Distributed Gaussian Learning over Time-varying Directed Graphs

1 Distrib uted Gaussian Learnin g ov er T ime-v arying Directed Graphs Angelia Nedi ´ c, Alex Olshe vsky and C ´ esar A. Uribe Abstract — W e present a distributed (n on-Bay esian) learning algorithm f or the problem of p aram eter estimation with Gaus- sian noise. The algorithm i s expressed as ex plicit updates on the parameters of the Gaussian beliefs (i.e. means and precision). W e sh o w a con vergence rate of O ( 1 /k ) with the constant term dependin g on the number of age nts and the topology of the network. Mor eove r , we show almost sure conv ergence to the optimal solution of th e e stimation problem f or the general case of time-varying d irec ted graphs. I . I N T R O D U C T I O N The analysis of distrib uted (non-Bayesian) lear ning alg o- rithm gain ed p opularity since the seminal work of Jad babaie et al. [ 1]. The ability of no n-Bayesian upda tes to combine distributed op timization and learning alg orithms make them especially useful for the d esign of distributed estimation algorithm s with provable per formance. In the distributed learning setup, a group of agents re- peatedly receive signals about a certain unknown state of the world or parameter . No single ag ent has enough info r - mation to accurately estimate the unknown state and, thu s, interaction with other age nts is need ed. Se veral results are readily a vailable for perfo rmance ev a luation of distributed learning algorithms for a variety of scen arios. Asymp totic exponential co n vergen ce rates where de veloped in [2], [3], [4], no n-asymptotic bo unds in [5], time-varying directed graphs in [ 6], co nflicting hyp otheses and line ar rates in [7], no-reca ll a pproaches to belief sharin g in [ 8] and adversarial cases in [9], [10]. This list is ne cessarily inc omplete, an d th e reader is refe rred to [11] fo r an extended set of ref erences. Most of the previously propo sed m odels assume that the parameter space of the estimation process is finite. Initial approa ches to the study of contin uum sets of hy potheses were developed in [12], where explicit non-asym ptotic rates we re derived. A similar setup with Gau ssian noisy observations with n onlinear function o f the p arameter to be estimated h as been considered in [13], [14], where almost sure con vergence and asymptotic expo nential rates for fixed und irected graphs were established . Allowing the hypo theses set to b e infin ite (e.g. a c ompact sub set of R m ) enables the explor ation of traditional estimation problem s in a distributed mann er . On e of such prob lems is the param eter estimation with Gaussian noise, which is the m ain con cern of this man uscript. In particu lar , we focus on the Gaussian ca se o f the distributed (non-Bayesian) lea rning setup in [12]. W e analyze A. Nedi ´ c ( angelia.nedi ch@asu.edu ) is with the ECEE Department, Ari- zona Stat e Uni versity . A. Olshe vsky ( al exols@b u.edu ) is with th e ECE Departmen t, Boston Uni ver sity . C.A. Uri be ( caurib e2@illino is.edu ) is with the Coordinated Scie nce Laboratory , Uni versi ty of Illi nois. This research is support ed partially by the Nati onal Science Foundat ion under grants no. CNS 15-44953 and no. CMMI-1463 262, and by the Office of Na va l Researc h under grant no. N000 14-12-1-0998. the belief update algorithm where agents observe a parameter corrup ted by Gaussian noise and likelihood models are Gaus- sian functio ns, which results in Ga ussian b eliefs . W e presen t explicit updates for the beliefs’ mean and variances, thus providing an algorithm for the d istrib uted estimation pro cess. W e sho w almost sure con vergence to an op timal p arameter and establish a con vergence rate of O (1 /k ) . W e also p rovide simulation results for o ur algor ithm and co mpare it with two a pproaches p roposed in [1 5], [ 16 ]. Ou r results ho ld for the gen eral case of time-varying dire cted graphs, which are established b y using ideas from the push-sum a lgorithm in [6], [17]. This paper is organized a s follows. Section II describes the p roblem setup, as well as the proposed algo rithm and main results. Sec tion III provides a detailed comp arison with results f rom [15], [16] for the case o f identically distributed observations for all age nts. Section I V shows simu lation results and comp arison with other algorith ms. Finally , con- clusions and futu re work ar e p resented in Sectio n V. Notatio n : superscr ipts refer to agen ts wh ich are usually indexed b y th e letters i or j . Subscripts indicate instants of time wh ich are de noted by the letter k . Random variables are denoted by capital letters, e.g. S i k , and their co rresponding realizations b y lower ca se letters, e. g. s i k . Th e transpose of a vector x is deno ted as x ′ . The term [ A ] ij denotes th e en try of a matrix A at th e i - th row and the j -th colum n. For a se- quence { A k } o f ma trices we let A k : t = A k A k − 1 · · · A t +1 A t for k ≥ t . W e den ote the Gaussian fu nction by N ( θ, σ 2 ) = 1 √ 2 π σ 2 exp  − ( x − θ ) 2 2 σ 2  . I I . P RO B L E M S E T U P , A L G O R I T H M A N D R E S U LT S Consider a group of n agents whose goal is to collectively solve the following op timization pro blem min θ ∈ Θ F ( θ ) , n X i =1 D K L  f i k ℓ i ( ·| θ )  , (1) where D K L  f i k ℓ i ( ·| θ )  is the Kullback- Leibler divergence between an un known distribution f i and a p arametrized distribution ℓ i ( ·| θ ) . Each agen t i has access to realizatio ns of a rand om variable S i k ∼ f i and a local family of parametrize d distributions { ℓ i ( ·| θ ) | θ ∈ Θ } , where Θ is a set of parameters. In other words, th e a gents want to determine a p arameter θ ∗ ∈ Θ correspo nding to a distribution Q n i =1 ℓ i ( ·| θ ∗ ) that is the closest to the distribution Q n i =1 f i in the sense of Kullback-Leib ler div ergence. Moreover , the agents a re allowed to interact over a sequ ence of time- varying d irected graph s {G k } , wher e G k = ( { 1 , . . . , n } , E k ) and E k is a set of edges such th at ( j, i ) ∈ E k indicates that agent j can com municate with agen t i at time k . 2 In [12], th e auth ors propo sed an alg orithm for solving the general pro blem of Eq. ( 1) for com pact sets Θ ⊂ R d . Th e algorithm generates no n-Bayesian posteriors beliefs b ased on local o bservations and shared beliefs fro m neighboring agents. Each ag ent i constru cts a sequence { µ i k } ∞ k =1 of be- liefs about the hypo thesis set Θ , where µ i k maps measurable subsets of Θ to real values indicating the belief that the unknown parameter θ ∗ is in the given subset. The algorithm propo sed in [12] is gi ven by ¯ µ i k +1 ∝ n Y j =1  ¯ µ j k  a ij ℓ i ( s i k +1 |· ) , (2) where ¯ µ i k = dµ i k /d ¯ λ is a belief d ensity function , see [18], with respect to a refer ence measur e ¯ λ . Effecti vely , fo r a measurable su bset D ⊆ Θ , we have that the belief that θ ∗ is in D is giv en by µ i k ( D ) = R θ ∈ D ¯ µ i k ¯ λ ( θ ) . Addition ally , the scalar a ij is non negati ve and indicates how mu ch agen t i weights the belief s c oming from its neighbor j , with an understan ding that a ij = 0 if n o inter action between them occurs. In this manuscript, we assum e that the observations have Gaussian distribution and that th e likelihoods models are Gaussian, both with bo unded secon d o rder mo ments, i.e. S i k ∼ N ( θ i , ( σ i ) 2 ) and ℓ i ( ·| θ, σ i ) = N ( θ , ( σ i ) 2 ) wher e σ i > 0 f or e very i . This setting correspon ds to the case of having m easurements o f the true para meter θ ∗ corrup ted by som e Gaussian noise a nd the agents being infor med that the noise is Gaussian with a kn o wn variance. The Kullbac k-Leibler distance between two univ ariate Gaussian distributions p and q , wh ere p = N ( θ 1 , ( σ 1 ) 2 ) and q = N ( θ 2 , ( σ 2 ) 2 ) is given by D K L ( p k q ) = log σ 2 σ 1 + ( σ 1 ) 2 + ( θ 1 − θ 2 ) 2 ( σ 2 ) 2 − 1 2 . Thus, in this c ase, the p roblem in Eq. (1) is equ i valent to min θ ∈ Θ ˆ F ( θ ) , n X i =1 ( θ − θ i ) 2 2( σ i ) 2 , (3) which is conve x with a u nique solution θ ∗ = n X i =1 θ i / ( σ i ) 2 n P j =1 1 / ( σ j ) 2 . (4) Howe ver , the exact value o f θ i is u nknown and each agent i has access only to noisy observations o f the form S i k = θ i + ǫ i , where ǫ i ∼ N (0 , ( σ i ) 2 ) . Moreover , variances are only known loca lly , i.e. ag ent i o nly knows σ i . W e pr opose the follo wing distributed algorithm for solving the pro blem in Eq. (3) over time- v arying d irected grap hs τ i k +1 = n X j =1 [ A k ] ij τ j k + τ i (5a) θ i k +1 = n P j =1 [ A k ] ij τ j k θ j k + s i k +1 τ i τ i k +1 (5b) where τ i = 1 / ( σ i ) 2 is refereed as the precision of the observations. The weigh ts [ A k ] ij are chosen as [ A k ] ij = ( 1 d j k +1 if ( j, i ) ∈ E k , 0 otherwise (6) where d j k is the out-degree of n ode j at time k . W itho ut loss of generality , we assume th at τ i 0 = τ i for all i . Remark 1 It is not necessary for each agent to h ave some form o f informative observations. In deed, ther e might be agents with no observation s working as buffer s for informa- tion for which we a lso e xpect correct estimates of θ ∗ . These “blind” agents depend on co mmunicating with othe r agents to construct its estimates. Remark 2 While our fo cus is in on the univariate Gaussian case, extensions to the mu ltivariate ar e similarly possible us- ing the r esults of con jugate priors for multivariate Gau ssian distributions. The next prop osition shows that the algor ithm in E q. (5) is a specific realization of Eq. (2) for the case of Gaussian distributions in the priors and likelihoo d models. Proposition 1 Let the prior belief density ¯ µ i 0 of every agent be a Gaussian fu nction, i.e. ¯ µ i 0 ( θ ; θ i 0 , σ i ) = N ( θ i 0 , ( σ i ) 2 ) and let th e p arametric fa mily o f distributions fo r the likeli- hood models be Ga ussian fun ctions, i.e. ℓ i ( s | θ ; ( σ i ) 2 ) = N ( θ , ( σ i ) 2 ) . Then, for any k ≥ 1 , the po sterior belief d ensity ¯ µ i k , given by Eq. ( 2) , is also a Ga ussian function. Mor eover , if the weights a ij ar e chosen to be 1 / ( d j k + 1) , th en th e mea n and the standa r d deviation o f the p osterior follow Eq. (5) . Before pr esenting our main results, we state two au xiliary lemmas f rom [17] that describe the g eometric convergence for the prod uct of column stochastic m atrices. Lemma 2 [Cor o llary 2.a in [17]] Let the gr aph sequence {G k } be B-str ongly conn ected 1 . Then , ther e is a sequ ence { φ k } o f stochastic vectors such that | [ A k : t ] ij − φ i k | ≤ C λ k − t for all k ≥ t ≥ 0 wher e { A k } is as in Eq. (6) and the constants C and λ satisfy the following relations: (1) F or general B -str on gly co nnected graph seque nces { G k } C = 4 , λ =  1 − 1 n nB  1 B . (2) If every graph G k is r e gular with B = 1 C = √ 2 , λ = 1 − 1 / 4 n 3 . 1 There is an inte ger B ≥ 1 such that the graph  V , S ( k +1) B − 1 i = kB E i  is strongly connec ted for all k ≥ 0 3 Lemma 3 [Cor o llary 2.b in [17]] Let the gr aph sequence {G k } b e B -str ongly con nected, and defin e δ , inf k ≥ 0  min 1 ≤ i ≤ n [ A k :0 1 n ] i  . Then, δ ≥ 1 /n nB . Moreover , if every G k is re gula r and str ong ly connecte d (i.e. B = 1 ), the n δ = 1 . Fu rthermor e, the sequence { φ k } fr om Lemma 2 satisfies φ j k ≥ δ /n for all k ≥ 0 a nd j = 1 , . . . , n . Now , we p roceed to state our two main r esults showing the convergence proper ties o f the a lgorithm in Eq. (5 ). Lemma 4 The e xpected mean pr ocess { E [ θ i k ] } co n ver ges to θ ∗ for all i with a con ve r gence rate o f O (1 /k ) . Moreo ver , the co nstant terms depend on the topology o f the network, the precision of the ob servations and the initial guess. Pr oo f: In fact, we will p rove the bo und   E [ θ i k +1 ] − θ ∗   ≤ τ max τ min k δ  k θ 0 − θ ∗ 1 k 1 + 2 C k θ − θ ∗ 1 k 1 1 − λ  (7) with τ max = max j τ j , and τ min is th e smallest non-zero precision among all a gents. First, define a new variable as x i k = τ i k θ i k , then fr om Eq. (5b) it f ollo ws that x k +1 = A k x k + diag ( τ ) s k +1 = A k :0 x 0 + k X t =1 A k : t diag ( τ ) s t + diag ( τ ) s k +1 where diag ( τ ) is a diago nal matrix with [ diag ( τ )] ii = τ i and x k = [ x 1 k , . . . , x n k ] ′ , τ = [ τ 1 , . . . , τ n ] ′ , s k = [ s 1 k , . . . , s n k ] ′ . Adding and subtracting P k t =1 φ k τ ′ s t from the preced ing relation we ob tain x k +1 = A k :0 x 0 + k X t =1 D k : t diag ( τ ) s t + diag ( τ ) s k +1 + k X t =1 φ k τ ′ s t with D k : t = A k : t − φ k 1 ′ , and φ k is as in Lemm a 2. Follo wing a similar pro cedure, from Eq. (5a) it ho lds that τ k +1 = A k :0 τ 0 + k X t =1 D k : t τ + k φ k 1 ′ τ + τ . Going back to the original variable θ k , we have that E [ θ i k +1 ] = [ A k :0 diag ( τ ) θ 0 ] i + P k t =1 [ D k : t diag ( τ ) θ ] i + τ i θ i + k φ i k τ ′ θ [ A k :0 τ 0 ] i + P k t =1 [ D k : t τ ] i + k φ i k 1 ′ τ + τ i By subtracting θ ∗ on bo th sides of the previous relation an d taking the absolu te value, we obta in   E [ θ i k +1 ] − θ ∗   ≤      [ A k :0 diag ( τ 0 ) ( θ 0 − θ ∗ 1 )] i P k t =1 [ D k : t τ ] i + k φ i k 1 ′ τ      +      τ i  θ i − θ ∗  P k t =1 [ D k : t τ ] i + k φ i k 1 ′ τ      +      P k t =1 [ D k : t diag ( τ ) ( θ − θ ∗ 1 )] i P k t =1 [ D k : t τ ] i + k φ i k 1 ′ τ      where the terms in volving k φ i k τ ′ θ cancel out an d the fol- lowing positive terms are re moved from the denomin ator [ A k :0 τ 0 ] i + τ i > 0 . Then b y the fact that [ D k : t 1 ] i + φ i k n > δ on the deno mi- nator and using L emma 2 on th e third term it fo llo ws th at   E [ θ i k +1 ] − θ ∗   ≤     [ A k :0 diag ( τ 0 ) ( θ 0 − θ ∗ 1 )] i k δ τ min     + τ i | θ i − θ ∗ | k δ τ min + C τ max k θ − θ ∗ 1 k 1 k δ τ min (1 − λ ) . Finally , the desired result follows by H ¨ o lders ineq uality in the first term with k [ A k :0 diag ( τ )] i k ∞ = τ max and g rouping the second and thir d terms since C 1 − λ > 1 .   E [ θ i k +1 ] − θ ∗   ≤ max j [ A k :0 ] ij τ j k θ 0 − θ ∗ 1 k 1 k δ τ min + + 2 C τ max k θ − θ ∗ 1 k 1 k δ τ min (1 − λ ) . The first term in Eq. ( 7) shows th e dep endency on the initial estimates θ 0 while th e secon d term shows d epends on the hetero geneity of mean of local observations. The network topolog y and the numbe r of agents is charac terized by λ an d δ . W e are now ready to state ou r main result about the a lmost sure conv ergence of the pr oposed algorith m. Theorem 5 Let the graph sequence of interactions {G k } ∞ k =1 be B-str on gly connected . Mor eover , assume S i k ∼ N ( θ i , ( σ i ) 2 ) an d ℓ i ( ·| θ ) = N ( θ , ( σ i ) 2 ) for a ll i . Then, the sequence { θ i k } generated by Eq. (5) conver ges almost sur ely to θ ∗ , i.e. lim k →∞ θ i k = θ ∗ a.s. ∀ i A proof o f Theorem 5 is not shown due to space constraints. Noneth eless, its r esult follo ws by the boun ded variance assumptio n of the ob servations and the weighted law o f large nu mbers in [1 9]. Remark 3 The spe cific selec tion of weights as 1 / ( d j k + 1 ) is a design choice. Theor em 5 st ill h olds for a ny sequence of column stochastic ma trices { A k } with every non-zer o entry bou nded fr om be llow away fr om zer o, and with po sitive diagonal entries. I I I . I D E N T I C A L D I S T R I B U T I O N S F O R A L L AG E N T S A specific version of the pr oposed pro blem is the case when all agents o bserve independent realizations of the same random variable, i.e. S i k ∼ N ( θ ∗ , ( σ 2 ) ∗ ) . Recently , autho rs in [1 5], [16] hav e explored this case. Specifically , in [16] the au thors are concerned with the effects of the netw ork topolog y on the conver gence rate o f the distrib uted mean estimation prob lem. They show me an square consistency o f the following algorith m θ i k +1 = k k + 1 n X j =1 a ij θ j k + 1 k + 1 s i k +1 , (8) and provide explicit rates for different netw ork topologies. Note that the algorithm in Eq. (8 ) re duces to Eq. (5) when 4 τ i = 1 in such a way that τ i k = k for all i , and the grap h is static with a do ubly stocha stic weight matrix. In [15], the autho rs p roposed a new distributed Gaussian learning algo rithm wh ere com munication between agents is noisy . Follo wing the non-Bayesian learning without recall approa ch pro posed in [8] they dev elop the specific realization for Gaussian random variables. Add itionally , they consider the seq uence of observations { s i k } as coming fr om an agen t, denoted as n + 1 , an d thus a different weightin g strategy is propo sed. T heir algo rithm is τ i k +1 = τ i k + d j k τ (9a) θ i k +1 = P n +1 j =1 τ j k a j k τ i k +1 (9b) with the specific condition that τ j k = τ fo r all j 6 = i , a j k = θ i k for j = i and a j k = θ j k + ǫ with ǫ ∼ N (0 , τ ) , with a n +1 k = s i k . The authors showed a lmost sure con vergenc e of the algorithm . Mo reover , a c on vergence rate of O ( k − γ 2 d ) was d eri ved, wher e γ is a bound o n th e un iform conn ecti vity to the truth ob serv ations and d is th e maximal degree over all the network s. One par ticular characteristic of the alg orithm propo sed in [15] is that, apart fro m traditional literature on distrib uted learning, th e authors d o not assume age nts communica te over a sufficiently connected network ( B -strong conn ecti vity in Theor em 5). They r eplace th is assumption by a so- called truth-hearing assumption which w orks as a 1 /γ -stron g connectivity with the n + 1 no de that provides direct noisy observations o f θ ∗ . Th us, it is r equired that every no de receives signals fro m n ode n + 1 a t least once in ev ery time interval of length 1 / γ . If all ag ents r ecei ve in dependent observations fr om iden tical distributions, connectivity of the network and tr uth hearing assumption s bo th serve the same purpo se of guara ntying th e diffusion of the inf ormation over the n etwork, o therwise some f orm of con necti vity between agents is nee ded. In addition to dif feren t conne cti vity assump tions, one m ain characteristic of the algorithm in Eq. (9 ) is tha t a gents do not differentiate the signal S i k coming from the ob servations of the parameter, and the signals { a j k } coming from other agents. Every agent tre ats both signals similarly . The weig hts for ob servati ons of S i k and ne ighbors signals { θ i k } n i =1 decay . Whereas in ou r appr oach in Eq. (5) the weight for S i k decays to zero an d th e weigh t fo r the conv ex combinatio n of { θ i k } n i =1 goes to on e. This in deed shows that we d o requir e th e identification of signals coming from either agents or the noisy par ameter observations. This extra inf ormation could explain why o ur app roach has better perfo rmance in terms of conv ergence rates. I V . S I M U L AT I O N S In this section, we provide simulation results f or o ur propo sed algorithm and we comp are its p erformance with results in [15], [16]. Initially , we will co nsider the same scenario as in [15], [16] with static undirec ted grap hs with all agents having identical distributions in their no iseless beliefs sharing . W e will ev aluate th e pe rformance of the algorithms for two d if ferent gra phs topolog ies, namely: p ath/line graph and a lattice/grid g raph. 10 0 10 1 10 2 10 3 10 4 10 5 10 −6 10 −4 10 −2 10 0 10 2 k | θ k i − θ * | Eq. (5) O(1/k 1/d ) O(1/k) Eq. (9) Fig. 1. Simulati ons results of algorit hms in Eq. (5) and Eq. (9) for a latti ce/grid graph of 25 node s for an av erage be havior over 500 Monte Carlo simulati ons. Figure 1 shows th e ab solute erro r o f th e estimated value θ ∗ for the lattice/grid grap h with 25 agents. It is assumed that S i k ∼ N (4 , 1) . An a verag e over 500 M onte C arlo simulation s is shown for one arbitrary ag ent. In a ddition, the theoretical conv ergence r ates are also shown fo r comp arison purp oses. No simu lation of th e algorith m in Eq. (8) is shown since it reduces to the same algo rithm as in Eq. (5) for the simula ted scenario. Figure 2 s hows the simulation results for th e sam e scen ario as in Figure 1 but now for a path/line gra ph of 15 agents. As predicted by the theoretical convergence rate bou nds, the propo sed algor ithm in Eq. (5) de cays as O (1 / k ) wh ere the topolog y of the network affects on ly th e constant whereas the p roposal in E q. (9) d epends explicitly on the maxim um degree amo ng all grap hs as O (1 /k 1 /d ) . 10 0 10 1 10 2 10 3 10 4 10 5 10 −6 10 −4 10 −2 10 0 10 2 k | θ k i − θ * | Eq. (5) O(1/k 1/d ) O(1/k) Eq. (9) Fig. 2. Simulati ons results of algori thms in Eq. (5) and Eq. (9) for a path graph of 25 nodes. A ver age behavio r over 500 Monte Carlo simulations. Next, we will sh o w that f or the case o f e ach agent having noise with d if ferent standard d e viations, by u sing informa tion about the curren t estimate precision (i.e. τ i k ) a better pe rformance is achieved. Figu re 3 shows the abso lute error on the e stimation of θ ∗ for the algorithm in Eq . (5) that uses pre cision in formation and the prop osal in Eq. (8) that assumes uniform precision. In this simulation, agents have heteroge neous pr ecisions such th at S i k ∼ N (4 , i ) . Tha t is, in the path graph , th e first a gent has τ 1 = 1 , th e last agent, on the o ther hand, h as τ n = n . This implies that agent 1 has the highest v ariance in its observations. W e have ch osen to show the results f or agent 1 o nly . 5 10 0 10 1 10 2 10 3 10 −6 10 −4 10 −2 10 0 10 2 k | θ 1 k − θ * | Eq. (5) Eq. (8) Fig. 3. Simulati ons results of algorit hms in Eq. (5) and Eq. (8) for a path graph of 25 nodes with heterogeneous precisions (i.e. τ ′ s ). A verage beha vior over 500 Monte Carlo simulations. Finally , we will present the simulatio n results for a di- rected static graph which has been sho wn t o be a pathological case fo r the p ush-sum algorithm , see Figure 4. Each agent receives si gnals of the form S i k ∼ N ( i, n − i + 1) . Thus every agent has different m easurement precisions an d d if ferent θ i . The optimal θ ∗ as defin ed in Eq. (4). Fig. 4. Dire cted graph for simulatio n of the algori thm in Eq. (5). Figure 5 shows the simulation results for the algo- rithm i n Eq. ( 5) to the specific set of observations S i k ∼ N ( i, n − i + 1) on the graph in Figur e 4. The av erage over 10 Monte Carlo simulatio n is shown. The p redicted O (1 / k ) behavior is o bserved, after a tran sition time that depend s on the n umber o f agents in the network, (i.e. the effects on n and λ in Lem ma 4). 10 0 10 1 10 2 10 3 10 4 10 5 10 −4 10 −2 10 0 10 2 k | θ k 1 − θ * | 10 Agents 20 Agents 30 Agents 40 Agents Fig. 5. Simulatio ns results of algorithms in Eq. (5) for the graph depicte d in Figure 4. Four dif ferent result s are sho wn, for 10 , 20 , 30 and 40 agent s respect iv ely . V . C O N C L U S I O N S W e d e veloped an a lgorithm fo r distributed parameter estimation with Gaussian noise over time-varying directed graphs. The proposed algo rithm is sh o wn to be a s pecific case of a m ore g eneral class of distributed (n on-Bayesian) learning methods. Almost sure conver ge as well as an explicit conv ergence rate is shown in terms of the network topol- ogy and the number of agents. Comparisons with recently propo sed approach es are presented . Future work sho uld consider no nlinear o bservations of the p arameter θ , that is S i k ∼ N ( g i ( θ ) , ( σ i ) 2 ) for some f unction g : Θ → R . Ongoing work develops similar par ameter estimation ap - proach es f or the larger case of the exponential family of distributions on the n atural parameter space. A particularly interesting case is wh en th e parameter θ ∗ is chang ing with time, either arbitrarily , on som e f orm o f Ma rkov p rocess or other d ependencies. 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