Weighted Gossip: Distributed Averaging Using Non-Doubly Stochastic Matrices

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1 Weghted Gossp: Dstrbuted Averagng Usng Non-Doubly Stochastc Matrces Florence Bénézt ENS-INRIA, France Vncent Blondel UCL, Belgum Patrc Thran EPFL, Swtzerland John Tstsls MIT, USA Martn Vetterl EPFL, Swtzerland Abstract Ths paper presents a general class of gosspbased averagng algorthms, whch are nspred from Unform Gossp [1] Whle Unform Gossp wors synchronously on complete graphs, weghted gossp algorthms allow asynchronous rounds and converge on any connected, drected or undrected graph Unle most prevous gossp algorthms [2] [6], Weghted Gossp admts stochastc update matrces whch need not be doubly stochastc Double-stochastcty beng very restrctve n a dstrbuted settng [7], ths novel degree of freedom s essental and t opens the perspectve of desgnng a large number of new gossp-based algorthms To gve an example, we present one of these algorthms, whch we call One-Way Averagng It s based on random geographc routng, just le Path Averagng [5], except that routes are one way nstead of round trp Hence n ths example, gettng rd of double stochastcty allows us to add robustness to Path Averagng I INTRODUCTION Gossp algorthms were recently developed to solve the dstrbuted average consensus problem [1] [6] Every node n a networ holds a value x and wants to learn the average x ave of all the values n the networ n a dstrbuted way Most gossp algorthms were desgned for wreless sensor networs, whch are usually modeled as random geometrc graphs and sometmes as lattces Ideally a dstrbuted averagng algorthm should be effcent n terms of energy and delay wthout requrng too much nowledge about the networ topology at each node, nor sophstcated coordnaton between nodes The smplest gossp algorthm s Parwse Gossp, where random pars of connected nodes teratvely and locally average ther values untl convergence to the global average [2] Parwse local averagng s an easy tas, whch does not requre global nowledge nor global coordnaton, thus Parwse Gossp fulflls the requrements of our dstrbuted problem However, the convergence speed of Parwse Gossp suffers from the localty of the updates, and t was shown that averagng random geographc routes nstead of local neghborhoods s an order-optmal communcaton scheme to run gossp Let n be the number of nodes n the networ On random geometrc graphs, Parwse Gossp requres Θ(n 2 ) messages whereas Path Averagng requres only Θ(n log n) messages under some condtons [5] The prevous algorthm ganed effcency at the prce of more complex coordnaton At every round of Path Averagng, a random node waes up and generates a random route Values are aggregated along the route and the destnaton node computes the average of the values collected along the route Then the destnaton node sends the average bac through the same route so that all the nodes n the route can update ther values to the average Path Averagng s effcent n terms of energy consumpton, but t demands some long dstance coordnaton to mae sure that all the values n the route were updated correctly Routng nformaton bac and forth mght as well ntroduce delay ssues, because a node that s engaged n a route needs to wat for the update to come bac before t can proceed to another round Furthermore, n a moble networ, or n a hghly dynamc networ, routng the nformaton bac on the same route mght even not succeed Ths wor started wth the goal of desgnng a undrectonal gossp algorthm fulfllng the followng requrements: Keep a geographc routng communcaton scheme because t s hghly dffusve, Avod routng bac data: nstead of long dstance agreements, only agreements between neghbors are allowed, Route crossng s possble at any tme, wthout ntroducng errors n the algorthm As we were desgnng One-Way Averagng, we happened to prove the correctness of a broad set of gossp-based algorthms, whch we present n ths paper along wth One-Way Averagng These algorthms can be asynchronous and they use stochastc dffuson matrces whch are not necessarly doubly stochastc, as announced by the ttle of the paper In Secton II, we gve some bacground on gossp algorthms, and we explan why Unform Gossp s a ey algorthm to get nspred from when buldng a undrectonal gossp algorthm In Secton III, we present Weghted Gossp, an asynchronous generalzaton of Unform Gossp, whch was already suggested n [1] but had remaned unnamed We show n Secton IV that weghted gossp algorthms converge to x ave, whch s a novel result to the best of our nowledge In Secton V, we descrbe n detal One-Way Averagng and we show on smulatons that the good dffusvty of geographc routes n Path Averagng perssts n One-Way Averagng Computng the speed of convergence of weghted gossp algorthms remans open and s part of future wor II BACKGROUND ON GOSSIP ALGORITHMS The values to be averaged are gathered n a vector x(0) and at any teraton t, the current estmates of the average x ave are gathered n x(t) Gossp algorthms update estmates lnearly At any teraton t, there s a matrx W(t) such that: x(t) T x(t 1) T W(t)

2 In gossp algorthms that converge to average consensus, W(t) s doubly stochastc: W(t)1 1 ensures that the global average s conserved, and 1 T W(t) 1 T guarantees stable consensus To perform averagng on a one way route, W(t) should be upper trangular (up to a node ndex permutaton) But the only matrx that s both doubly stochastc and upper trangular matrx s the dentty matrx Thus, undrectonal averagng requres to drop double stochastcty Unform Gossp solves ths ssue n the followng way Instead of updatng one vector x(t) of varables, t updates a vector s(t) of sums, and a vector ω(t) of weghts Unform Gossp ntalzes s(0) x(0) and ω(0) 1 At any tme, the vector of estmates s x(t) s(t)/ω(t), where the dvson s performed elementwse The updates are computed wth stochastc dffuson matrces {D(t)} t>0 : s(t) T s(t 1) T D(t), (1) ω(t) T ω(t 1) T D(t) (2) Kempe et al [1] prove that the algorthm converges to a consensus on x ave (lm t x(t) x ave 1) n the specal case where for any node, D (t) 1/2 and D j (t) 1/2 for one node j chosen d unformly at random As a ey remar, note that here D(t) s not doubly stochastc The algorthm s synchronous and t wors on complete graphs wthout routng, and on other graphs wth routng We show n ths paper that the dea wors wth many more sequences of matrces {D(t)} t>0 than just the one used n Unform Gossp III WEIGHTED GOSSIP We call Weghted Gossp the class of gossp-based algorthms followng the sum and weght structure of Unform Gossp descrbed above (Eq (1) and (2)) A weghted gossp algorthm s entrely characterzed by the dstrbuton of ts dffuson matrces {D(t)} t>0 Let P(s,t) : D(s)D(s + 1)D(t) and let P(t) : P(1,t) Then s(t) T x(0) T P(t), (3) ω(t) T 1 T P(t) (4) If a weghted gossp algorthm s asynchronous, then, D (t) 1 and D j,j (t) 0 for the nodes that do not contrbute to teraton t If D j (t) 0, then node sends (D j (t)s (t 1),D j (t)ω (t 1)) to node j, whch adds the receved data to ts own sums j (t 1) and weght ω j (t 1) At any teraton t, the estmate at node s x (t) s (t)/ω (t) Because 1 T D(t) 1 T, sums and weghts do not reach a consensus However, because D(t)1 1, sums and weghts are conserved: at any teraton t, s (t) x (0) nx ave, (5) ω (t) n (6) Ths mples that Weghted Gossp s a class of non-based estmators for the average (even though x (t) s not conserved through tme!): Theorem 31 (Non-based estmator): If the estmates x(t) s(t)/ω(t) converge to a consensus, then the consensus value s the average x ave Proof: Let c be the consensus value For any ǫ > 0, there s an teraton t 0 after whch, for any node, x (t) c < ǫ Then, for any t > t 0, s (t) cω (t) < ǫω (t) (weghts are always postve) Hence, summng over, (s (t) cω (t)) s (t) cω (t) < ǫ ω (t) Usng Eq (5), (6), the prevous equaton can be wrtten as nx ave nc < nǫ, whch s equvalent to x ave c < ǫ Hence c x ave In the next secton, we show that, although sums and weghts do not reach a consensus, the estmates{x (t)} 1 n converge to a consensus under some condtons IV CONVERGENCE In ths secton we prove that Weghted Gossp succeeds n other cases than just Unform Gossp Assumpton 1: {D(t)} t>0 s a statonary and ergodc sequence of stochastc matrces wth postve dagonals, and E[D] s rreducble Irreducblty means that the graph formed by edges (, j) such that P[D j > 0] > 0 s connected, whch requres the connectvty of the networ Note that d sequences are statonary and ergodc Statonarty mples that E[D] does not depend on t Postve dagonals means that each node should always eep part of ts sum and weght:,t,d (t) > 0 Theorem 41 (Man Theorem): Under Assumpton 1, Weghted Gossp usng {D(t)} t>0 converges to a consensus wth probablty 1, e lm t x(t) x ave 1 To prove Th 41, we wll start by upper boundng the error x(t) x ave 1 wth a non-ncreasng functon f(t) (Lemma 41): letη j (t) P j (t) j1 P j(t)/n P j (t) ω (t)/n, then f s defned as f(t) max 1 n f (t), where f (t) j1 η j(t) /ω (t) Then, we wll prove that f(t) vanshes to 0 by showng that η j (t) vanshes to 0 (wea ergodcty argument of Lemma 43) and that ω (t) s bounded away from 0 nfntely often (Lemma 44) Lemma 41: If {D(t)} t>0 s a sequence of stochastc matrces, then the functon f(t) s non ncreasng Furthermore, x(t) x ave 1 x(0) f(t) (7) Proof: By Eq (3), for any node, j1 x (t) x ave P j(t)x j (0) x ave ω (t) j1 (ω (t)/n+η j (t))x j (0) x ave ω (t) j1 η j(t)x j (0) ω (t) x(0) j1 η j(t) ω (t) x(0) f (t),

3 whch proves Eq (7) Next, we need to prove that f(t) s a non-ncreasng functon For any node, by Eq (1) and (2), η j (t) n f (t) ω j1 (t) η j(t 1)D (t) j1 ω (t 1)D (t) η j(t 1) D (t) ω (t 1)D (t) j1 max max max j1 η j(t 1) D (t) ω (t 1)D (t) j1 η j(t 1) D (t) ω (t 1)D (t) j1 η j(t 1) ω (t 1) f (t 1) f(t 1), whch mples that f(t) f(t 1) Eq (8) comes from the followng equalty: for any {a } 1 n 0,{b } 1 n > 0, a b b a a n j1 b max j b b The followng lemma s useful to prove Lemmas 43 and 44 Lemma 42: Under Assumpton 1, there s a determnstc tme T and a constant c such that P[D(1)D(2)D(T) > c] > 0, where A > c means that every entry of A s larger than c Proof: The proof of ths lemma can be found n [8] For the case where {D(t)} t>0 s d, a smpler proof can be found n [9] Note that the theorems proven n [8] and [9] are slghtly dfferent than our lemma because the authors multply matrces on the left, whereas we multply them on the rght However the multplcaton sde does not change the proof For completeness and smplcty, we gve the proof n the d case E[D] beng rreducble and havng a postve dagonal, t s prmtve as well: there s an m > 0 such that E[D] m > 0 (elementwse) {D(t)} t 1 s d, [ hence E[D(1)D(2)D(m)] ] E[D] m > 0, and P (D(1)D(2)D(m)) j > 0 > 0 for any entry (,j) For any tme t, the dagonal coeffcents of D(t) are nonzero, thus, f the (,j) th entry of P(, + m 1) D()D( +1)D(+m 1) s postve, then P j (t) > 0 for all t + m 1 Now tae T n(n 1)m The probablty that P(T) > 0 s larger than or equal to the jont probablty that P 12 (1,m) > 0, P 13 (m + 1,2m) > 0,, P n,n 1 (T m+1,t) > 0 By ndependence of {D(t)} t 1, P[P(T) > 0] P[P 1,2 (1,m) > 0]P[P 1,3 (m+1,2m) > 0] (8) P[P n,n 1 (T m+1,t) > 0] > 0 Therefore, there s a c > 0 such that P[D(1)D(2)D(T) > c] > 0 Lemma 43 (Wea ergodcty): Under Assumpton 1, {D(t)} t 1 s wealy ergodc Wea ergodcty means that when t grows, P(t) tends to have dentcal rows, whch may vary wth t It s weaer than strong ergodcty, where P(t) tends to a matrx 1π T, where π does not vary wth t Interestngly, smple computatons show that f P(t) has dentcal rows, then consensus s reached All we need to now n ths paper s that wea ergodcty mples that lm max t,j P (t) P j (t) 0, and we suggest [10] for further readng about wea ergodcty Proof: Let Q be a stochastc matrx The Dobrushn coeffcent δ(q) of matrx Q s defned as: δ(q) 1 2 max j Q Q j One can show [10] that 0 δ(q) 1, and that for any stochastc matrces Q 1 and Q 2, δ(q 1 Q 2 ) δ(q 1 )δ(q 2 ) (9) Another useful fact s that for any stochastc matrx Q 1 δ(q) maxmnq j mn Q j j (10),j A bloc crteron for wea ergodcty [10] s based on Eq (9): {D(t)} t 1 s wealy ergodc f and only f there s a strctly ncreasng sequence of ntegers { s } s 1 such that (1 δ(p( s +1, s+1 ))) (11) s1 We use ths crteron wth s st, where T was defned n Lemma 42 A jont consequence of Lemma 42 and of Brhoff s ergodc theorem [11], [8] (n the d case, one can use the strong law of large numbers nstead) s that the event {D( s + 1)D( s + 2) D( s+1 ) > c} happens nfntely often wth probablty 1 Hence, usng Eq (10), the event {1 δ(p( s +1, s+1 )) > c} happens nfntely often wth probablty 1 We can thus conclude that the bloc crteron (11) holds wth probablty 1 and that {D(t)} t 1 s wealy ergodc The next lemma shows that, although weghts can become arbtrarly small, they are unformly large enough nfntely often Lemma 44: Under Assumpton 1, there s a constant α such that, for any tme t, wth probablty 1, there s a tme t 1 t at whch mn ω (t 1 ) α Proof: As mentoned n the proof of Lemma 43, the event {D( s + 1)D( s + 2) D( s+1 ) > c}, where s st, happens nfntely often wth probablty 1 Let t 1 be the frst tme larger than t such that D(t 1 T + 1)D(t 1 T + 2)D(t 1 ) > c Then the weghts at tme t 1 satsfy ω(t 1 ) T ω(t 1 T) T D(t 1 T +1)D(t 1 ) > cω(t 1 T) T 11 T,

4 because weghts are always postve Now, because the sum of weghts s equal to n, ω(t 1 T) T 1 n Hence ω(t 1 ) T > cn1 T Tang α cn concludes the proof To prove Theorem 41, t remans to show that f(t) converges to 0 Proof: (Theorem 41) For any ε > 0, accordng to Lemma 43, there s a tme t 0 such that for any t t 0, max,j P (t) P j (t) < ε As a consequence P (t) P j (t) < ε for any,j, Hence η j (t) < ε as well Indeed, η j (t) P P (t) j(t) n P j (t) P (t) n P j (t) P (t) ε < n n ε Therefore, for any t t 0 and any 1 n, f (t) < nε ω (t), and therefore nε f(t) < mn ω (t) Usng Lemma 44, there s a constant α such that, wth probablty 1, there s a tme t 1 t 0 at whch mn ω (t 1 ) α Then, for any ε, t suffces to tae ε αε /n to conclude that there s a tme t 1 wth probablty 1 such that f(t 1 ) < ε Snce f s non ncreasng (Lemma 41), for all tme t t 1, f(t) < ε ; n other words f(t) converges to 0 Usng (7) concludes the proof Remar: A smlar convergence result can be proved wthout Assumpton 1 (statonarty and ergodcty of the matrces D(t)), n a settng where the matrces are chosen n a perhaps adversaral manner One needs only some mnmal connectvty assumptons, whch then guarantee that there exsts a fnte number T such that, for all t, all entres of D(t+1) D(t+T) are bounded below by a postve constant c (see, eg, Lemma 521 n [12]) V ONE-WAY AVERAGING In ths secton, we descrbe n detal a novel weghted gossp algorthm, whch we call One-Way Averagng A Assumptons and Notatons Assume that the networ s a random geometrc graph on a convex area A, wth a connecton radus r(n) large enough to enable geographc routng [3] For every node, let T be a dstrbuton of ponts outsde of the area A, and let H be a dstrbuton of ntegers larger than 2 Each node has an ndependent local exponental random cloc of rate λ, and ntates an teraton when t rngs Equvalently, tme s counted n terms of a global and vrtual exponental cloc of rate nλ Each tme the global cloc rngs, a node waes up ndependently and unformly at random In the analyss, t ndcates how many tmes the global cloc rang A detaled analyss of ths tme model can be found n [2] B Descrpton of One-Way Averagng Each node ntalzes ts sum s (0) x (0) and ts weght ω (0) 1 For any teraton t > 0, let be the node whose cloc rngs Node draws a target Z accordng to dstrbuton Z and a number H 2 of hops accordng to dstrbuton H Node chooses unformly at random a neghbor whch s closer to the target Z than tself If there s no such neghbor then the teraton termnates If such a neghbor j exsts, then node dvdes ts sum s (t 1) and ts weght ω (t 1) by H and sends (s (t 1),ω (t 1)) (H 1)/H to node j It also sends the remanng number H 1 of hops and the target Z Node j adds the receved sum and weght to ts sum s j (t 1) and ts weght ω j (t 1) Then t performs the same operaton as node towards a node that s closer to the target, except that t dvdes ts new sum and weght by H 1 nstead of H (formally, H H 1) Messages are greedly sent towards the target,h beng decremented at each hop The teraton ends when H 1 or when a node does not have any neghbor to forward a message to At any tme, the estmate of any node s the rato between ts sum and ts weght C Dffuson Matrces Suppose that at round t, a whole route of H nodes s generated Then, after re-ndexng nodes startng wth the nodes n the route, the dffuson matrx D(t) can be wrtten as: 1/H 1/H 1/H 1/H 0 0 1/(H 1) 1/(H 1) 1/(H 1) /2 1/ Id, where Id denotes the dentty matrx If the route stops early and has for example only 3 nodes whle H 4, then, after re-ndexng the nodes, D(t) can be wrtten as: 1/4 1/4 1/ /3 2/ Id Note that D(t) s ndeed stochastc for all t It s uppertrangular as well: One-Way Averagng does not requre to route nformaton bacwards along the path Furthermore, {D(t)} t>0 verfes Assumpton 1 Frst, {D(t)} t>0 s an d sequence Second, {D(t)} t>0 have postve dagonals Thrd, f the networ s connected and f the routes generated by dstrbutons {Z } 1 n and {H } 1 n connect the networ, then E[D] s rreducble Therefore, One-Way Averagng s a successful dstrbuted averagng algorthm Fnally, routes can cross each other wthout corruptng the algorthm (the resultng dffuson matrces are stll stochastc) D Smulaton One-Way Averagng and Path Averagng were run (Matlab) on random geometrc graphs on the unt square, usng the

5 same routes for a far comparson At each teraton t, the number H(t) of hops was generated wth H unform n [ 1/ 2r(n), 2/r(n) ] and the target Z(t) was drawn n the followng way: let I be the coordnates of the woen node, and let U be a pont drawn unformly at random n the unt square, then Z(t) I +3 U I U I 2 Let C(t 1,t 2 ) be the message cost of a gven algorthm from teraton t 1 to teraton t 2 For One-Way Averagng, C(t 1,t 2 ) t 2 tt1 R(t), where R(t) H(t) s the effectve route length at teraton t Because Path Averagng routes nformaton bac and forth, the cost of one teraton s taen to be equal to twce the route length: C(t 1,t 2 ) 2 t 2 tt 1 R(t) Let ǫ(t) x(t) x ave 1 The emprcal consensus cost s defned as: so that C emp (t 1,t 2 ) C(t 1,t 2 ) log ǫ(t 1 ) log ǫ(t 2 ), ( ǫ(t 2 ) ǫ(t 1 ) exp C(t ) 1,t 2 ) C emp (t 1,t 2 ) In Fg 1, we dsplay the emprcal consensus cost of both algorthms, wth t and t 2 growng lnearly wth n We can see that One-Way Averagng performs better than Path Averagng on ths example Although One-Way Averagng converges slower n terms of teratons, spendng twce as few messages per teraton s suffcent here to outperform Path Averagng The speed of convergence depends on the networ but also on {Z } 1 n and {H } 1 n, whch we have not optmzed It would be nterestng n further wor to compute the speed of convergence of Weghted Gossp, and to derve optmal dstrbutons {Z } 1 n and {H } 1 n for a gven networ usng One-Way Averagng As a concluson, One-Way Averagng seems to have the same dffusve qualtes as Path Averagng whle beng more robust at the same tme ACKNOWLEDGEMENTS The wor presented n ths paper was supported (n part) by the Natonal Competence Center n Research on Moble Informaton and Communcaton Systems (NCCR-MICS), a center supported by the Swss Natonal Scence Foundaton under grant number , and by the NSF under grant ECCS VI CONCLUSION We proved that weghted gossp algorthms converge to average consensus wth probablty 1 n a very general settng, e n connected networs, wth statonary and ergodc teratons, and wth a smple stablty condton (postve dagonals) We beleve that droppng double stochastcty opens great opportuntes n desgnng new dstrbuted averagng algorthms that are more robust and adapted to the specfctes of each networ One-Way Averagng for example s more C emp (t1,t2) One Way Averagng Path Averagng networ sze n Fg 1 Comparson of the consensus cost for One-Way Averagng and Path Averagng n random geometrc graphs of ncreasng szes n The connecton radus scales as r(n) 6logn/n Dsplay of C emp (t 1,t 2 ) averaged over 15 graphs and 4 smulaton runs per graph robust than Path Averagng, and t surprsngly consumes fewer messages on smulatons Also, double stochastcty s dffcult to enforce n a dstrbuted manner n drected graphs usng undrectonal communcatons Wth Weghted Gossp, one could easly buld averagng algorthms for drected networs that are relable enough not to requre acnowledgements The next step of ths wor s to compute analytcally the speed of convergence of Weghted Gossp In classcal Gossp, double stochastcty would greatly smplfy dervatons, but ths feature dsappears n Weghted Gossp, whch maes the problem more dffcult REFERENCES [1] D Kempe, A Dobra, and J Gehre, Gossp-based computaton of aggregate nformaton, n FOCS, vol 44 IEEE, 2003, pp [2] S Boyd, A Ghosh, B Prabhaar, and D Shah, Gossp algorthms : Desgn, analyss and applcatons, n IEEE, INFOCOM, 2005 [3] A G Dmas, A D Sarwate, and M J Wanwrght, Geographc gossp: effcent aggregaton for sensor networs, n ACM/IEEE Symposum on Informaton Processng n Sensor Networs, 2006 [4] B Nazer, A Dmas, and M Gastpar, Local nterference can accelerate gossp algorthms, n Allerton Conference on Communcaton, Control, and Computng, 2008, pp [5] F Bénézt, A Dmas, P Thran, and M Vetterl, Order-optmal consensus through randomzed path averagng, submtted for publcaton [6] A Nedc, A Olshevsy, A Ozdaglar, and J Tstsls, On dstrbuted averagng algorthms and quantzaton effects, n IEEE Conference on Decson and Control, 2008, pp [7] B Gharesfard and J Cortés, When does a dgraph admt a doubly stochastc adjacency matrx? n submtted to the Amercan Control Conference, 2010 [8] A Tahbaz-Saleh and A Jadbabae, Consensus over ergodc statonary graph processes, IEEE Transactons on Automatc Control, 2009 [9], Necessary and suffcent condtons for consensus over random ndependent and dentcally dstrbuted swtchng graphs, n IEEE Conference on Decson and Control, 2007, pp [10] P Brémaud, Marov Chans Gbbs Felds, Monte Carlo Smulaton, and Queues Sprnger, 1999 [11] R Durrett, Probablty: theory and examples Duxbury Press Belmont, CA, 1996 [12] J Tstsls, Problems n decentralzed decson mang and computaton, PhD dssertaton, M I T, Dept of Electrcal Engneerng and Computer Scence, 1984