On the Convergence Rate of Average Consensus and Distributed Optimization over Unreliable Networks

Transcription

1 On he Convergence Rae of Average Consensus and Disribued Opimizaion over Unreliable Neworks Lili Su EECS, MIT Absrac We consider he problems of reaching average consensus and solving consensus-based opimizaion over unreliable communicaion neworks wherein packes may be dropped accidenally during ransmission. Exising work eiher assumes ha he link failures affec he communicaion on boh direcions or ha he message senders know exacly heir ougoing degrees in each ieraion. In his paper, we consider direced links, and we do no require each node know is curren ougoing degree. We characerize he convergence rae of reaching average consensus in he presence of packe-dropping link failures. Then we apply our robus consensus updae o he classical disribued dual averaging mehod as he informaion aggregaion primiive. We show ha he local ieraes converge o a common opimum of he global objecive a rae O( ), where is he number of ieraions, maching he failure-free performance of he disribued dual averaging mehod. I. INTRODUCTION Reaching consensus and solving disribued opimizaion are wo closely relaed global asks of muli-agen neworks. In he former, every agen has a privae inpu, and he goal of he neworked agens is o reach an agreemen on a value ha is a funcion of hese privae inpus such as maximum, minimum, average, ec; in he laer, ypically, every agen has a privae cos funcion, and he goal is o collaboraively minimize a global objecive which is a proper aggregaion of hese privae cos funcions. Average consensus has received inensive aenion [8], [0], [2] parially due o he fac ha one can use average consensus as a way o aggregae agens privae informaion. Differen sraegies o robusify reaching average consensus agains unreliable neworks have been proposed [7], [7], [4], [20], [6]. Specifically, undireced graphs were considered in [7], [6], where he link failures affec he communicaion in boh direcions; dynamically changing daa and neworks are considered in [6]. Direced graphs were firs considered in [7], however, only biased average was achieved. This bias was laer correced in [4], [20] via inroducing auxiliary variables a each agen; however, only asympoic convergence was shown. To he bes of our knowledge, he characerizaion of nonasympoic convergence rae is sill lacking. Consensus-based muli-agen opimizaion is an imporan family of disribued opimizaion algorihms. In a ypical consensus-based muli-agen opimizaion problem [5], [4], [3], [9], each agen i keeps a privae cos funcion h i : X R, and he neworked agens, as a whole, wan o reach agreemen on a global decision x X such ha he average of hese privae cos funcions is minimized, i.e., x argmin x X h i (x), n where n is he oal number of agens in he sysem. The applicaions of such disribued opimizaion problems include disribued machine learning and disribued resource allocaion. Robusifying disribued opimizaion agains link failures has received some aenion recenly [5], [3]. Duchi e al. [5] assumed ha each realizable link failure paern admis a doubly-sochasic marix. In he case wherein each agen knows he number of reliable ougoing links [3], he requiremen for he doubly sochasic marices was removed by incorporaing he push-sum mechanism. However, he implemenaion of push-sum in [3] implicily assumed he adopion of acknowledgemen mechanism. In his work, we consider direced links, and we do no require each node know is curren ougoing degree. As a resul of his, if a message packe is dropped over a link, he sender is no aware of his loss. This scenario arises frequenly in real sysems. Alhough acknowledge mechanisms can be incorporaed o improve reliabiliy, his may slow down he convergence due o he need for message reransmission (requiring more ime for each ieraion of he algorihm). We characerize he convergence rae of reaching average consensus in he presence of packe-dropping link failures, which is, o he bes of our knowledge, lacking in lieraure. Then we apply our robus consensus updae o he classical disribued dual averaging mehod as he informaion aggregaion primiive. We show ha he local ieraes converge o a common opimum of he global objecive a rae O( ), where is he number of ieraions, maching he failure-free performance of he disribued dual averaging mehod. II. NETWORK MODEL AND NOTATION We consider a synchronous sysem ha consiss of n neworked agens. The nework srucure is represened as a srongly conneced graph G(V, E), where V = {,..., n} is he collecion of agens, and E is he collecion of direced communicaion links. Le I i = {j (j, i) E} and O i = {j (i, j) E} be he ses of incoming neighbors and ougoing neighbors, respecively, of agen i. For ease of exposiion, we assume here exis no self-loops, i.e., i / I i O i, i V. For i V, le d o i = O i. The communicaion links are

2 unreliable in ha packes may be dropped during ransmission unexpecedly. However, a given link is operaional a leas once during B consecuive ieraions, where B. Similar assumpion is adoped in [3], [4]. III. ROBUST AVERAGE CONSENSUS Reaching average consensus in direced neworks has been inensively sudied [6], [], [2]. In paricular, in Push- Sum [2], [2], each neworked agen updaes wo coupled ieraes and he raio of hese wo ieraes approaches he average asympoically. The correcness of Push-Sum relies crucially on mass preservaion (specified laer) of he sysem. However, when he communicaion links suffer packedropping failures, he desired mass preservaion may no hold, since he ransmied mass may be dropped wihou even being noified by he senders. Robusificaion mehod has been inroduced o recover he dropped mass [0], [9], where auxiliary variables are inroduced o record he oal mass sen and delivered, respecively, hrough a given communicaion link. Only asympoic convergence is provably guaraneed [0], [9]. In his secion, we focus on characerizing he convergence rae of robus average consensus. To do ha, we need o modify he robus Push-Sum proposed in [0]. A. Robus Push-Sum In his subsecion, we briefly review he Push-Sum algorihm [2], [2] and is robus varian [0]. In sandard Push- Algorihm : Push-Sum [2], [2] Iniializaion: z i [0] = y i R d, w i [0] = R. 2 for do 3 Broadcas zi[ ] wi[ ] d o i + and d o i + o all ougoing neighbors; 4 z i [] z j[ ] j I i {i} +, and 5 end w i [] j I i {i} d o j w j[ ] d o j +. Sum [2], [2], described in Algorihm, each agen i runs wo ieraes: value sequence {z i []} =0, and weigh sequence {w i []} =0, where z i [0] = y i R d is he privae inpu, and w i [0] = R is he iniial weigh of agen i. The weigh sequences {w i []} =0 are inroduced o relax he need for doubly sochasic marices. Inuiively speaking, he weighs are used o correc he bias caused by he nework srucure. In each ieraion of Algorihm, each agen i divides boh he local value z i and local weigh w i by d o i +, recalling ha do i is he ou-degree of agen i in he fixed G(V, E). Among he d o i + z pars of he values fracions i d o i + and he weigh fracions w i d o+, agen i sends do i pars o is ougoing neighbors and i one par o iself. Upon receiving he value fracions and he weigh fracions from is incoming neighbors, agen i sums hem up respecively. When he communicaion nework is reliable, i has been shown ha, a each agen, he raio of he value and he weigh converges o he average of he privae inpus, i.e., z i [] lim w i [] = y j, i =,, n. () n The correcness of Push-Sum algorihm relies crucially on he mass preservaion of he sysem [3], [2], which says ha he oal weighs kep by he agens in he sysem sum up o n a every ieraion, i.e., w i [] = n,. (2) Unforunaely, (2) does no hold in he presence of packedropping link failures. Neverheless, as illusraed in [0] (also described below in Algorihm 2), if we are able o keep rack of he dropped mass, we are able o show ha he oal mass is preserved in some augmened graph. And, running Algorihm 2 can be viewed as running sandard push-sum on his augmened graph. Algorihm 2: Robus Push-Sum [0] Iniializaion: z i [0] = y i R d, w i [0] = R, σ i [0] = 0 R d, σ i [0] = 0 R, and ρ ji [0] = 0 R d, ρ ji [0] = 0 R for each incoming link, i.e., j I i. 2 for do 3 σ i [] σ i [ ] + zi[ ] d o i +, σ i[] σ i [ ] + wi[ ] d o i + ; 4 Broadcas (σ i [], σ i []) o ougoing neighbors; 5 for each incoming link (j, i) do 6 if message (σ j [], σ j []) is received hen 7 ρ ji [] σ j [], ρ ji [] σ j []; 8 else 9 ρ ji [] ρ ji [ ], ρ ji [] ρ ji [ ]; 0 end z i [] j I (ρ i {i} ji[] ρ ji [ ]), and w i [] j I ( ρ i {i} ji[] ρ ji [ ]). 2 end 3 end Similar o he sandard Push-Sum, in Algorihm 2, each agen i wans o share wih is ougoing neighbors of is value fracion zi d o i + and weigh fracion wi d o i +. If agen i sends hese wo fracions ou direcly, he oal mass will no be preserved. In order o recover he mass dropped by an incoming link, in addiion o z i [] and w i [], in Algorihm 2 each agen i uses variable σ i [] o record he cumulaive weigh (up o ieraion ) sen hrough each ougoing link; uses variable σ i [] for he corresponding quaniy of he value sequence. In paricular, σ i [] = σ[ ] + z i[ ] d o i +, and σ i [] = σ i [ ] + w i[ ] d o i +, (3) 2

3 wih σ i [0] = 0 R d, and σ i [0] = 0 R. In each ieraion, agen i broadcass he uple (σ i [], σ i []) o all of is ougoing neighbors. To record he cumulaive informaion delivered via he link (i, k), he ougoing neighbor k uses a pair of variables ρ ik [] and ρ ik [], wih ρ ik [0] = 0 R d and ρ ik [0] = 0 R. If he link (i, k) is operaional, i.e., he uple (σ i [], σ i []) is successfully delivered, hen ρ ik [] = σ i [], and ρ ik [] = σ i []. Oherwise, since no new message is delivered, boh ρ ik [] and ρ ik [] are unchanged. In summary, if he link is operaional a a given ieraion, hen Oherwise, oal mass sen = oal mass delivered; oal mass sen oal mass delivered. In addiion, if he link (i, k) is operaional a ieraion, i holds ha ρ ik [] ρ ik [ ] = ρ ik [] ρ ik [ ] = r= r= z i [r] and (4) +, d o i w i [r] (5) +, where is he immediaely preceding ieraion of such ha link (i, k) is operaional. As a link is reliable a leas once during B consecuive ieraions, i holds ha B. Under Algorihm 2, i has been shown ha [0], a each agen i, z i [] w i [] a.s. n d o i y i, as. However, no convergence rae (asympoic or non-asympoic) is given. Informally speaking, his is because he dynamics of he sysem under Algorihm 2 is no sable enough. In paricular, in he augmened graph consruced in [0] (formally defined laer), he wo ieraes kep by he virual agens are rese o zero periodically and unexpecedly. This rese causes non-rivial echnical challenges he corresponding marix produc does no converge o a rank one marix. B. Convergen Robus Push-Sum In his subsecion, we propose a simple varian of Algorihm 2. We refer o our algorihm as Convergen Robus Push-Sum, described in Algorihm 3 simply o emphasize he fac ha a finie-ime convergence rae is derived. Noe ha his does no mean ha our Algorihm 3 is superior o Algorihm 2 [0]. Our Algorihm 3 has he same se of variables as ha in Algorihm 2. For ease of exposiion, we use σ + i [], σ+ i [], z+ i [], and w + i [] o emphasize he fac ha hey are inermediae values of corresponding quaniies in an ieraion. In each ieraion of our Algorihm 3, he cumulaive ransmied value and weigh (σ i, σ i ), and he local value and weigh (z i, w i ) are updaed wice, wih he firs updae being idenical o ha in Algorihm 2. As menioned before, wih Algorihm 3: Convergen Robus Push-Sum Iniializaion: z i [0] = y i R d, w i [0] = R, σ i [0] = 0 R d, σ i [0] = 0 R, and ρ ji [0] = 0 R d, ρ ji [0] = 0 R for each incoming link, i.e., j I i. 2 for do 3 σ + i [] σ i[ ] + zi[ ] d o i +, σ + i [] σ i[ ] + wi[ ] d o i + ; 4 Broadcas ( σ + i [], σ+ i []) o ougoing neighbors; 5 for each incoming link (j, i) do 6 if message ( σ + j [], σ+ j []) is received hen 7 ρ ji [] σ + j [], ρ ji[] σ + j []; 8 else 9 ρ ji [] ρ ji [ ], ρ ji [] ρ ji [ ]; 0 end z + zi[ ] i [] (ρ ji [] ρ ji [ ]), w + wi[ ] i [] ( ρ ji [] ρ ji [ ]). 2 end 3 σ i [] σ + i [] + z+ i [] d o+, σ i[] σ + i [] + w+ i [] i d o i +, 4 end z i [] z+ i [] d o i +, w i[] w+ i [] d o i +. only his firs updae, he dynamics in he sysem is no sable enough, as he wo ieraes kep by he virual agens are rese o zero periodically and unexpecedly. This rese is prevened by he second updae in our Algorihm 3. Inuiively speaking, in he second updae, each agen pushes nonzero mass o he virual agens on is ougoing links. As a resul of his, he wo ieraes kep by a virual agen will never be zero a he end of an ieraion. C. Augmened Graph The augmened graph of a given G(V, E), denoed as G a (V a, E a ), is consruced as follows [20]: ) V a = V E, i.e., E addiional auxiliary agens are inroduced, each of which represens a link in G(V, E). For ease of noaion, we use n ij o denoe he virual agen corresponding o edge (i, j). 2) E E a, i.e., he edge se in G a (V a, E a ) preserves he opology of G(V, E); 3) Addiionally, auxiliary edges are inroduced: each auxiliary agen n ij has one incoming neighbor agen i and one ougoing neighbor agen j. As shown in Fig., in he augmened graph (i.e., Fig. (b)), four addiional agens are inroduced, each of which corresponds o a direced edge of he original graph. D. Marix Represenaion For each link (j, i) E, and, define he indicaor variable B (j,i) [] as follows: {, if link (j, i) is reliable a ime ; B (j,i) [] (6) 0, oherwise. 3

4 (a) Original graph (b) Augmened graph Fig. : For each direced link, a buffer agen is added. Recall ha z i and w i are he value and weigh for i V = {,, n}. For each (j, i) E, we define z nji and w nji as z nji [] σ j [] ρ ji [], and (7) w nji [] σ j [] ρ ji [], (8) wih z nji [0] = 0 R d and w nji [0] = 0 R. Le m = n + E. We nex show ha he evoluion of z and w can be described in a marix form. Since he updae of value z and weigh w are idenical, for ease of exposiion, henceforh, we focus on he value sequence z. From Algorihm 3, we know ρ ji [] = B (j,i) []σ + j [] + ( B (j,i)[])ρ ji [ ],. (9) By (6), (7) and (9), for each i V, he updae of z i is z + i [] z i [] Thus, = zi[ ] d o i + + ( ) zj[ ] j I i B (j,i) [] d o j + + z n ji [ ], = z+ i [] d o i +. (0) z i [] = z i[ ] (d o i + )2 + j I i B (j,i)[] (d o i + ) ( d o j + )z j[ ] Similarly, we ge + j I i B (j,i) [] (d o i + )z n ji [ ]. () ( z nji [] = ( d o j + ) 2 + B ) (j,i)[] d o j + z j [ ] + k I j B (k,j) [] (d o k + ) ( d o j + )z k[ ] + k I j B (k,j) [] d o j + z n kj [ ] + ( B (j,i) [] ) z nji [ ]. (2) Thus, we consruc a marix M[] R m m wih he following srucure: M i,i [] (d o i + ; )2 M j,i [] B (j,i) [] (d o i + ) ( d o j + ), j I i; M nji,i[] B (j,i)[] d o i +, j I i; M j,nji [] M k,nji [] ( d o j + ) 2 + B (j,i)[] d o j + ; B (k,j) [] (d o k + ) ( d o j + ), k I j; M nkj,n ji [] B (k,j)[] d o j +, k I j; M nji,n ji [] B (j,i) []. (3) and any oher enry in M[] be zero. I is easy o check ha he obained marix M[] is row sochasic. Le Ψ(r, ) be he produc of r + row-sochasic marices Ψ(r, ) M[τ] = M[r]M[r + ] M[], τ=r wih r. In addiion, Ψ( +, ) I by convenion. For ease of exposiion, wihou loss of generaliy, le us fix a one-o-one mapping beween {n+,, m} and (j, i) E. Thus, for each non-virual agen i V = {,, n}, we have m z i [] = z j [0]Ψ ji (, ) = y j Ψ ji (, ), (4) where he las equaliy holds due o z j [0] = y j for i V and z j [0] = 0 for j / V. Similar o (4), for he weigh evoluion, for each i {,, m}, we have w i [] = w j [0]Ψ ji (, ), (5) Using ergodic coefficiens and some celebraed resuls obained by Hajnal [], we show he following hoerem. Theorem. Under Algorihm 3, a each agen i V = {,, n}, z i [] w i [] n k= y k y k γ nb+. n nβnb+ k= Here we use o denoe l 2 norm. All he missing proofs can be found in he full version [8]. IV. ROBUST DISTRIBUTED DUAL AVERAGING METHOD We apply Algorihm 3 o disribued dual averaging mehod as informaion fusion primiive. Throughou his secion, we assume ha each agen i knows a privae cos funcion h i : X R, where (A) X R d is nonempy, convex and compac; and 4

5 (B) h i is convex and L Lipschiz coninuous wih respec o l 2 norm, i.e., for all x, y X, h i (x) h i (y) L x y, i V (6) We are ineresed in solving min x X h(x) n h i (x). (7) using a muli-agen nework where he communicaion links may suffer packe-dropping failures. Le X be he collecion of opimal soluions of h subjec o X. Since X R d is a nonempy, convex and compac, X is also nonempy, convex and compac. In addiion o he esimae sequence {x[]} =0, in dual averaging mehod, here is an addiional sequence {z[]} =0 in he dual space ha essenially aggregaes all he subgradiens generaed so far. In addiion, he dual averaging scheme involves a proximal funcion ψ : R d R ha is srongly convex. In his paper, we choose ψ o be srongly convex wih respec o l 2 norm, ha is ψ(y) ψ(x) + ψ(x), y x + 2 x y 2, for x, y R d. In addiion, we assume ha ψ 0 and argmin x ψ(x) = 0 R d, which is also referred as proximal cener. This choice of ψ is raher sandard [5], [9]. As i can be seen laer, his proximal funcion can be used o smooh he updae of he primal sequence {x[]} =0. One ypical ierae sequence under dual averaging mehod is as follows. Iniializing z[0] = x[0] = 0 R d, for ieraion ( 0), compue g[] h(x[]), and updae z and x as z[ + ] = z[] + g[], (8) x[ + ] = ψ (z[ + ], α[]), (9) x Rd where ψ x R ( ) is he projecion operaor defined as d ψ (z, α) x Rd argmin x R d { z, x + α ψ(x) }. (20) From (9), we know ha he updae of x is based on all he subgradiens generaed so far, and all hese subgradiens are weighed equally. The convergence rae of he dual averaging mehod is O( ), which is faser han he subgradien mehod whose convergence rae is O( log ). Besides, he consans of he dual averaging mehod are ofen smaller [5]. Nex we presen our Robus Push-Sum Disribued Dual Averaging (RPSDA) mehod. In our RPSDA, each agen i locally keeps esimae sequence {x i []} =0, gradien aggregaion (value) sequence {z i []} =0, and weigh sequence {w i []} =0, where x i [0] = z i [0] = 0 R d and w i [0] = R. In addiion, le {α[]} =0 be a sequence of posiive sepsizes. We will specify he choice of α[] in our saemen of heorem. Noe ha he only difference beween Algorihm 4 and Algorihm 3 Algorihm 4: RPSDA Iniializaion: z i [0] = x i [0] = σ i [0] = 0 R d, σ i [0] = 0 R, w i [0] = R, ρ ji [0] = 0 R d and ρ ji [0] = 0 R for each incoming link, i.e., j I i. 2 for do 3 σ + i [] σ i[ ] + zi[ ] d o i +, σ + i [] σ i[ ] + wi[ ] d o i + ; 4 Broadcas ( σ + i [], σ+ i []) o ougoing neighbors; 5 for each incoming link (j, i) do 6 if message ( σ + j [], σ+ j []) is received hen 7 ρ ji [] σ + j [], ρ ji[] σ + j []; 8 else 9 ρ ji [] ρ ji [ ], ρ ji [] ρ ji [ ]; 0 end z + zi[ ] i [] (ρ ji [] ρ ji [ ]), 2 end w + i [] wi[ ] ( ρ ji [] ρ ji [ ]). 3 σ i [] σ + i [] + z+ i [] d o i +, σ i[] σ + i [] + w+ i [] d o i +, z i [] z+ i [] d o+, w i[] w+ i [] i d o i +. 4 Compue a subgradien g i [ ] h i (x i [ ]); 5 z i [] z i [] + g i [ ]; 6 x i [] ψ X 7 end ( zi[] w i[], α[ ] ); is ha a subgradien is compued and added o he local value z. One imporanly, he local esimae x is updaed using dual averaging updae. For ease of exposiion, le g i [r] = 0 for each virual agen i {n +,, m} and r 0. Similar o (4) and (5), we have z i [] = w i [] = r=0 g j [r]ψ j,i (r, ) Ψ j,i (, ). Le z[] n n z i[]. We have z[] = n m z i [] = n r=0 g i [r]. (2) Le {α[]} =0 be a sequence of non-increasing sepsizes. For each agen i V, we define he running average of x i [], denoed by ˆx i [T ], as follows: ˆx i [T ] = T T x i []. = Theorem 2. Le x X, and suppose ha ψ(x ) R 2. Le {α[] = A } = wih α[0] = A be he sequence of sepsizes 5

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