Decentralized Decision-Making Over Multi-Task Networks
aa r X i v : . [ m a t h . O C ] D ec Decentralized Decision-Making OverMulti-Task Networks
Sahar Khawatmi, Abdelhak M. Zoubir Technische Universit¨at Darmstadt, Signal Processing Group64283 Darmstadt, GermanyEmail: { khawatmi, zoubir } @spg.tu-darmstadt.de Ali H. Sayed Ecole polytechnique f ´ ed ´ erale de Lausanne EPFL, Adaptive Systems LaboratoryCH-1015 Lausanne, SwitzerlandEmail: ali.sayed@epfl.ch Abstract
In important applications involving multi-task networks with multiple objec-tives, agents in the network need to decide between these multiple objectivesand reach an agreement about which single objective to follow for the network.In this work we propose a distributed decision-making algorithm. The agents areassumed to observe data that may be generated by different models. Throughlocalized interactions, the agents reach agreement about which model to trackand interact with each other in order to enhance the network performance. Weinvestigate the approach for both static and mobile networks. The simulationsillustrate the performance of the proposed strategies.
Keywords:
Decentralized processing, decision-making, multi-task networks,adaptive learning.
1. Introduction and Related Work
Bio-inspired systems are designed to mimic the behavior of some animal groupssuch as bee swarms, birds flying in formation, and schools of fish [1, 2, 3, 4, 5, Member of the European Association for Signal Processing (EURASIP). Member of the European Association for Signal Processing (EURASIP).
Preprint submitted to EURASIP signal processing journal December 27, 2018 ]. Diffusion strategies can be used to model some of these coordinated typesof behavior, as well as solve inference and estimation tasks in a distributedmanner over networks [7, 8]. We may distinguish between two types of networks:single-task and multi-task networks. In single-task implementations [7, 8], thenetworks consist of agents that are interested in the same objective and sensedata that are generated by the same model. An analogy would be a school offish tracking a food source: all elements in the fish school sense distance anddirection to the same food source and are interested in approaching it. On theother hand, multi-task networks [9, 10, 11, 12, 13, 14, 15, 16, 17] involve agentssensing data arising from different models and different clusters of agents maybe interested in identifying separate models. A second analogy is a school offish sensing information about multiple food sources.In the latter case, agents need to decide between the multiple objectives andreach agreement on following a single objective for the entire network. In theearlier works [18, 19], a scenario was considered where agents were assumed tosense data arising from two models, and a diffusion strategy was developed toenable all agents to agree on estimating a single model. The algorithm developedin [18] relied on binary labeling and is applicable only to situations involvingtwo models. In this work, we propose an approach for more than two models.We consider a distributed mean-square-error estimation problem over an N -agent network. The connectivity of the agents is described by a graph (seeFig. 1). Data sensed by any particular agent can arise from one of differentmodels. The objective is to reach an agreement among all agents in the networkon one common model to estimate. Two definitions are introduced: the observedmodel, which refers to the model from which an agent collects data, and thedesired model, which refers to the model the agent decides to estimate. Theagents do not know which model generated the data they collect; they also do notknow which other agents in their neighborhood sense data arising from the samemodel. Therefore, each agent needs to determine the subset of its neighbors thatobserves the same model. This initial step is referred to as clustering . Since thedecision-making objective depends on the clustering output, errors made during2he clustering process have an impact on the global decision. In this work, werely on the clustering technique proposed in[20] to reduce this effect.The paper is organized as follows. The network and data model are describedin Section II. We illustrate the local labeling system and the decision-makingalgorithm in Sections III and IV, respectively. A special case when the entirenetwork follows the model of a specific agent is studied in Section V. Simulationresults and discussion are presented in Sections VI and VII, respectively. Notation . We use lowercase letters to denote vectors, uppercase letters for ma-trices, plain letters for deterministic variables, and boldface letters for randomvariables. E denotes the expectation operator and k ·k the Euclidean norm. Thesymbols and I denote the all-one vector and identity matrix of appropriatesizes, respectively. The k − th row (column) of matrix X is denoted by [ X ] k, : ([ X ] : ,k ).
2. Network and Data Model
Consider a collection of N agents distributed in space, as illustrated in Fig. 1.We represent the network topology by means of an N × N adjacency matrix E whose entries e ℓk are defined as follows: e ℓk = , ℓ ∈ N k , , otherwise (1)where N k is the set of neighbors of agent k (we denote its size by n k ). We alsowrite N − k to denote the same neighborhood excluding agent k .Figure 1 shows the network structure where agents with the same colorobserve the same model. We denote the unknown models by { z ◦ , . . . , z ◦ C } , eachof size M × C ≤ N . Each agent k observes data generated by one ofthese C unknown models. We denote the model observed by agent k by w ◦ k .Figure 1 shows that agent k collects data from model z ◦ , in which case w ◦ k = z ◦ .For any other agent ℓ observing the same model z ◦ , it will hold that w ◦ ℓ = z ◦ .3e stack the { w ◦ k } into a column vector: w ◦ , col { w ◦ , w ◦ , · · · , w ◦ N } , w ◦ ∈ R MN × . (2)At every time instant i , every agent k has access to a scalar measurement d k ( i ) and a 1 × M regression vector u k,i . The measurements across all agentsare assumed to be generated via linear regression models of the form: d k ( i ) = u k,i w ◦ k + v k ( i ) . (3)All random processes are assumed to be stationary. Moreover, v k ( i ) is a zero-mean white measurement noise that is independent over space and has variance σ v,k . The regression data u k,i is assumed to be a zero-mean random process,independent over time and space, and independent of v ℓ ( j ) for all k, ℓ, i, j . Wedenote the covariance matrix of u k,i by R u,k = E u ⊺ k,i u k,i .Agents do not know which model is generating their data. They also do notknow which models are generating the data of their neighbors. Still, we wouldlike to perform a learning strategy that allows agents to converge towards one ofthe models, while also learning which of their neighbors share the same model.Using the algorithm proposed in [20], each agent k repeats the following stepsinvolving an adaptation step followed by an aggregation step: ψ k,i = ψ k,i − + µ k u ⊺ k,i ( d k ( i ) − u k,i ψ k,i − ) (4) φ k,i = N X ℓ =1 a ℓk ( i ) ψ ℓ,i (5)where µ k is the step-size used by agent k . These updates generate two iteratesby agent k at time i , and which are denoted by ψ k,i and φ k,i , respectively. Wecollect the estimated vectors across all agents into the aggregate vector: φ i , col { φ ,i , φ ,i , · · · , φ N,i } . (6)In a manner similar to [20], we introduce a clustering matrix E i . Its structureis similar to the adjacency matrix E , with ones and zeros, except that the valueat location ( ℓ, k ) will be set to one if agent k believes at time instant i that its4 igure 1: Example of a network topology, agents with the same color observe the same model. neighbor ℓ belongs to the same cluster, i.e., observes the same model: e ℓk ( i ) = , if ℓ ∈ N k and k believes that w ◦ k = w ◦ ℓ , , otherwise . (7)These entries help define the neighborhood set N k,i , which consists of all neigh-bors at time instant i that agent k believes share the same model. To learn thematrix E i over time, we apply the clustering technique proposed in [20] to cre-ate the estimated clustering matrix F i of size N × N as follows: we initialize ψ k, − = 0 and B − = F − = E − = I N . Where the matrix B i is of size N × N . Each entry e ℓk ( i ) is designed using the following steps from [20], where ℓ ∈ N k : b ℓk ( i ) = , if || ψ ℓ,i − φ k,i − || ≤ α , otherwise (8) f ℓk ( i ) = ν × f ℓk ( i −
1) + (1 − ν ) × b ℓk ( i ) (9) e ℓk ( i ) = ⌊ f ℓk ( i ) ⌉ (10)where α >
0, 0 ≤ ν ≤
1, and the notation ⌊·⌉ denotes rounding to the nearestinteger. Using the evolving neighborhoods N k,i , the entries { a ℓk ( i ) } in (5) are5on-negative scalars that satisfy a ℓk ( i ) = 0 for ℓ / ∈ N k,i , N X ℓ =1 a ℓk ( i ) = 1 . (11)Although there is a multitude of models generating the data that is feeding intothe agents, namely, { z ◦ , z ◦ , . . . , z ◦ C } , the objective is to develop a strategy thatwill allow all agents to converge towards one of these models. We refer to thisparticular choice as the desired model and denote it by z ◦ d .In this way, an agent whose source (observed) model agrees with the desiredmodel, i.e., w ◦ k = z ◦ d , will end up tracking its own source. On the other hand,an agent whose source model is not the desired model, i.e., w ◦ k = z ◦ d , will track z ◦ d instead although it is sensing data generated by a different model.We define the estimate vector of agent’s k desired model by w k,i . The reasonbehind indicating w k,i as the estimate vector of agent’s k desired model insteadof the network’s desired model is that the agents may have different desiredmodels before convergence (steady-state). Once the agents reach agreementamong themselves on a single model, we can then refer to w k,i as the estimatevector by agent k of the network’s desired model. For the initialization at timeinstant i = 1, each agent assigns w k, = ψ k, (i.e., at time instant i = 1,the desired model of each agent is a rough estimate of its own source model).The decision-making process drives the desired models of all agents to converge.For example, if the agents observe C = 5 different models, the number of thedesired models in the network will decrease with iterations gradually form 5models down to one model. This is achieved by switching the estimate w k,i ofsome agents during the decision-making process according to some conditionsthat are explained later. However, agents do not know which models are desiredby their neighbors at each time instant i . Thus, we need to develop a learningstrategy that allows the agents to distinguish the individual desired models oftheir neighbors.It turns out that in order for the objective of the network to be met, it isimportant for agents to combine the estimates of their neighbors in a judiciousmanner because, unbeknown to the agents, some of their neighbors may be6ishing to estimate different models. If cooperation is performed blindly withall neighbors, then performance can deteriorate with agents converging to non-existing locations. For this reason, and motivated by the discussion from [18], weadd the step (12) below after (4) and (5), which involves two sets of combinationcoefficients from two matrices ˙ A i and ¨ A i . There are two main ideas behind theconstruction (12). First, it is meant to let each agent k cooperate only withthe subset of neighbors that share the same desired model as it does. Second,it also lets each agent k combine φ ℓ,i if the desired model of agent k at timeinstant i is the same as ℓ ’s observed model: w k,i = N X ℓ =1 ˙ a ℓk ( i ) φ ℓ,i + N X ℓ =1 ¨ a ℓk ( i ) w ℓ,i − . (12)Note that the matrices ˙ A i and ¨ A i are not constructed from matrix A i . Theselection of the non-negative coefficients { ˙ a ℓk ( i ) } and { ¨ a ℓk ( i ) } is explained inSection IV.We summarize the main five steps of the approach:1. Learning the observed models of the neighbors . This step is performed bybuilding the matrix E i in step (7). The information provided by eachentry e ℓk ( i ) is whether the corresponding agents ℓ and k have the sameobserved model or not.2. Learning and labeling the desired model of the neighbors at each time in-stant i . This step allows the agents to distinguish the individual desiredmodels of their neighbors at time instant i . The information provided bythis step is the number of different models that are desired by neighborsand how many times each model is repeated at time i among neighbors.3. Decision-making step by switching the desired model of some agents to letthe network converge to only one model.4.
Learning the desired models of the neighbors after the switching step . Thisstep is performed by building the matrix H i in step (19) in Section IV. Theinformation provided by each entry h ℓk ( i ) is whether the correspondingagents ℓ and k have the same desired model or not after the switchingstep. 7 igure 2: Example of an agent k and its neighborhood N k . The inner color indicates theobserving model while the outer one indicates the current desired model. Updating the estimate vectors { w k,i } by sharing data thoughtfully withthe subset of the neighbors that share the same desired model.
3. Local Labeling
Each agent needs to learn the desired models of its neighbors to proceed withthe decision-making process and let the network converge to only one model. Inthis step, instead of only estimating whether two agents have the same desiredmodel or not, the construction involves a local labeling procedure that enablesevery agent to estimate in real-time how many different models are desired byits neighborhood.For this purpose, we associate with each agent k an n k × n k matrix Y ki withentries { y kℓm ( i ) } given by: y kℓm ( i ) = , if k w m,i − − w ℓ,i − k ≤ β, , otherwise (13)for some small threshold β >
0. Whenever y kℓm ( i ) = 1, agent k believes attime instant i that its neighbors ℓ and m wish to estimate the same desiredmodel. On account of that, the variables w m,i − and w ℓ,i − which are usedin the test (13) are presenting the current desired model of agents m and ℓ ,8espectively. It is clear from (13) that the matrix Y ki is symmetric and has oneson the diagonal. An example is depicted in Fig. 2 where agents having the sameinner color observe the same model, while the outer color indicates the model inwhich the agent is interested (or towards which the agent is moving in mobilenetworks). The corresponding matrix Y ki has the following entries: Y ki = kℓmnoq k ℓ m n o q . (14)From (14) agents that share the same desired model will have identical columnsin matrix Y ki , namely, if agents m and ℓ have the same desired model at timeinstant i , this implies that: [ Y ki ] : ,m = [ Y ki ] : ,ℓ . We denote the local label of eachagent ℓ ∈ N k by agent k as l kℓ ( i ). The local label l kℓ ( i ) is updated at each timeinstant i using the following relation: l kℓ ( i ) = B ([ Y ki ] : ,ℓ ) (15)where B ( · ) is a function that converts the input sequence from binary to decimal.For the example in (14), we have l kk ( i ) = B (100101) = 37 , l kℓ ( i ) = B (011000) = 24 , l km ( i ) = B (011000) = 24 , l kn ( i ) = B (100101) = 37 , l ko ( i ) = B (000010) = 2 , l kq ( i ) = B (100101) = 37 . We define the number of desired models within N k at time instant i by C k ( i ).After updating matrix Y ki and generating the local labels { l kℓ ( i ) } , agent k counts9ow many models are desired by its neighborhood to update C k ( i ). In the exam-ple (14), agent k distinguishes at time instant i three desired models { , , } ,i.e., C k ( i ) = 3. Agent k labels these three different models locally as: { , , } .In addition, agent k determines which model among these C k ( i ) models hasthe maximum number of followers. A follower of a model is an agent thatwishes to estimate and track this model. We define the largest set of agentsbelonging to N k and following the same desired model at time instant i by Q k,i . In the example, agent k assigns the majority set at time instant i asfollows: Q k,i = { k, n, q } which has the label 37 and is repeated three timesamong other labels.
4. Decision-Making Over Multi-Task Networks
Using the information provided by matrix Y ki , agent k can capture how manyagents within its neighbors follow the same desired model at time instant i .Once agent k and all its neighbors agree on a single desired model, the matrix Y ki will end up being of the following form with unit entries everywhere: Y ki = kℓmnoq k ℓ m n o q . (16)We define the degree of agreement by each agent k among its neighbors N k as p k ( i ) = [ Y ki ] k, : n k . (17)Equally, having p k ( i ) = 1 means that agent k and all of its neighbors haveagreed on a common desired model. On the other hand, if p k ( i ) = 1, then the10ollowing switching step is applied: w k,i − ← w ℓ,i − , if k / ∈ Q k,i for any ℓ ∈ Q k,i , w n,i − , if k ∈ Q k,i and C k ( i ) = 2 , ∀ n ∈ N k , w k,i − , otherwise . (18)The main idea of the switching step is for each agent k to make a new decisionor to keep the previous one. The first case of (18) implies that agent k doesnot belong to the majority desired model set Q k,i at time instant i . Therefore,agent k changes its decision and switches into the desired model of the majorityset Q k,i . The second case in (18) is applied to prevent an unwanted equilibriumsituation. This problem may arise when only two desired models remain in N k . In this case, if all agents in N k belong to the majority set, this leads to asituation in which no agent in N k will change its decision anymore. An exampleis shown in Fig. 3 where the outer color of the agents indicate the desired model.We indicate only the desired model of agent’s k neighbors and their neighbors.Figure 3 shows that all agents within N k belong to a majority set and no agent in N k will change its decision anymore, e.g. agents q and ℓ belong to the majorityset among their neighbors, as well as agents k , m , n , and o . Namely, k ∈ Q k,i , m ∈ Q m,i , n ∈ Q n,i , and o ∈ Q o,i (with z ◦ ) ,ℓ ∈ Q ℓ,i and q ∈ Q q,i (with z ◦ ) . To break the equilibrium, an agent that recognizes these two models picks ran-domly one of the two desired models.From (18), we can conjecture that the network will probably converge tothe most observable model, since the initial desired model by each agent is itsown observed model. This fact remains true even with the random switching inthe second case of (18), because in that case the more repeated desired modelwithin N k has the highest probability to be picked.To proceed with the cooperation and sharing information among the agentswithin the subset that has the same desired model at time instant i , we definean N × N matrix H i . The coefficients { h ℓk ( i ) } are updated after the switching11 igure 3: Example of the equilibrium case. All agents within N k belong to the majority setsamong their neighbors. step (18) using a test that is quite similar to (13) and is applied between eachagent k and its neighbors as follows: h ℓk ( i ) = , if k w k,i − − w ℓ,i − k ≤ β, , otherwise . (19)According to matrix H i , each agent knows which subset of its neighbors has thesame desired model as it does after the switching step at time instant i . Having h ℓk ( i ) = 1 means that ℓ and k have the same desired model at time instant i .We define an N × N combination matrix G i as follows: G i = F ( H i ) (20)where F ( · ) is some function which satisfies g ℓk ( i ) = 0 if h ℓk ( i ) = 0 , N X ℓ =1 g ℓk ( i ) = 1 (21)An example of F ( · ) is the uniform function which generates a left-stochasticmatrix G i where each entry g ℓk ( i ) is given by g ℓk ( i ) = P Nn =1 h nk ( i ) , if h ℓk ( i ) = 0 , , otherwise . (22)12atrix G i by itself does not have enough information for proceeding and updat-ing the estimate w k,i . The agents still need knowledge about which data to becombined from each neighbor. Therefore, matrix G i is split into two matrices ˙ A i and ¨ A i . The weight of the entry g ℓk ( i ) goes to ˙ a ℓk ( i ) if the desired model ofagent k at time instant i is the same as ℓ ’s observed model. Otherwise, ¨ a ℓk ( i )obtains the weight g ℓk ( i ). The coefficients { ˙ a ℓk ( i ) } and { ¨ a ℓk ( i ) } for ℓ ∈ N k areupdated using the following steps: ˙ a ℓk ( i ) = g ℓk ( i ) , if k w k,i − − ψ ℓ,i k ≤ β, , otherwise . (23) ¨ a ℓk ( i ) = g ℓk ( i ) , if ˙ a ℓk ( i ) = 0 , , otherwise . (24)In (23), the case that ψ ℓ,i is close to w ℓ,i − implies that the observed modelof agent ℓ is the same as the desired model of agent k at time instant i . Theestimate w k,i is updated using (12). Algorithm 1 summarizes the decision-making scheme.
5. Following the Observed Model of a Specific Agent
In this section the goal is to let the whole network follow the observed modelof some specific agent m , as shown in Fig. 4 where agent m observes model z ◦ (red), therefore, the network converges in a distributed manner to estimate themodel z ◦ . The first step is to spread the ψ m,i among agents and keep updatingit over time. This step aims at having a copy (reference) of ψ m,i by all agentsin the network. Agents keep updating the copy of ψ m,i for two reasons. First,to have a more accurate version of the vector ψ m,i , which indicates the desiredmodel of the network. Second, to endow the algorithm to work in non-stationarysituations, if drift is happening in agent m ’s model.We denote the copy vector of ψ m,i by agent k by ˘ ψ k,i and refer to it asthe anchor vector . Agents are informed beforehand about the index m of thespecific agent that they should follow. If m ∈ N k , this implies that agent k lgorithm 1 (Decentralized decision-making scheme)Initialize A = ˙ A = ¨ A = E = H = G = I Initialize ψ = φ = 0 and p = 0 for i > dofor k = 1 , . . . , N do ψ k,i = ψ k,i − + µ k u ⊺ k,i ( d k ( i ) − u k,i ψ k,i − ) (25)assign w k, = ψ k, at i = 1update { a ℓk ( i ) } according to (11) φ k,i = N X ℓ =1 a ℓk ( i ) ψ ℓ,i (26)update Y ki according to (13)find Q k,i and C k ( i )update p k ( i ) according to (17) if p k ( i ) = 1 then switch w k,i − according to (18)resend w k,i − end iffor ℓ ∈ N k do update { h ℓk ( i ) } according to (19)update { g ℓk ( i ) } according to (20)update { ˙ a ℓk ( i ) } according to (23)update { ¨ a ℓk ( i ) } according to (24) end for w k,i = N X ℓ =1 ˙ a ℓk ( i ) φ ℓ,i + N X ℓ =1 ¨ a ℓk ( i ) w ℓ,i − (27) end forend for igure 4: Final decision of a network after following the model of the specific agent m . Theinner color indicates the observing model while the outer one indicates the desired model.The arrows represent the spreading process of ψ m,i through the network. (a) (b) (c) Figure 5: Example of the spreading process of ψ m,i from agent m to agent k over time. Theinner color indicates the observing model while the outer one indicates the desired model. m . If not, i.e., m / ∈ N k , thenagent k depends on another agent ℓ ∈ N k that has already a copy of ψ m,i .Agent k stores the index of this source agent. The index of the source agent ofagent k is denoted by s k ( i ). Note that the anchor vector ˘ ψ k,i is not the finalestimate of the desired model.The circulation process of ψ m,i in a distributed manner needs cooperationamong agents. In case that agent k has no direct link to receive data fromagent m , i.e., m
6∈ N k , agent k gets one of the ˘ ψ ℓ,i − provided that s ℓ ( i ) = 0.If s ℓ ( i ) = 0 this implies that agent ℓ has already a source to update its ˘ ψ ℓ,i ,regardless whether m ∈ N ℓ or not. In other words, s ℓ ( i ) = 0 means that agent ℓ finds a direct or indirect link to agent m . Therefore, it is important for eachagent k to store the agent’s index of its source. An example is shown in Fig. 5where m ∈ N ℓ but m / ∈ N k . First, the anchor vectors and the source agents foragents k and ℓ at time instant i = 0 (Fig. 5(a)) are given, respectively, by ˘ ψ k, = 0 , s k (0) = 0 , ˘ ψ ℓ, = 0 , s ℓ (0) = 0 . (28)The anchor vectors and the source agents for agents k and ℓ at time instants i = { , } (Fig. 5(b) and (c)) are given, respectively, by ˘ ψ k, = 0 , s k (1) = 0 , ˘ ψ ℓ, = ψ m, , s ℓ (1) = m, (29) ˘ ψ k, = ˘ ψ ℓ, , s k (2) = ℓ, ˘ ψ ℓ, = ψ m, , s ℓ (2) = m. (30)Agents update their anchor vectors { ˘ ψ k,i } at each time instant i by the followingstep: ˘ ψ k,i = ψ m,i , if m ∈ N k , ˘ ψ ℓ,i − , if ℓ ∈ N k and s k ( i ) = 0 and s ℓ ( i ) = 0 , ˘ ψ ℓ,i − , if ℓ ∈ N k and s k ( i ) = ℓ, ˘ ψ k,i − , otherwise (31)where ˘ ψ m,i = ψ m,i for agent m itself. The source of the anchor vector is16pdated simultaneously as follows: s k ( i ) = m, if m ∈ N k ,ℓ, if s k ( i ) = 0 and s ℓ ( i ) = 0 , s k ( i − , otherwise . (32)Similarly to the previous section, the next step is to update the coefficients { h ℓk ( i ) } using the following test: h ℓk ( i ) = , if s ℓ ( i ) = 0 and s k ( i ) = 0 , , otherwise . (33)Again, having s k ( i ) = 0 leads to the situation that agent k has the anchorvector and has been informed about the decision of the network, therefore,agent k can start sharing information with the other agents whose s ℓ ( i ) = 0 aswell to estimate the desired model. The matrix G i will be generated using (20).Agents update the coefficients of both matrices ˙ A i and ¨ A i using the followingsteps: ˙ a ℓk ( i ) = g ℓk ( i ) , if k ˘ ψ k,i − ψ ℓ,i k ≤ β, , otherwise . (34) ¨ a ℓk ( i ) = g ℓk ( i ) , if ˙ a ℓk ( i ) = 0 , , otherwise . (35)Then, the estimate w k,i is updated using Eq. (12). According to (34) and (12),agent k combines φ ℓ,i if the desired model of the network (which is representedby the anchor vector ˘ ψ k,i of agent k ) is close to the observed model of agent ℓ that is represented by ψ ℓ,i . Algorithm 2 summarizes the steps of the approachfor following the observed model of a specific agent m .
6. Simulation Results and Discussion
We consider a connected network with 80 randomly distributed agents. Theagents observe data originating from C = 3 different models. Each model17 lgorithm 2 (Following the observed model of a specific agent)Initialize A = ˙ A = ¨ A = E = H = G = I Initialize ψ = ˘ ψ = φ = 0 and s = 0 for i > dofor k = 1 , . . . , N do ψ k,i = ψ k,i − + µ k u ⊺ k,i ( d k ( i ) − u k,i ψ k,i − ) (36)assign w k, = ψ k, at i = 1update { a ℓk ( i ) } according to (11) φ k,i = N X ℓ =1 a ℓk ( i ) ψ ℓ,i (37)update ˘ ψ k,i according to (31)update s k ( i ) according to (32) for ℓ ∈ N k do update { h ℓk ( i ) } according to (33)update { g ℓk ( i ) } according to (20)update { ˙ a ℓk ( i ) } according to (34)update { ¨ a ℓk ( i ) } according to (35) end for w k,i = N X ℓ =1 ˙ a ℓk ( i ) φ ℓ,i + N X ℓ =1 ¨ a ℓk ( i ) w ℓ,i − (38) end forend for T r a ce ( R u , k ) (b)Agent index, k
10 20 30 40 50 60 70 80 σ v , k (a) PSfrag replacements σ v,k Figure 6: Statistical noise and signal profiles over the network. z ◦ j ∈ R M × is generated as follows: z ◦ j = [ r , . . . , r M ] ⊺ where r m ∈ [1 , − M = 2. The assignment of the agents to models is random. The maximumnumber of neighbors is n k = 7. We set { α, β, ν, µ } = { . , . , . , . } .We use the uniform combination policy to generate the coefficients { a ℓk ( i ) } and { g ℓk ( i ) } .Figure 6 shows the statistical profile of the regressors and noise across theagents. The regressors are of size M = 2 zero-mean Gaussian, independent intime and space, and have diagonal covariance matrices R u,k . Figure 7 showsthe topology of one of 100 Monte Carlo experiments. Agents having the sameinner color observe the same model, while the outer color indicates the desiredmodel at steady-state.The transient network mean-square deviation (MSD) regarding each ob-served model z ◦ j at each time instant i is defined byMSD j ( i ) , |C j | X k ∈C j || z ◦ j − φ k,i || (39)where j = 1 , . . . , C and each MSD j is computed for agents belonging to C j .The transient network mean-square deviation (MSD) for the whole network19 a) (b) Figure 7: Network topology (a) and final decision of the agents where the bold (dashed) linksrepresent { ˙ a ( i ) } ( { ¨ a ( i ) } ) at steady-state (b). Time, i M S D ( d B ) -40-30-20-10010 MSD MSD MSD MSD d Figure 8: Transient mean-square deviation (MSD). (a) m (b) Figure 9: Network topology (a) and final decision of the agents to follow the model of agent m where the bold (dashed) links represent { ˙ a ( i ) } ( { ¨ a ( i ) } ) at steady-state (b). regarding the desired model at each time instant i is defined byMSD d ( i ) , N N X k =1 || z ◦ d − w k,i || (40)where z ◦ d is the desired model when the whole network agrees on one common de-sired model, i.e., MSD d ( i ) is only computed at the instants when all { p k ( i ) } = 1.Figure 8 depicts the simulated transient mean-square deviation (MSD) of thenetwork for all observed models and for the network desired model. Table 1 dis-plays the success rate of the decision-making to agree on one model for differentnumbers of observed models, C ∈ { , , , } . The proposed strategy appears toprovide good success rate under the simulated conditions. Table 1
Decision-making success rate for different C . C Success rate
99% 98% 99% 99%Regarding the application of following the observed model of a specific agent m , Fig. 9 shows the topology of one case from 100 different experiments. Agents21 ime, i M S D ( d B ) -40-30-20-10010 MSD MSD MSD MSD MSD d Figure 10: Transient mean-square deviation (MSD). are observing C = 4 different models. Agent m = 10, which is represented bya square, is the specific agent whose observed model the whole network wishesto follow. Figure 10 shows the transient mean-square deviation MSD of 100different experiments when a change in the model assignment occurs suddenlyat time instant i = 600. The success rate of the decision-making to agree onthe observed model of agent m was observed to be 100% in this simulation. We consider a network with 80 randomly distributed mobile agents [2]. Theagents observe data originating from four different models (sources) C = 4,where w r m ∈ [50 , − k updates itslocation according to the motion mechanism described in [19].Figure 12 shows the maneuver of the agents over time where the models(sources) are represented by squares. Figure 13 represents the transient net-work mean-square deviation (MSD) obtained by averaging over 100 independentMonte Carlo experiments. 22 T r a ce ( R u , k ) (b)Agent index, k
10 20 30 40 50 60 70 80 σ v , k (a) PSfrag replacements σ v,k Figure 11: Statistical noise and signal profiles over the mobile network. -50 0 50-50050 (a) -50 0 50-50050 (b) -50 0 50-50050 (c) x-axis (body length) -50 0 50 y - a x i s ( b o d y l e n g t h ) -50050 (d) Figure 12: Maneuver of the agents with four sources over time (a) i =1, (b) i =200, (c) i =500,and (d) i =1000. The unit length is the body length of a agent. ime, i M S D ( d B ) -40-2002040 MSD MSD MSD MSD MSD d Figure 13: Transient mean-square deviation (MSD) of the mobile network.
7. Conclusion
We have proposed a distributed algorithm that allows agents over multi-tasknetworks to follow only one common model while proceeding with the estimationprocess. Agents use a local labeling step to distinguish the multiple desiredmodels of their neighbors. Simulation results illustrate the operation of thealgorithms and its performance.
Acknowledgements
The research for this paper was financially supported by Technische Universit¨atDarmstadt, Signal Processing Group and NSF grant CCF-1524250.
ReferencesReferences [1] F. S. Cattivelli, A. H. Sayed, Modeling bird flight formations using diffusionadaptation, IEEE Trans. Signal Process. 59 (5) (2011) 2038–2051.[2] S. Y. Tu, A. H. Sayed, Mobile adaptive networks, IEEE J. Sel. Topics SignalProcess. 5 (4) (2011) 649–664. 243] A. Avitabile, R. A. Morse, R. Boch, Swarming honey bees guided bypheromones, Ann. Entonomol. Soc. Am. 68 (1975) 1079–1082.[4] H. Berg, Motile behavior of bacteria, Physics Today 53 (1) (2000) 24–29.[5] F. Dressler, O. B. Akan, Bio-inspired networking: from theory to practice,IEEE Communications Magazine (2010) 176–183.[6] S. Camazine, J. L. Deneubourg, N. R. Franks, G. T. J. Sneyd, E. Bonabeau,Self-Organization in Biological Systems, Princeton, NJ: Princeton Univ.Press, 2003.[7] J. C. X. Z. A. H. Sayed, S.Y. Tu, Z. Towfic, Diffusion strategies for adapta-tion and learning over networks: an examination of distributed strategiesand network behavior, IEEE Signal Processing Magazine 30 (3) (2013)155–171.[8] A. H. Sayed, Adaptation, learning, and optimization over networks, Found.Trends in Mach. Learn. 7 (4–5) (2014) 311–801.[9] X. Zhao, A. H. Sayed, Clustering via diffusion adaptation over networks,in: Proc. International Workshop on Cognitive Inform. Processing (CIP),Baiona, Spain, 2012, pp. 1–6.[10] J. Chen, C. Richard, A. H. Sayed, Diffusion LMS over multitask networks,IEEE Trans. Signal Processing 63 (11) (2015) 2733–2748.[11] X. Zhao, A. H. Sayed, Distributed clustering and learning over networks,IEEE Trans. Signal Processing 63 (13) (2015) 3285–3300.[12] S. Khawatmi, A. M. Zoubir, Decentralized partitioning over adaptive net-works, in: Proc. IEEE International Workshop on Machine Learning forSignal Processing (MLSP), Vietri sul Mare, Salerno, Italy, 2016, pp. 1–6. doi:10.1109/MLSP.2016.7738880doi:10.1109/MLSP.2016.7738880