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Success and Failure of Adaptation-Diffusion Algorithms for Consensus in Multi-Agent Networks Gemma Morral, Pascal Bianchi, Gersende Fort

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Success and Failure of Adaptation-Diffusion Algorithms for Consensus in Multi-Agent Networks Gemma Morral, Pascal Bianchi, Gersende Fort Success and Failure of Adaptation-Diffusion Algorithms for Consensus in Multi-Agent Networks Gemma Morral, Pascal Bianchi, Gersende Fort To cite this version: Gemma Morral, Pascal Bianchi, Gersende Fort. Success and Failure of Adaptation-Diffusion Algo- rithms for Consensus in Multi-Agent Networks. 2014. hal-01078466 HAL Id: hal-01078466 https://hal.archives-ouvertes.fr/hal-01078466 Preprint submitted on 29 Oct 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Success and Failure of Adaptation-Diffusion Algorithms for Consensus in Multi-Agent Networks Gemma Morral∗, Pascal Bianchi and Gersende Fort Abstract—This paper investigates the problem of distributed of some other j’s and computes the weighted average: stochastic approximation in multi-agent systems. The algorithm N under study consists of two steps: a local stochastic approxi- ˜ mation step and a diffusion step which drives the network to θn,i = wn(i,j) θn,j , (2) a consensus. The diffusion step uses row-stochastic matrices to Xj=1 weight the network exchanges. As opposed to previous works, where the wn(i,j)’s are scalar non-negative random coeffi- exchange matrices are not supposed to be doubly stochastic, and N may also depend on the past estimate. cients such that j=1 wn(i,j) = 1 for any i. The sequence N We prove that non-doubly stochastic matrices generally in- of random matricesP Wn := [wn(i,j)]i,j=1 represents the fluence the limit points of the algorithm. Nevertheless, the limit time-varying communication network between the nodes. One points are not affected by the choice of the matrices provided that simply set wn(i,j) = 0 whenever nodes i and j are unable to the latter are doubly-stochastic in expectation. This conclusion communicate at time n. The aim of this paper is to investigate legitimates the use of broadcast-like diffusion protocols, which are easier to implement. Next, by means of a central limit theorem, the almost sure (a.s.) convergence of this algorithm as n tends we prove that doubly stochastic protocols perform asymptotically to infinity as well as the convergence rate. Our goal is in as well as centralized algorithms and we quantify the degradation particular to quantify the effect of the sequence of matrices caused by the use of non doubly stochastic matrices. Throughout (Wn)n 1 on the convergence. The algorithm is initialized at the paper, a special emphasis is put on the special case of some arbitrary≥ Rd-valued vectors θ , ,θ . distributed non-convex optimization as an illustration of our 0,1 ··· 0,N results. Application to distributed optimization. The algo- rithm (1)-(2) under study is not new. The idea beyond the algorithm traces back to [5], [6] where a network of processors seeks to optimize some objective function known by all agents I. INTRODUCTION (possibly up to some additive noise). More recently, numerous Distributed stochastic approximation has been recently pro- works extended this kind of algorithm to more involved multi- posed using different cooperative approaches. In the so-called agent scenarios, see [7]–[19] as a non-exhaustive list. In this incremental approach (see for instance [1]–[4]) a message context, one seeks to minimize a sum of local private cost containing an estimate of the quantity of interest iteratively functions fi of the agents: travels all over the network. This paper focuses on another N cooperative approach based on average consensus techniques min fi(θ) , (3) θ Rd where the estimates computed locally by each agent are ∈ Xi=1 combined through the network. where for all i, the function fi is supposed to be unknown Consider a network composed by N agents, or nodes. by any other agent j,j = i. To address this question, it is Agents seek to find a consensus on some global parameter by assumed that 6 means of local observations and peer-to-peer communications. Yn,i = fi(θn 1,i)+ ξn,i (4) The aim of this paper is to analyze the behavior of the follow- −∇ − ing distributed algorithm. Node i (i = 1,...,N) generates a where is the gradient operator and ξn,i represents some d ∇ R -valued stochastic process (θn,i)n 0. At time n, the update random perturbation which possibly occurs when observing ≥ is in two steps: the gradient. In this paper, we handle the case where functions ˜ [Local step] Node i generates a temporary iterate θn,i fi are not necessarily convex. Of course, in that case, there is given by generally no hope to ensure the convergence to a minimizer to (3). Instead, a more realistic objective is to achieve crit- θ˜n,i = θn 1,i + γn Yn,i , (1) − ical points of the objective function i.e., points θ such that where γn is a deterministic positive step size and where the i fi(θ) = 0. d ∇ R -valued random process (Yn,i)n 1 represents the observa- PConvergence to a global minimizer is shown in [20] as- tions made by agent i. ≥ suming convex utility functions and bounded (sub)gradients. [Gossip step] Node i is able to observe the values θ˜n,j The results of [20] are extended in [21] to the stochastic descent case i.e., when the observation of utility functions *This work is supported by DGA (French Armement Procurement Agency), is perturbed by a random noise. More recently, [14] investi- the Institut Mines-Tel´ ecom´ and by the ANR grant ODISSEE of program gated distributed stochastic approximation at large, providing ASTRID (ANR-13-ASTR-0030). stability conditions of the algorithm (1)-(2) while relaxing the G. Morral, P. Bianchi and G. Fort are with LTCI, Tel´ ecom´ Paris- Tech & CNRS, 46 rue Barrault, 75634 Paris Cedex 13, France bounded gradient assumption and including the case of random [firstname].[lastname]@telecom-paristech.fr communication links. In [14], it is also proved under some hypotheses that the estimation error is asymptotically normal: necessarily doubly stochastic. We show that non-doubly the convergence rate and the asymptotic covariance matrix are stochastic matrix have an influence on the asymptotic characterized. An enhanced averaging algorithm a` la Polyak error covariance (even if they are doubly stochastic in is also proposed to recover the optimal convergence rate. average). On the other hand, we prove that when the Doubly and non-doubly stochastic matrices. In most matrix Wn is doubly stochastic for all n, the asymptotic works (see for instance [20]–[22]), the matrices (Wn)n 1 are covariance is identical to the one obtained in a central- T 1 1≥ 1 assumed doubly stochastic, meaning that Wn = Wn = ized setting. where 1 is the N 1 vector whose components are all The paper is organized as follows. Section II is a gentle × equal to one and where T denotes transposition. Although presentation of our results in the special case of distributed 1 1 row-stochasticity (Wn = ) is rather easy to ensure in optimization (see (3)) assuming in addition that sequence T 1 1 practice, column-stochasticity (Wn = ) implies more strin- (Wn) is i.i.d. In Section III we provide the general setting gent restrictions on the communication protocol. For instance, to study almost sure convergence. Almost sure convergence in [23], each one-way transmission from an agent i to another is studied in Section IV. Section V investigates convergence agent j requires at the same time a feedback link from j rates. Conclusions and numerical results complete the paper. to i. As a matter of fact, double stochasticity prevents from Notations: Throughout the paper, the vectors are column using natural broadcast schemes, in which a given node may vectors. The random variables W RN N and Y := transmit its local estimate to all neighbors without expecting n × n (Y T ,...,Y T )T RdN , n 1, are∈ defined on the same any immediate feedback. n,1 n,N measurable space equipped∈ with≥ a probability P; E denotes Remarkably, although generally assumed, double stochastic- the associated expectation. For any n 1, define the σ- ity of the matrices W is in fact not mandatory. A couple of n field F := σ(θ ,W ,...,W ,Y ,...,Y≥) where θ is the works (see e.g. [14], [24]) get rid of the column-stochasticity n 0 1 n 1 n 0 (possibly random) initial point of the algorithm. condition, but at the price of assumptions that may not always It is assumed that for any i 1,...,N, (θn,i)n 0 satisfies the be satisfied in practice. Other works ([17], [25], [26]) manage ≥ update equations (1)-(2); and∈ we set to circumvent the use of feedback links by coupling the gradient descent with the so-called push-sum protocol [27]. T T T θn := (θn,1,...,θn,N ) . The latter however introduces an additional communication of weights in the network in order to keep track of some For any vector x Rℓ, x represents the Euclidean norm of ∈ | | T summary of the past transmissions. In this paper, we address x. IN is the N N identity matrix. J := 11 /N denotes the the following questions: What conditions on the sequence orthogonal projector× onto the linear span of the all-one N 1 × (Wn)n 1 are needed to ensure that Algorithm (1)-(2) drives vector 1, and J := IN J.
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