacy preservation properties. Motivated by practical
t considerations, we analyze thealgorithm implementation with discrete-time communication. We provide an upperbound on
a
thestepsizethatguaranteesexponentialconvergenceoverconnectedgraphsforimplementationswithperiodiccommunication.
m
Building on this result, we design a provably-correct centralized event-triggered communication scheme that is free of Zeno
[ behavior.Finally,wedevelopadistributed,asynchronousevent-triggeredcommunicationschemethatisalsofreeofZenowith
asymptoticconvergenceguarantees. Severalsimulations illustrate ourresults.
3
v
2 Keywords: cooperative control, distributedconvexoptimization, weight-balanced digraphs, event-triggered control.
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4
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. 1 Introduction Literature review: In distributed convex optimiza-
1
0 An important class of distributed convex optimiza- tion, most coordination algorithms are time-varying,
4 consensus-baseddynamics[Boydetal.,2010,Duchietal.,
tion problems consists of the (un-)constrained net-
1 2012, Johanssonetal., 2009, Nedi´c andOzdaglar,
work optimization of a sum of convex functions, each
: 2009, ZhuandMart´ınez, 2012] implemented in dis-
v one representing a local cost known to an individual
crete time. Recent work [GharesifardandCort´es, 2014,
i agent. Examples include distributed parameter estima-
X LuandTang,2012,WangandElia,2011,Zanellaetal.,
tion[Rametal.,2010,WanandLemmon,2009],statis-
2011]hasintroducedcontinuous-timedynamicalsolvers
r tical learning [Boydetal., 2010], and optimal resource
a whose convergence properties can be analyzed via
allocation over networks [MadanandLall, 2006]. To
classical stability analysis. This has the added advan-
find the networkoptimizers, we propose a coordination
tage of facilitating the characterization of properties
modelwhereeachagentrunsapurelylocalcontinuous-
such as speed of convergence, disturbance rejection,
time evolution dynamics and communicates at discrete
and robustness to uncertainty. WangandElia [2011]
instantswithitsneighbors.Wearemotivatedbythede-
establish asymptotic convergence under connected
sire of combining the conceptual ease of the analysis of
graphs and GharesifardandCort´es [2014] extend the
continuous-timedynamicsandthepracticalconstraints
design and analysis to strongly connected, weight-
imposed by real-time implementations. Our design is
balanced digraphs. The continuous-time algorithms
basedonanovelcontinuous-timedistributedalgorithm
in [LuandTang, 2012, Zanellaetal., 2011] require
whose stability can be analyzedthrough standard Lya-
twice-differentiable, strictly convex local cost func-
punovfunctions.
tions to make use of the inverse of their Hessian and
need a careful initialization to guarantee asymptotic
⋆ convergence under undirected connected graphs. The
Corresponding author:S.S.Kia
novelclassofcontinuous-timealgorithmsproposedhere
Emailaddresses: [email protected] (Solmaz S.Kia),
[email protected] (Jorge Cort´es), [email protected] (upon which our implementations with discrete-time
communicationarebuilt)do notsufferfromthe limita-
(SoniaMart´ınez).
Preprintsubmitted toAutomatica 25August 2014
tions discussed above and have, under some regularity 2 Preliminaries
assumptions, exponential convergence guarantees. Our
Inthissection,weintroduceournotationandsomebasic
work here also touches, albeit slightly, on the concept
concepts from convex functions and graph theory. Let
of privacy preservation, see e.g., [Weeraddanaetal.,
RandZ denote,respectively,thesetofrealandnon-
2013,Yanetal.,2013]forrecentworksinthecontextof ≥0
negative integer numbers. We use () to represent the
distributed multi-agent optimization. Regarding event- ℜ ·
realpart of a complex number. The transpose of a ma-
triggered control of networked systems, recent years
trix A is A⊤.We let 1 (resp. 0 ) denote the vectorof
haveseenanincreasingbodyofworkthatseekstotrade n n
nones(resp.n zeros),anddenotebyI the n niden-
computation and decision making for less communica- n
tion, sensing or actuator effort, see e.g. [Heemels etal., titymatrix.WeletΠn =In− n11n1⊤n.Whenc×learfrom
2012, MazoandTabuada, 2011, Wang andLemmon, the context, we do not specify the matrix dimensions.
2011].Closesttotheproblemconsideredhereareworks ForA Rn×m andB Rp×q,weletA Bdenotetheir
∈ ∈ ⊗
that study event-triggered communication laws for av- Kronecker product. For u Rd, u = √u⊤u denotes
∈ k k
erage consensus, see e.g., [Dimarogonasetal., 2012, the standard Euclidean norm. For vectors u , ,u ,
1 m
···
Garciaetal., 2013, NowzariandCort´es, 2014]. The u = (u , ,u ) is the aggregated vector. In a net-
1 m
···
strategies proposed in [WanandLemmon, 2009] save workedsystem,wedistinguishthelocalvariablesofeach
communication effort in discrete-time implementations agentbya superscript,e.g.,xi isthe state ofagenti.If
by using local triggering events but are not guaranteed pi Rd is a variable of agent i, the aggregated pi’s of
to avoid Zeno behavior, i.e., an infinite number of trig- the∈network of N agents is p = (p1, ,pN) (Rd)N.
gered events in a finite period of time. Our goal is to Forconvenience,we use r RN andR··· RN×(∈N−1),
combine the best of both approaches by synthesizing ∈ ∈
1
provably-correct continuous-time distributed dynam- r= 1 , r⊤R=0, R⊤R=I , RR⊤=Π . (1)
N N−1 N
ical systems which only require communication with √N
neighbors at discrete instants of time. We are partic-
ularly interested in the opportunistic determination of A differentiable function f : Rd R is strictly convex
this communicationtimes viaeventtriggeringschemes. overaconvexsetC Rdiff(z x→)⊤( f(z) f(x))>
⊆ − ∇ −∇
Statementof contributions:We proposea novelclassof 0forallx,z C andx=z,anditism-stronglyconvex
continuous-time, gradient-based distributed algorithms (m > 0) iff ∈(z x)⊤( 6 f(z) f(x)) m z x 2,
for network optimization where the global objective for all x,z C−. A fun∇ction f−:∇Rd R≥d iskLip−schkitz
∈ →
functionisthesumoflocalcostfunctions,oneperagent. withconstantM >0,orsimplyM-Lipschitz,overaset
We prove that these algorithms converge exponentially C Rd iff f(x) f(y) M x y , for x,y C.
⊂ k − k ≤ k − k ∈
under strongly connected and weight-balanced agent Foraconvexfunctionf withM-Lipschitzgradient,one
interactions when the local cost functions are strongly has f(z) f(x) 2 M(z x)⊤( f(z) f(x))
convexandtheirgradientsaregloballyLipschitz.Under forakll∇x,z −C∇ Rd.k ≤ − ∇ −∇
connected, undirected graphs, we establish exponential ∈ ⊂
We briefly review basic concepts from algebraic graph
convergence when the local gradients are just locally
theoryfollowing[Bulloetal.,2009].Adigraph, isapair
Lipschitz, and asymptotic convergence when the local
= ( , ), where = 1,...,N is the node set and
cost functions are simply convex and the global cost G V E V { }
is the edge set.An edgefromi to j,denoted
function is strictly convex. We also study convergence E ⊆V ×V
by (i,j), means that agent j can send information to
under networks with time-varying topologies and char-
agenti.Foranedge(i,j) , iis calledanin-neighbor
acterizethetopologicalrequirementsonthecommunica- ∈E
ofj andj iscalledanout-neighbor ofi.Agraphisundi-
tiongraph,algorithmparameters,andinitialconditions
rected if (i,j) anytime (j,i) . A directed path
necessaryforanagenttoreconstructthelocalgradients ∈ E ∈ E
is a sequence of nodes connected by edges. A digraph
of local cost functions of other agents. Our technical
is strongly connected if for every pair of nodes there is
approachbuildsontheidentificationofstrictLyapunov
a directed path connecting them. A weighted digraph is
functions. The availability of these functions enable
a triplet = ( , ,A), where ( , ) is a digraph and
our ensuing design of provably-correctcontinuous-time A RN×GN is aVwEeighted adjaceVncEy matrix such that
implementations with discrete-time communication. In
a ∈> 0 if (i,j) and a = 0, otherwise. A weighted
particular, for networks with connected graph topolo- ij ij
digraphisundir∈ecEted ifa =a foralli,j .Werefer
gies,weobtainanupperboundonthesuitablestepsizes ij ji
∈V
to a stronglyconnected andundirected graphas a con-
that guarantee exponential convergence under periodic
nectedgraph.Theweightedin-andout-degrees ofanode
communication.Buildingonthisresult,wedesignacen-
i are, respectively, di = ΣN a and di = ΣN a .
tralized, synchronous event-triggered communication in j=1 ji out j=1 ij
scheme with an exponential convergence guarantees A digraph is weight-balanced if at each node i ,
∈ V
andZeno-freebehavior.Finally,we developa Zeno-free the weighted out-degree and weighted in-degree coin-
asynchronous event-triggered communication scheme cide (although they might be different across different
whose execution only requires agents to interchange nodes). Any undirected graph is weight-balanced. The
informationwiththeirneighborsandestablishitsexpo- (out-)Laplacian matrix isL=Dout A,whereDout =
nential convergence to a neighborhood of the network Diag(d1 , ,dN ) RN×N. Note−that L1 = 0. A
optimizer.Severalsimulationsillustrateourresults. digraphouits·w··eighotu-tbal∈anced iff 1TL = 0 iff SNym(L) =
N
2
(L+L)/2 is positive semi-definite. Based on the struc- in[GharesifardandCort´es, 2014,Wang andElia,2011]
tureofL,atleastoneoftheeigenvaluesofLiszeroand require the communication of the corresponding vari-
therestofthemhavenonnegativerealparts.Wedenote ables in both x and v. The synthesis of algorithm (4)
the eigenvalues of L by λ ,...,λ , where λ = 0 and is inspired by the following feedback control consid-
1 N 1
(λ ) (λ ),fori<j,andtheeigenvaluesofSym(L) erations. In (4b), each agent follows a local gradient
i j
ℜ ≤ℜ
by λˆ ,...,λˆ . For a strongly connected and weight- descentwhiletryingtoagreewithitsneighborsontheir
1 N
balanceddigraph, zero is a simple eigenvalue of both L estimate of the final value. However, as the local gra-
and Sym(L). In this case, we order the eigenvalues of dients are not the same, this dynamics by itself would
Sym(L) asλˆ =0<λˆ λˆ λˆ andwehave never converge. Therefore, to correct this error, each
1 2 ≤ 3 ≤···≤ N agentusesanintegralfeedbacktermvi whoseevolution
0<λˆ I R⊤Sym(L)R λˆ I. (2) isdrivenby the agentdisagreementaccordingto(4a).
2 N
≤ ≤
Ouranalysisofthe algorithmconvergenceis structured
Notice that for connected graphs λˆ = λ for i
1, ,N . For convenience, we definie L = Li I an∈d intwoparts,dependingonthedirectedcharacterofthe
{Π=··Π· }I todealwithvariablesofdimensio⊗ndd N. interactions. Section 4.1 deals with strongly connected
N ⊗ d ∈ andweight-balanceddigraphsandSection4.2dealswith
connected graphs. In each case, we identify conditions
3 ProblemDefinition
on the agent cost functions that guarantee asymptotic
Consider a network of N agents interacting over a convergence.Giventhechallengesposedbydirectedin-
strongly connected and weight-balanced digraph . formation flows, it is not surprising that we can estab-
G
Eachagenti hasadifferentiablelocalcostfunction lish stronger results under less restrictive assumptions
fi : Rd R.∈TVhe global cost function of the network, forthe caseofundirectedtopologies.
whichwe→assumetobestrictlyconvex,isf =ΣN fi(x).
i=1 4.1 StronglyConnected,Weight-Balanced Digraphs
Our aim is to design a distributed algorithm such that
eachagentsolves Here, we study the convergence of the distributed op-
min f(x) timization algorithm (3) over strongly connected and
x∈Rd
weight-balanced digraph topologies. We first consider
using only its own local data and exchanged informa-
thecasewheretheinteractiontopologyisfixed,andthen
tionwithitsneighbors.Weassumetheaboveoptimiza-
discussthetime-varyinginteractiontopologies.Thefol-
tionproblemis feasible (which,together withthe strict
lowingresultidentifiesconditionsonthelocalcostfunc-
convexityoff,impliestheuniquenessoftheglobalopti- tions fi N andthe parameterβ toguaranteethe ex-
mizer,whichwedenotex⋆ Rd).Wearealsointerested { }i=1
ponential convergence of (3) to the solution of the dis-
∈
in characterizing the privacy preservation properties of
tributedoptimizationproblem.
thealgorithmicsolutiontothisdistributedoptimization
Theorem 1 (Convergence of (3) over strongly con-
problem.Specifically,weaimtoidentifyconditionsguar-
nectedandweight-balanceddigraphs): Let beastrongly
anteeing that no information about the local costfunc-
G
connected and weight-balanced digraph. Assume each
tion of an agent is revealed to, or can be reconstructed
fi, i , is mi-strongly convex, differentiable, and
by,any otheragentinthe network.
its gra∈dieVnt is Mi-Lipschitz on Rd. Given α > 0,
4 Distributed Continuous-Time Algorithm for m = min m1,...,mN and M = max M1,...,MN ,
{ } { }
Convex Optimization letβ,φ>0satisfy φ+1>4M and
In this section, we provide a novelcontinuous-time dis- γ=α2(φ+1)m+9βλˆ φα 4α2(Mm+(φ+1)2)>0, (5)
2
−
tributed coordination algorithm to solve the problem
statedinSection3andanalyzeindetailitsconvergence Then, for each i , starting from xi(0),vi(0) Rd
prov˙pie=rtiαesβ.ForNj=i∈1aVija(xnid−wxitjh),α,β>0,consider (3a) lxweiis(tsht)thP→anNi=x1⋆veix(p0o)n=e∈n0tVdia,lltyhefaasltgoarsitthm→(∞3)woivtehraGram∈teakneos
7 1
x˙i = αXfi(xi) β N a (xi xj) vi, (3b) min , γ /(2λ¯F). (6)
− ∇ − j=1 ij − − {16 9 }
X Here,λ¯F is themaximumeigenvalue of
Innetworkvariablesx,v (Rd)N,thealgorithmreadsas 1α(φ+1)I 0 0
v˙ =αβLx∈, (4a) F= 19 0 d α(φ+1)I I . (7)
x˙ =−α∇f˜(x)−βLx−v. (4b) 2 0 I(N−1()Nd−1)d α1(IN(N−−11)d)d
 
Here, f˜: (Rd)N R is defined by f˜(x) = ΣN fi(xi).
→ i=1 PROOF. For weight-balanced digraphs, we have
This algorithm is distributed because each agent only
1⊤L=0.Thus,leftmultiplying(4a)by1⊤ I resultsin
needs to receive information from its out-neighbors N N⊗ d
about their corresponding variables in x. In con- N v˙i=0 N vi(t)= N vi(0)=0, t 0. (8)
trast, the continuous-time coordination algorithms i=1 ⇒ i=1 i=1 ∀ ≥
X X X
3
Next,weobtaintheequilibriumpointof(3),(x¯,v¯),from Next, we show that under φ+1 > 4M and (5), V˙ is
negative definite. Invoking the assumptions on the lo-
cal cost functions in the statement, and using the M-
0=αβLx¯, (9a)
Lipschitzness of f˜and the m-strongly convexity of f˜
0= α f˜(x¯) βLx¯ v¯. (9b) alongwith R⊤ ∇I =1 and z = y ,we have
− ∇ − − k ⊗ dk k k k k
(R⊤ I )h 2 h 2 My⊤h, (15a)
Given(9a),x¯belongstothenull-spaceofL.Forstrongly k ⊗ d k ≤k k ≤
connecteddigraphsthenull-spaceofLisspannedby1N. y⊤h m y 2 =m z 2. (15b)
Thus,x¯ =1 θ,whereθ Rd.Leftmultiplying (9b) ≥ k k k k
N ⊗ ∈
by1⊤ I andusing(8),weobtain0= N fi(x¯i). Giventhese relationsandinvoking (2),wehave
ThenN,t⊗hedoptimalitycondition f(x⋆)=0i=a1lo∇ngwith
∇f(θ)= Ni=1∇fi(θ)imply ∇ Pd V˙ ≤−α2((φ+1)9−4M)mz⊤z−176w⊤2:Nw2:N
P x¯i =x⋆, i . φαβλˆ z⊤ z + 4α2(1+φ)2z⊤ z
∈V − 2 2:N 2:N 9 2:N 2:N
3 2α 2α(φ+1)
Substituting this valuein(9b),weobtain w + (R⊤ I )h+ z ) 2.
−k4 2:N 3 ⊗ d 3 2:N k
v¯i = α fi(x⋆), i . (10)
− ∇ ∈V Since z⊤z = z⊤z + z⊤ z , it follows that V˙ <
1 1 2:N 2:N
Tostudythestabilityof (3),wetransfertheequilibrium min 7 ,1γ p 2 < 0, where γ is a shorthand nota-
− {16 9 }k k
pointtotheoriginandthenapplyachangeofvariables tion for the expression in (5). Thus, z 0 as t ,
equivalently xi x⋆, for all i , is e→xponentia→l w∞ith
u=v−v¯, y=x−x¯, (11a) ratenolessthan→(6)(cf.[Khalil,∈20V02,Theorem4.10]).✷
u=([r R] I )w, y=([r R] I )z, (11b)
⊗ d ⊗ d
InTheorem1,therequirement N vi(0)=0 istriv-
whereweused(1).Wepartitionthenewvariablesasfol- i=1 d
lows:w=(w ,w )andz=(z ,z ),wherew ,z ially satisfiedby eachagentwith vi(0)=0d. This is an
1 2:N 1 2:N 1 1 P
Rd.Inthese new variables,the algorithm(3)readsas∈ advantagewithrespecttothecontinuous-timecoordina-
tionalgorithmsproposedin[LuandTang,2012],which
w˙1 =0d, requiresthenontrivialinitialization N fi(xi(0))=
i=1∇
w˙2:N =αβ(R⊤LR⊗Id)z2:N, 0tida,liaznatdioinn [oZnanaelsltaaetetaclo.,m2m01u1n]i,cawtheidPchamreoqnugirenseitghhebionris-
z˙ = α(r⊤ I )h, (12)
1 − ⊗ d andis subjectto communicationerror.
z˙2:N =−α(R⊤⊗Id)h−β(R⊤LR⊗Id)z2:N −w2:N, Remark2 (Designparametersin(3)):Weprovidehere
severalobservationsontheroleofthedesignparameters
where αandβ.First,notetherealwaysexistα,βsatisfying(5),
e.g., any β > 4(φ+1)2α/(9φλˆ ). The determination of
h= f˜(y+x¯) f˜(x¯). (13) 2
∇ −∇ theseparameterscanbeperformedbyindividualagents
iftheyknowanupperboundonM,alowerboundonm,
Note that the first equation in (12) corresponds to the andhaveknowledgeofλˆ ,eitherthroughadedicatedal-
constantofmotion(8).Tostudythestabilityintheother 2
gorithmtocomputeit,seee.g.,Yangetal.[2010],oruse
variables,considerthe candidate Lyapunovfunction
alowerboundonit,seee.g.Mohar[1991].We haveob-
1 φα servedinsimulationthat(5)is onlysufficientandthat,
V(z,w2:N)= α(φ+1)z1⊤z1+ z2:N⊤z2:N (14) in fact, the algorithm (3) converges for any positive α
18 2
1 andβ inournumericalexamples.Althoughnotevident
+ (αz2:N +w2:N)⊤(αz2:N +w2:N), in(6),onecanexpectthelargerαandβ are,thehigher
2α
the rate of convergence of the algorithm (3) is. A coef-
ficient α > 1 can be interpreted as a way of increasing
with φ>0 as in the statement. Note that V λ¯F p 2, the strong convexity coefficient of the local cost func-
≤ k k
withp=(z,w ).TheLiederivativeofV along(12)is
2:N tions.Acoefficientβ >1canbeinterpretedasameans
1 7 of increasing the graph connectivity. The relationship
V˙ =− 9α2(φ+1)y⊤h− 16w⊤2:Nw2:N between these parameters and the rate of convergence
of (3) is more evidentfor quadraticlocalcostfunctions
−φαβz2:N⊤(R⊤Sym(L)R⊗Id)z2:N fi(x)= 1(x⊤x+x⊤ai+bi), i . In this case,the al-
4 4 2 ∈V
+ α2 (R⊤ I )h 2+ α2(1+φ)2z⊤ z gorithm (3) is a linear time-invariant system where the
9 k ⊗ d k 9 2:N 2:N eigenvaluesofthesystemmatrixare α,withmultiplic-
3w + 2α(R⊤ I )h+ 2α(φ+1)z ) 2. ityofNd,and βλi,i (λi’sareth−eeigenvaluesofL),
−k4 2:N 3 ⊗ d 3 2:N k with multiplic−ity d. Th∈eVrefore, (3) converges regardless
4
of the value of α, β > 0 with an exponential rate equal for all t t⋆, it canuse the time history ofxj(t) to nu-
≤
tomin α,β (λ2) .Whendiscussingdiscrete-timecom- mericallyreconstructx˙j(t⋆).Becauseiisthein-neighbor
{ ℜ }
munication,sometrade-offsariseregardingthechoiceof of j and its out-neighbors, it can use its knowledge of
the parameters,aswe explainlaterin Section5. a ,k toreconstruct N a (xj(t) xk(t))forall
Remark3 (Semiglobal convergence of (3) under loca•l tjk t⋆.∈AVgenticanreconstrukc=t1vjjk(t)from−(3a)uniquely
gradientsthatarelocallyLipschitz):Theconvergencere- as≤it knows vj(0). Then,Pagent i has all the elements
sult in Theorem 1 is semiglobal [Khalil, 2002] if the lo- to solve for fj(xj(t⋆)) in (3b). The lack of knowledge
cal gradients are only locally Lipschitz or, equivalently, ∇
about any of this information would prevent i from re-
Lipschitzoncompactsets.Infact,onecanseefromthe constructingexactlythe localgradientofj. ✷
proofofthe result that, for any compactsetcontaining
the initial conditions xi(0) Rd andvi(0)=0 , i ,
d
∈ ∈V 4.2 ConnectedGraphs
onecanfindφ>0andβ >0sufficientlylargesuchthat
the compactset is containedin the regionof attraction Here,westudytheconvergenceofthealgorithm(3)over
ofthe equilibriumpoint. connectedgraphtopologies.Whiletheresultsofthepre-
•
Next,westudytheconvergenceof (3)overdynamically vious section are of course valid for these topologies,
changing topologies. Since the proof of Theorem 1 re- here,usingthestructuralpropertiesoftheLaplacianma-
liesonaLyapunovfunctionwithnodependencyonthe trix of undirected graphs, we establish the convergence
system parameters and its derivative is upper bounded of (3)foralargerfamilyoflocalcostfunctions.Indoing
byaquadraticnegativedefinitefunction,wecanreadily so,wearealsoabletoanalyticallyestablishconvergence
extend the convergence result to dynamically changing foranyα,β >0,asweshownext.
networks.The proofdetails areomittedforbrevity. Theorem 6 (Exponential convergence of (3) over con-
Proposition 4 (Convergence of (3) over dynamically nected graphs): Let be a connected graph. Assume the
changinginteractiontopologies): Let beatime-varying local cost function fGi, i , is mi-strongly convex and
digraph which is stronglyconnected aGnd weight-balanced differentiableonRd,and∈itVsgradientislocallyLipschitz.
at all times and whose adjacency matrix is uniformly Then, for any α,β > 0 and each i , starting from
bounded and piecewise constant. Assume the local cost xi(0),vi(0) Rdwith N vi(0)=0∈,tVhealgorithm (3)
function fi, i , is mi-strongly convex, differentiable, over satisfi∈es xi(t) xi=⋆1as t exponentially fast.
anditsgradien∈tVisMi-Lipschitz onRd.Givenα>0,let G →P →∞
β,φ>0satisfy φ+1>4M and (5)withλˆ replaced by
2 PROOF. We usethe equivalentrepresentation(12)of
(λˆ ) =min λˆ (L ) , where is the index set of all
2 min 2 p the algorithm (3) obtained in the proof of Theorem 1.
p∈P{ } P
possible realizations of . Then, for each i ,starting Considerthe followingcandidate Lyapunovfunction
from xi(0),vi(0) RdGwith N vi(0)=∈0 V, the algo- 1 φα
rithm (3)over m∈akes xi(t) i=x1⋆ exponentdiallyfast as V(z,w2:N)= 2α(φ+1)z⊤1z1+ 2 z⊤2:Nz2:N (16)
t . G P→ 1
→∞ + (αz2:N +w2:N)⊤(αz2:N +w2:N)
Ourfinalresultofthissectioncharacterizesthetopolog- 2α
ical requirements on the communication graph and the + 1 (φ+1)w⊤ (R⊤LR)−1 I w ,
knowledgeaboutthealgorithm’sparametersandinitial 2β 2:N ⊗ d 2:N
conditionsthatallowapassiveagent(i.e.,anagentthat (cid:0) (cid:1)
does not interfere in the algorithmexecution) to recon- withφ 1definedbelow.Given(2),V ispositivedefinite
structthelocalgradientsofotheragentsinthenetwork. and rad≥iallyunbounded and satisfies V λ¯E p 2, with
Proposition 5 (Privacy preservation under (3)): Let p=(z,w2:N) andλ¯E>0is the maximum≤eigeknvkalueof
G
beastronglyconnectedandweightbalanceddigraph.For
α,β > 0, consider any execution of the coordination al- α(φ+1)Id 0 0
gorithm (3) over starting from xi(0),vi(0) Rd with E=1 0 α(φ+1)I I .
N vi(0) = 0 .GThen, an agent i can re∈construct 2 (N−1)d (N−1)d
thei=lo1cal gradiendt of another agent ∈j =V i only if j and  0 I(N−1)d α1I+(φ+β1)(R⊤LR)−1⊗Id
aPll its out-neighbors are out-neighbors6 of i, and agent i  
knows vj(0) and a , k (here we assume that the The Lie derivativeofV alongthe dynamics(12)is
jk
∈ V
agent i is aware of the identity of neighbors of agent j
V˙ = α2(φ+1)y⊤h φαβz ⊤(R⊤LR I )z
andithasmemorytosavethetimehistoryofthedatait − − 2:N ⊗ d 2:N
receives from its out-neighbors). w ⊤w αw ⊤(R⊤ I )h.
− 2:N 2:N − 2:N ⊗ d
PROOF. Consideranarbitrarytimet⋆.Letibeanin- Next, we show that V˙ is upper bounded by a negative
neighborofagentj andallofits out-neighbors.The al- definite function.We startbyidentifying acompactset
gorithm(3)requireseachagenttocommunicateitscom- whose definition is independent of φ and contains the
ponentofxtotheirin-neighbors.Sinceagentihasmem- set = (z,w ) RNd R(N−1)d V(z,w )
0 2:N 2:N
S { ∈ × | ≤
orytosaveinformationitreceivesfromitsout-neighbors V(z(0),w (0)) . For any given initial condition,
2:N
}
5
let ρ = 1αz (0)⊤z (0) + (α+1)z (0)⊤z (0) + on Rd, and the global cost function f is strictly con-
0 2 1 1 2 2:N 2:N
( 1 + 1 + 1)w (0)⊤w (0)anddefine ¯ = z vex. Then, for any α,β > 0 and each i , starting
sRe2Ntβλdis2|co12mα2αzp⊤1aczt12.+No412t:αeNzt⊤2h:Natz(2z:N2,:Nw≤2:Nρ0)}. O0bsiemrvpelSiet0shazt{th¯i∈0s frriothmmx(i3()0)o,vveir(0G)m∈aRkedxwii(tth)P→Nix=⋆1vasi(t0)→=∞∈0.dV, the algo-
∈S ∈S
becauseV(z(0),w (0)) (φ+1)ρ and
2:N 0
≤ PROOF. We use again the equivalent representa-
1 1 tion (12) of the algorithm (3) obtained in the proof
2α(φ+1)z⊤1z1+ 4α(φ+1)z⊤2:Nz2:N ≤ of Theorem 1. To study the stability of this system,
considerthe followingcandidate Lyapunovfunction
1 1
2α(φ+1)z⊤1z1+ 2αφz⊤2:Nz2:N ≤V(z,w2:N). V(z,w )=1z⊤z+ 1 w ⊤ (R⊤LR)−1 I w .
2:N 2 2αβ 2:N ⊗ d 2:N
Here, we used φ 1 in the first inequality. Since the (cid:16) (cid:17)
≥
change of variables (11a) and (11b) are linear, the cor- Given(2),V ispositivedefiniteandradiallyunbounded.
responding x and y for z ¯ belong to compact sets, TheLie derivativeofV along (12)is givenby
0
∈ S
aswell.Then,theassumptiononthegradientsofthelo-
V˙ = αy⊤( f(y+x¯) f(x¯)) βy⊤Ly
calcostfunctionsimplies thatthereexistsM0 >0with − ∇T −∇T −
h M0 y =M0 z ,forall z ¯0.Consequently, = α N yi⊤( fi(yi+x⋆) fi(x⋆)) βy⊤Ly.
k k≤ k k k k ∈S
− i=1 ∇ −∇ −
k(R⊤⊗Id)hk2≤M02kzk2, ∀(z,w2:N)∈S0. To obtainXthe second summand, we have used z2:N =
(R⊤ I )y and RR⊤ = I rr⊤. Since the lo-
Thenwecanshow−αw2:N⊤(R⊤⊗Id)h≤ 12w⊤2:Nw2:N+ cal c⊗ostd functions are convNex−, the first summand
12α2M02z⊤z for (z,w2:N) ∈ S0. Using this inequality, of V˙ is non-positive for all y. Because the graph
factthatthelocalcostfunctionsaremi-stronglyconvex is connected, the second summand is non-positive
(andhence (15b)holds)andinvoking(2)we deduce with its null-space spanned by θ⊗1N, θ ∈ Rd.
On this null-space, the first summand becomes
V˙ ≤−α12w(φ⊤+w1)m(z+⊤1z11α+2Mz⊤22:(Nzz⊤2z:N)+−z⊤φαβzλ2z)⊤2.:Nz2:N b−eαθze⊤ro wNi=h1e(n∇θfi(=θ +0,xb⋆e)c−au∇sefi(xN⋆)), wfih(ixc)h=can fon(xly)
− 2 2:N 2:N 2 0 1 1 2:N 2:N and thPe global cost function is stir=ic1t∇ly convex b∇y as-
sumption. Then, the two summPands of V˙ can be zero
Letφ+1=21mM02+2m1α2δ0,whereδ0>0issuchthatφ≥1 simultaneously only when yi = 0, for all i ,
(sinceM0doesnotdependonφ,thischoiceisalwaysfea- which is equivalent to z = 0. Thus, V˙ is neg∈atiVve
sible). Then, we have V˙ ≤−21min{1,δ0}kpk2. As such, semi-definite, with V˙(z,w2:N) = 0 happening on the
the Lyapunov function (16) satisfies all the conditions set = (z,w ) RNd R(N−1)d z = 0 .
2:N
of [Khalil, 2002, Theorem 4.10], for the dynamics (12). NoteSthat{(12) on r∈educes t×o w˙ = |0, z˙ = 0},
2:N 1
Therefore,the convergenceofz 0 as t , equiva- and z˙ = w S. Therefore, the only trajectory
lently xi x⋆,for alli ,is ex→ponentia→l. ∞ ✷ of (122):Nthat r−ema2:iNns in is the equilibrium point
→ ∈V S
(z = 0,z = 0,w = 0). The LaSalle invariance
1 2:N 2:N
Remark7 (Boundonexponentialrateofconvergence): principle (cf. [Khalil, 2002, Theorem 4.4 and Corol-
One can see from the proof of Theorem 6 that, for a lary 4.2]) now implies that the equilibrium is globally
giveninitialcondition,therateofconvergenceisatleast asymptoticallystableor,inotherwords,xi x⋆, i
41(min{1,δ0})/λ¯E > 0. The guaranteed rate of conver- globallyasymptotically. → ∈✷V
genceisthereforenotuniform,unlessthelocalgradients
are globally Lipschitz. In this case, one recoversthe re- Remark9 (Simplification of (3)): For strictly convex
sultinTheorem1but forarbitraryα, β >0. local cost functions, using the LaSalle function of the
•
NotethatinTheorem6thelocalLipschitznessof fiis proof of Theorem 8, one can show that the asymptotic
triviallyheldiffi istwicedifferentiable.TheLya∇punov convergetothe optimizer,startingfromthe initialcon-
function (16) identified in the proof of this result plays dition stated in Theorem 8, is also guaranteed for the
akeyrolelaterinourstudyofthealgorithmimplemen- followingalgorithmoverconnectedgraphs
tation with discrete-time communication in Section 5. N
Next,westudytheconvergenceofthealgorithm(3)over v˙i = aij(xi xj),
j=1 −
connectedgraphswhenthelocalcostfunctionsareonly
x˙i =Xfi(xi) vi.
convex.Here,thelackofstrongconvexitymakesusrely −∇ − •
on a LaSalle function, rather than on a Lyapunov one,
toestablishasymptoticconvergencetothe optimizer. 5 Continuous-time Evolution with Discrete-
TimeCommunication
Theorem 8 (Asymptotic convergence of (3) over con-
nected graphs): Let be a connected graph. Assumethe Here,weinvestigatethedesignofcontinuous-timecoor-
G
local cost function fi, i , is convex and differentiable dination algorithms with discrete-time communication
∈V
6
to solve the distributed optimization problem of Sec- 1 (αM +1)ζ
τ= ln 1+ ,
tion 3. The implementation of (3) requires continuous- αM +1 αM +1+βλ √1+α2(1+ζ)
N
time communication among the agents. While this (cid:16) (cid:17)
(20)
abstraction is useful for analysis, in practical scenarios
communication is only available at discrete instants
where ζ2 = 2ǫ(1−ǫ)λ2min{δ,1} . Then, if ∆ (0,τ),
of time. This observation motivates our study here. αβλ2Nφ+4α2λ2(1+φ)2 ∈
Throughout the section, we deal with communication the algorithm evolution starting from initial conditions
topologies described by connected graphs. Our results xi(0),vi(0) Rd with N vi(0) = 0 makes xi(t)
build on the discussion of Section 4, particularly the x⋆ exponenti∈ally fast as ti=1 , for alldi with a ra→te
identificationofLyapunovfunctions forstability. ofnoless than 1ǫ(minP1,→δ )∞/λ¯E >0. ∈V
4 {2 }
Westartbyintroducingsomeusefulconventions.Atany
given time t R , let xˆj be the last known state of PROOF. We start by transferring the equilibrium
≥0
agent j t∈ransmitted to its in-neighbors. If ti point to the origin using (11a) and then apply the
R den∈oteVs the times at which agent i commu{nicka}te⊂s changeofvariables(11b)to write (18)as
≥0
with its in-neighbors, then one has xˆi = xi(tik) for t ∈ w˙1 =0d, (21a)
[tik,tik+1). Consider the next implementation of the al- w˙2:N =αβ(R⊤LR⊗Id)(z2:N +z˜2:N), (21b)
gorithm(3)withdiscrete-time communication, z˙ = α(r⊤ I )h, (21c)
N 1 − ⊗ d
v˙i =αβ a (xˆi xˆj), (18a) z˙ = α(R⊤ I )h (21d)
j=1 ij − 2:N − ⊗ d
x˙i = αXfi(xi) β N a (xˆi xˆj) vi. (18b) −β(R⊤LR⊗Id)(z2:N +˜z2:N)−w2:N,
ij
− ∇ − j=1 − −
where z˜ (t) = z (t ) z (t), for t [t ,t ),
X 2:N 2:N k 2:N k k+1
− ∈
Clearly,the evolutionof (18)depends onthe sequences and h is given by (13). To study the stability of (21b)-
of communication times for each agent. Here, we con- (21d),considertheLyapunovfunction(16)withφsatis-
sider three scenarios. Section 5.1 studies periodic com- fying(19).ItsLiederivativecanbeboundedby(details
municationschemeswhereallagentscommunicatesyn- similartotheproofofTheorem6areomittedforbrevity)
chronously at ∆ intervals of time, i.e., ti = t = ∆k
k k δ(1 ǫ) δǫ
for all i .We provide a characterizationof the peri- V˙ − (z⊤z +z⊤ z ) (z⊤z +z⊤ z )
ods that∈guVarantee the asymptotic convergence of (18) ≤− 2 1 1 2:N 2:N − 2 1 1 2:N 2:N
totheoptimizer.Ingeneral,periodicschemesmightre- (1−ǫ)w⊤ w ǫw⊤ w φαβλ (1 ǫ)z⊤ z
sult in a wasteful use of the communication resources − 2 2:N 2:N−4 2:N 2:N− 2 − 2:N 2:N
becauseoftheneedtoaccountforworst-casesituations 1 1
+ φαβλ2 ˜z⊤ z˜ + α2(φ+1)2z˜⊤ ˜z
indetermining appropriateperiods.This motivates our 4ǫλ N 2:N 2:N ǫ 2:N 2:N
2
study in Section 5.2 of event-triggered communication 1 1
schemes that tie the communication times to the net- ≤−φαβλ2(1−ǫ)z⊤2:Nz2:N − 2ǫmin{δ,2}p⊤p
work state for greater efficiency. We discuss two event-
1
triggeredcommunicationimplementations,acentralized − 2(1−ǫ)min{δ,1} p⊤p−ζ−2z˜⊤2:N˜z2:N , (22)
synchronousoneandadistributedasynchronousone.In
both cases, we pay special attention to ruling out the (cid:0) (cid:1)
where p = (z,w ) and ǫ and ζ are given in the the-
2:N
presence of Zeno behavior (the existence of an infinite
orem’sstatement.Observethatateachcommunication
numberofupdates ina finite intervaloftime).
time t , ˜z (t ) = 0, then, it grows until next com-
k 2:N k
k k
5.1 Periodic Communication municationattimetk+1whenitbecomeszeroagain.Our
proofproceedsbyshowingthatift <t +τ,whereτ
The following result provides an upper bound on the k+1 k
isgivenin(20),thenwe havethe guaranteethat
sizeofadmissiblestepsizesfortheexecutionof (18)over
˜z (t) <ζ p(t) , t [t ,t ), (23)
connectedgraphswithperiodiccommunicationschemes. 2:N k k+1
k k k k ∈
Theorem 10 (Convergence of (18) with periodic com-
(note that, from (22), this guarantee ensures that V˙ is
munication): Let beaconnectedgraph.Assumethelo-
G negativedefiniteforallt 0).Tothisend,westudythe
cal cost function fi, i ,is mi-stronglyconvex, differ- ≥
entiable, and its gradi∈enVt is Mi-Lipschitz on Rd. Given dynamicsofq =k˜z2:Nk/kpkandfindalowerboundon
the time that it takes for q to evolve from zero (recall
α,β >0,consideranimplementationof (18)withagents
˜z (t )=0)to ζ.Notice that
communicating over synchronously every ∆ seconds 2:N k
.
s0ta<rtǫin<g1atatn0d=δ >0,0i.esGu.,cthikth=attk = ∆k for all i ∈ V. Let q˙ = z˜⊤2:Nz˜2:N k˜z2:Nkp⊤p˙ (1+q)kp˙k (1+q)
˜z p − p 3 ≤ p ≤ ×
2:N
1 2 1 k kk k k k k k
φ= M + δ 1>0, (19) (αM+βλ √1+α2) p + w +βλ √1+α2 ˜z
2m 2mα2 − N k k k 2:Nk N k 2:Nk
p
k k
whereM andm aregiven in Theorem 1, anddefine (αM +1)(1+q)+βλ 1+α2(1+q)2.
N
≤
p
7
Here,weusedinthefirstinequalityd/dt(˜z )= z˙ uation of the triggering condition (23) requires knowl-
2:N 2:N
−
and z˙ p˙ and in the second one the evolution edge of the global optimizer. Our forthcoming discus-
2:N
k k ≤ k k
ofp˙ givenin(21b)-(21d), R⊤LR L =λ and sion shows how one can circumvent this problem. We
N
k k≤k k firstconsiderthedesignofcentralizedtriggersrequiring
βR⊤LR global network knowledge and then discuss triggering
(cid:13)"−αβR⊤LR#(cid:13)=βλN 1+α2. schemesthatonlyrelyoninter-neighborinteraction.
(cid:13)(cid:13) (cid:13)(cid:13) p 5.2.1 Centralized Synchronous Implementation
(cid:13) (cid:13)
UsingtheCo(cid:13)mparisonLem(cid:13)ma(cf.[Khalil,2002,Lemma Here,wepresentacentralizedevent-triggeredschemeto
3.4]),weconcludethatq(t,q ) ψ(t,ψ ),whereψ(t,ψ ) determine the sequence of synchronous communication
0 0 0
isthesolutionofψ˙ =(αM+1)≤(1+ψ)+βλ √1+α2(1+ times in (18). Our discussionbuilds upon the examina-
N
ψ)2 satisfyingψ(0,ψ )=ψ .Then, tionoftheLiederivativeoftheLyapunovfunctionused
0 0
in the proof of Theorem 10 and the observations made
q(t,0) ψ(t,0) aboveregardingthe lackof knowledgeof the globalop-
≤
(αM +1+βλ √1+α2)(eαMt+t 1) timizer. From (24), we see that an event-triggered law
N
= − . shouldnot employ p, but ratherrely on ˜z and z ,
2:N 2:N
βλN√1+α2eαMt+t+αM +1+βλN√1+α2 to be independent of x⋆. With this in mind, the exami-
−
nationofthe upper bound(22)onV˙ revealsthat,if
The time τ for ψ(τ,0)=ζ is given by (20). Then, for
{tk+1 − tk}k∈Z≥0 < τ, we have (23), and as a result k˜z2:N(t)k2 =kΠ(x(tk)−x(t))k2
from(22)wehaveV˙ < 1ǫmin δ,1 p⊤p.Thus,z 0, κ z2:N(t) 2 =κ Πx(t) 2, (25)
−2 { 2} → ≤ k k k k
ast ,whichisequivalenttoxi x⋆ ast ,ex-
pone→nti∞allyfastwiththe rategiven→inthe stat→em∞ent.✷ whereκ isshorthandnotationfor
ǫδλ +2φαβλ2ǫ2(1 ǫ)
Remark11 (Dependenceof thecommunication period κ=2 2 2 − , (26)
onthedesignparameters):Thevalueofτ inTheorem10 αβφλ2N +2λ2α2(1+φ)2
depends on the graph topology, the parameters of the
localcostfunctions,thedesignparametersαandβ,and (here0<ǫ<1andφis givenby (19)), thenwehave
the variables ǫ and δ. One can use this dependency to
maximizethevalueofτ.Notethattheargumentofln(.) 1
V˙ φαβλ (1 ǫ)2z ⊤z (1 ǫ)w ⊤w (27)
in (20) is a monotonically increasing function of ζ > 0. 2 2:N 2:N 2:N 2:N
≤− − −2 −
Therefore,thesmallerthevalueofβ,thelargerthevalue 1 1
of τ. However, the dependency of τ on the rest of the δz1⊤z1 min δ,2φαβλ2(1 ǫ)2,(1 ǫ) p⊤p.
− 2 ≤−2 { − − }
parameterslistedaboveismorecomplex.Forgivenlocal
cost functions, fixed network topology and fixed values
Then, we can reproduce the proof of Theorem 10 and
of α, β, the maximum value of ζ is when φ+1 is at its
concludetheexponentialconvergencetotheoptimalso-
minimumandǫλ min 1 ǫ,δ is atits maximum.
2 lution. Accordingly, the sequence of synchronous com-
{ − } •
5.2 Event-Triggered Communication munication times {tk}k∈Z≥0 ⊂ R≥0 for (18) should be
determinedby(25).However,foratrulyimplementable
This section studies the design of event-triggered com-
law,oneshouldruleoutZenobehavior,i.e.,thesequence
municationschemesfortheexecutionof(18).Incontrast
of times does not have any finite accumulation point.
toperiodicschemes,event-triggeredimplementationstie
However,observing(25),onecanseethatZenobehavior
the determination of the communication times to the
willariseatleastneartheagreementsurfaceΠx=0 .
currentnetworkstate,resultinginamoreefficientuseof dN
Thenextresultdetails howweaddressthis problem.
theresources.TheproofofTheorem10revealsthatthe
Theorem 12 (Convergence of (18) with Zeno-free cen-
satisfactionof condition(23)guaranteesthe monotonic
tralizedevent-triggeredcommunication): Let beacon-
evolution of the Lyapunov function, which in turn en-
G
suresthe correctasymptotic behaviorof the algorithm. nected graph. Assume the local cost function fi, i ,
Onecouldthereforespecifywhencommunicationshould is mi-strongly convex, differentiable, and its grad∈ieVnt
occur by determining the times when this condition is is Mi-Lipschitz on Rd. Consider an implementation
not satisfied. There is, however, a serious drawback to of (18)withagentscommunicatingover synchronously
this approach: the evaluation of the condition (23) re- at{tk}k∈Z≥0 ⊂R≥0,startingat t0 =0,G
quires the knowledge of the globalminimizer x⋆, which
t =argmax t [t +τ, )
isofcoursenotavailable.To seethis,note that k+1 { ∈ k ∞ |
Π(x(t ) x(t)) 2 κ Πx(t) 2 , (28)
˜z2:N = Π(x(tk) x) , (24a) k k − k ≤ k k }
k k k − k
p = x x¯ 2+ Π(v v¯) 2, (24b) where τ and κ < 1 are defined in (20) and (26), respec-
k k k − k k − k
tively. Then, for any given α,β > 0 and each i ,
where we have upsed (1) and (8) (recall Π=Π I ). the algorithm evolution starting from initial condi∈tioVns
From(10),v¯i= α fi(x⋆)fori ,andthustNhe⊗evdal- xi(0),vi(0) Rd with N vi(0) = 0 makes xi(t)
− ∇ ∈V ∈ i=1 d →
P
8
x⋆ exponentially fast as t with a rate no less than Given α>0,let β,φ>0 satisfy φ+1>4M and
14(min{δ,2φαβλ2(1−ǫ)2,(→1−∞ǫ),12ǫ})/λ¯E >0.
γ′=α2(φ+1)m+9βλˆ φα 4α2(Mm+(φ+1)2)>0. (30)
PROOF. We first show κ < 1. This is an important 2 2 −
propertyguaranteeingthat,ifagentsstartinagreement Then, for each i , the evolution starting from ini-
at a point other than the optimizer x⋆, then the con- tial conditions xi(∈0),Vvi(0) Rd with N vi(0) = 0
dition(25)iseventuallyviolated,enforcinginformation ∈ i=1 d
updates.Noticethat(a)4ǫ2(1−ǫ)λ22 <λ2N and(b)ǫδ < makes kxi(t)−x⋆k ≤ φ4αηβλλ¯FFkǫk2 as tP→ ∞ exponen-
α2(1+φ)2 imply that the numerator in (26) is smaller tially fast with a rate no less than η/λ¯ . (Here, η =
F
thanitsdenominator,andhenceκ<1.(a)followsfrom min 7 ,1γ′ , and λ and λ¯ are the minimum and
noting that the maximum of 4ǫ2(1 ǫ) for ǫ (0,1) is max{im16um9 ei}genvaluesFofFinF(7)).Moreover, theinter-
− ∈
16/27 < 1 and the fact that λ λ . We prove (b)
2 N executiontimes of iare lower boundedby
reasoning by contradiction. Assum≤e δ > α2(1+φ)2 or
equivalently α(1+φ) √δ < 0. Using (19) and multi- τi = 1 ln 1+ αMiǫi , (31)
plyingboth sidesofth−e inequalityby 2mα, weobtain αMi 2 di (αMi+2βdi +1)θ
out out
(cid:16) (cid:17)
α2M2+δ 2αm√δ=(√δ αm)2+α2(M2 m2)<0, whereθ = λ¯F x(0)p x¯ 2+ v(0) v¯ 2+φαβλ¯F ǫ 2.
− − − λF k − k k − k 4ηλF k k
which, since M m, is a contradiction. Having estab- p
lished the consis≥tency of (25), consider now the can- PROOF. Given an initial condition, let [0,T) be the
didate Lyapunov function V in (16) and let t be the maximal interval on which there is no accumulation
k
last time at which a communication among all neigh- pointin{tk}k∈Z¯ =∪Ni=1∪k∈Z¯i⊆Z≥0tik.NotethatT >0,
boring agentsoccurred.Fromthe proofof Theorem10, since the number ofagents is finite and, for eachi ,
we know that the time derivativeof V is negative,V˙ < ǫi >0andx˜i(0)=xˆi(0) xi(0)=0.Thedynamics(∈18V),
−21ǫmin{δ,21}p⊤p as long as t < tk + τ. After this under the event-triggere−d communication scheme (29),
time, (27) shows that as long as (25) is satisfied, V˙ is has a unique solution in the time interval [0,T). Next,
negative,andexponentialconvergencefollows. ✷ we use Lyapunov analysis to show that the trajectory
isboundedduring[0,T).ConsiderthefunctionV given
in(14),whoseLie derivativealong (21b)-(21d)is
Interestingly,giventhat(25)doesnotusethefullstateof
the networkbutinsteadreliesonthe disagreement,one 1 7
V˙ = α2(φ+1)y⊤h w⊤ w
can interpret it as an output feedback event-triggered −9 − 16 2:N 2:N
controller.Guaranteeingtheexistenceoflowerbounded 3 2α 2α
inter-executiontimes forsuchcontrollersis ingenerala −k4w2:N + 3 (R⊤⊗Id)h+ 3 (φ+1)z2:N)k2
difficultproblem,seee.g.,[DonkersandHeemels,2012].
4 4
Augmenting(25)withtheconditiont t +τ results + α2 (R⊤ I )h 2+ α2(1+φ)2z⊤ z
k+1 ≥ k 9 k ⊗ d k 9 2:N 2:N
in Zeno-free executions by lower bounding the inter-
φαβ φαβ
event times by τ. The knowledge of this value also al- z⊤ (R⊤LR I )z + s,
lows the designer to compute bounds on the maximum − 2 2:N ⊗ d 2:N 2
energyspent bythe networkoncommunication.
wheres= z⊤ (R⊤LR I )z 2z⊤ (R⊤LR I )˜z ,
5.2.2 DistributedAsynchronous Implementation − 2:N ⊗ d 2:N− 2:N ⊗ d 2:N
and z˜ = ˆz z . Using the assumptions on
2:N 2:N 2:N
Wepresentadistributedevent-triggeredschemeforde- the local cost funct−ions and following steps similar to
terminingthesequenceofcommunicationtimesin(18). those taken in the proof of Theorem 1 to lower bound
At each agent, the execution of the communication y⊤h and upper bound (R⊤ I )h along with using
schemedependsonlyonlocalvariablesandthetriggered λ z⊤ z z⊤ (R⊤LRk I )⊗z d ,konecanshowthat
statesreceivedfromitsneighbors.Thisnaturallyresults 2 2:N 2:N ≤ 2:N ⊗ d 2:N
inasynchronouscommunication.Wealsoshowthatthe φαβ
V˙ η p 2+ s,
resultingexecutionsarefree fromZenobehavior. ≤− k k 2
Theorem 13 (Convergence of (18) with Zeno-free dis-
tributedevent-triggeredcommunication): LetGbeacon- where p = (z,w2:N). Next, we show s ≤ 21kǫk2 for t ∈
nected graph. Assume fi, i , is mi-strongly convex, [0,T). Using RR⊤ = Π , LΠ = Π L = L, z =
differentiable, and its gradie∈ntVis Mi-Lipschitz on Rd. (R⊤ I )y = (R⊤ I )Nx, andNz˜ N= (R⊤ I2):Ny˜ =
Forǫ RN ,consideranimplementation of (18)where ⊗ d ⊗ d 2:N ⊗ d
agent∈i >0 communicates with its neighbors in at (R⊤⊗Id)x˜,wegets=−x⊤Lx−2x⊤Lx˜.Then,
times {tik∈}kV∈Z¯i⊆Z≥0 ⊂R≥0,startingat ti0 =0, G s=−(xˆ−x˜)⊤L(xˆ−x˜)−2(xˆ−x˜)⊤Lx˜
ti =argmax t [ti, ) (29) =x˜⊤Lx˜ xˆ⊤Lxˆ.
k+1 { ∈ k ∞ | −
N
4dioutkxˆi(t)−xi(t)k2 ≤ j=1aijkxˆi(t)−xˆj(t)k2+(ǫi)2}. GivenL=Dout AandDout+A 0,we have
− ≥
X
9
x˜⊤Lx˜ ≤2x˜⊤(Dout⊗Id)x˜ =2 Ni=1dioutkx˜ik2. kxˆi−xi(t)k≤ci(eαMi(t−tik)−1)/(αMi), t≥tik.
X
Therefore,we canwrite (recallx˜i =xˆi xi) Then, the time it takes kxˆi−xik to reach ǫi/(2 diout)
− is lower bounded by τi > 0 given by (31). To show
1 N N p
s= 4di xˆi xi 2 a xˆi xˆj 2 , T = , we proceed by contradiction. Suppose that
2Xi=1(cid:0) outk − k −Xj=1 ijk − k (cid:1) aTcc<um∞∞ul.aTtihonenp,otihnetasteqTu.eBnceecaoufseevweenthsa{vteka}kfi∈nZ¯itehansuman-
which,with(29),yieldss 1 ǫ 2 for t [0,T).Then, ber of agents, this means that there must be an agent
≤2k k ∈
V˙ η p 2+ φαβ ǫ 2, t [0,T), ait∈T,Vimfoprlywinhgicthha{ttika}gke∈nZt¯iihtraasnsamnitasccinufimnuitlaeltyioonftpenoinint
≤− k k 4 k k ∈
the time interval [T ∆,T) for any ∆ (0,T]. How-
− ∈
Recall from the proof of Theorem 1 that λ p 2 ever, this is in contradiction with the fact that inter-
V(p) λ¯ p 2. Then, using the ComparisonFkLekmm≤a eventtimesarelowerboundedbyτi >0on[0,T).Hav-
≤ Fk k ing established T = , note that this fact implies that
(cf.[Khalil, 2002,Lemma3.4]),wededuce that ∞
under the event-triggeredcommunication law (29), the
kp(t)k≤λ1 kV(0)ke−λ¯ηFt+φαβ4λη¯λFkǫk2(1−e−λ¯ηFt) amlgoorer,itfhromm(1(83)2)d,oweesndoedtuecxehitbhiattZ,efonroebaechhaivior.,Founrethhears-
≤λλ¯FFFkp(0)ke−λ¯ηFt+φαβ4λη¯λFFkǫFk2(1−e−λ¯ηFt), (32) ktixail(lyt)f−asxt⋆wki≤thkapr(at)tke≤noφw4αηoβλrλ¯FsFektǫhka2naηs/tλ¯→F.∞∈,Vexponen✷-
fort [0,T).Notice thatregardlessofvalue ofT, Regardingthe role of the designparametersand condi-
∈
λ¯ φαβλ¯ tion(30),weomitforspacereasonsobservationssimilar
p(t) F p(0) + F ǫ 2, (33) tothe onesmadeinRemark2.The lowerboundonthe
k k≤ λ k k 4ηλ k k
F F inter-eventtimesallowsthedesignertocomputebounds
onthemaximumenergyspentbyeachagentoncommu-
for t [0,T). Notice that the right-hand side corre-
nication during any given time interval. Since the total
∈
spondstoθ.Thiscanbeseenbynotingthat,from(11b),
number of agents is finite and each agent’s inter-event
we have z = x x¯ and w = w = v v¯
2:N times are lowerbounded, it follows that the total num-
k k k − k k k k k k − k
(recall Ni=1vi(0)=0 resultsinw1 =0 forall t≥0). berofevents inanyfinite time intervalis finite. Ingen-
Our final objective is to show that T = . To achieve eral,anexplicit expressionlowerbounding the network
this,wePfirtestablisalowerboundonthei∞nter-execution inter-eventtimes is notavailable.Itis alsoworthnotic-
times of any agent. To do this, we determine a lower ing that the farther away the agents start from the fi-
boundonthetimeittakesi tohave xˆi xi evolve nalconvergencepoint(largerθ in(31)),the smallerthe
∈V k − k guaranteed lower bound between inter-event times be-
from0 toǫi/(2 di ).Using (10)and(18b),wehave
out comes.Asbefore,τiin(31)dependsonthegraphtopol-
d xˆi xi =p (xˆi−xi)⊤x˙i x˙i opgayr,atmheetperasraαmaentedrsβo,fatnhdeltohcealvcaorsitabfulencǫtii.oOn,ntehecadnesuigsne
dtk − k − xˆi xi ≤k k
k − k thisdependencytomaximizethevalueofτi inasimilar
N
= α( fi(xi) fi(x⋆)) β a (xˆi xˆj) fashion as discussed in Remark 11. For quadratic local
ij
k− ∇ −∇ − j=1 − costfunctions of the form fi(x) = 1(x⊤x+x⊤ai+bi),
(vi+α fi(x⋆) X i the claim of Theorem 13 hol2ds for any α,β > 0.
− ∇ k ∈ V
N Finally, we point out that the discrete-time communi-
αMi xi x⋆ +β a xˆi xˆj + vi v¯i ,
ij cationstrategiesintroducedaboveenjoysimilarprivacy
≤ k − k j=1 k − k k − k
preservation properties as the ones stated in Proposi-
X
whichleadsto tion5,butwe omitthe details hereforbrevity.
d
xˆi xi αMi xˆi xi +αMi xˆi x⋆ + 6 Simulations
dtk − k≤ k − k k − k
N Here,weillustratetheperformanceofthealgorithm(3)
β a ( xˆi x⋆ + xˆj x⋆ )+ vi v¯i .
ij and its implementation with discrete-time communica-
j=1 k − k k − k k − k
tion (18). We consider a network of 10 agents, with
X
From (33), we have x(t) x¯ θ and v(t) v¯ stronglyconvexlocalcostfunctions onR givenby
θ. This implies vi(tk) v¯−i k ≤θ, xˆi xk⋆ −θ, kan≤d f1(x)=0.5e−0.5x+0.4e0.3x, f2(x)=(x 4)2,
Nj=1aij(kxˆi −kx⋆k+k−xˆj −kx≤⋆k) ≤k 2d−ioutθ,kfo≤r all i ∈ f3(x)=0.5x2ln(1+x2)+x2, f4(x)=x2−+e0.1x,
VP. Therefore,fromthe inequality above, ddtkxˆi−xik ≤ f5(x)=ln(e−0.1x+e0.3x)+0.1x2, f6(x)=x2/ln(2+x2),
αMikxˆi −xik+ci, where ci = (αMi +2βdiout +1)θ. f7(x)=0.2e−0.2x+0.4e0.4x, f8(x)=x4+2x2+2,
UsingtheComparisonLemma(cf.[Khalil,2002,Lemma
3.4])andthe factthat xˆi xi(ti) =0,we deduce f9(x)=x2/ x2+1+0.1x2, f10(x)=(x+2)2.
k − k k
p
10