Gaussian Learning-Without-Recall in a Dynamic Social Network
aa r X i v : . [ m a t h . O C ] S e p Gaussian Learning-Without-Recall in a Dynamic Social Network
Chu Wang and Bernard Chazelle Abstract — We analyze the dynamics of the Learning-Without-Recall model with Gaussian priors in a dynamic socialnetwork. Agents seeking to learn the state of the world, the“truth”, exchange signals about their current beliefs across achanging network and update them accordingly. The agents areassumed memoryless and rational, meaning that they Bayes-update their beliefs based on current states and signals, withno other information from the past. The other assumptionis that each agent hears a noisy signal from the truth at afrequency bounded away from zero. Under these conditions,we show that the system reaches truthful consensus almostsurely with a convergence rate that is polynomial in expectation.Somewhat paradoxically, high outdegree can slow down thelearning process. The lower-bound assumption on the truth-hearing frequency is necessary: even infinitely frequent accessto the truth offers no guarantee of truthful consensus in thelimit.
I. I
NTRODUCTION
People typically form opinions by updating their cur-rent beliefs and reasons in response to new signals fromother sources (friends, colleagues, social media, newspapers,etc.) [1], [2], [3]. Suppose there were an information sourcethat made a noisy version of the “truth” available to agentsconnected through a social network. Under which conditionswould the agents reach consensus about their beliefs? Whatwould ensure truthful consensus (meaning that the consensuscoincided with the truth)? How fast would it take for theprocess to converge? To address these questions requiresagreeing on a formal model of distributed learning. Fullyrational agents update their beliefs by assuming a prior andusing Bayes’ rule to integrate all past information available tothem [4], [5], [6], [7], [8], [9]. Full rationality is intractablein practice [10], [11], so much effort has been devoted todeveloping computationally effective mechanisms, includingnon- (or partially) Bayesian methods [12], [10], [3], [13],[14]. Much of this line of work can be traced back to theseminal work of DeGroot [15] on linear opinion pooling.This paper is no exception. Specifically, it follows the
Bayesian-Without-Recall (BWR) model recently proposed byRahimian and Jadbabaie in [11]; see also [16], [17], [18].The agents are assumed to be memoryless and rational:this means that they use Bayesian updates based on currentbeliefs and signals with no other information from the past.The process is local in that agents can collect informationonly from their neighbors in a directed graph. In this work,the graph is allowed to change at each time step. The BWR Nokia Bell Labs, 600-700 Mountain Avenue, Murray Hill, New Jersey07974, [email protected] Department of Computer Science, Princeton University, 35 Olden Street,Princeton, New Jersey 08540, chazelle @ cs.princeton.edu model seeks to capture the benefits of rational behavior whilekeeping both the computation and the information stored toa minimum [18].A distinctive feature of our work is that the social networkneed not be fixed once and for all. The ability to modifythe communication channels over time reflects the inherentlychanging nature of social networks as well as the realitythat our contacts do not all speak to us at once. Thus evenif the underlying network is fixed over long timescales,the model allows for agents to be influenced by selectedsubsets of their neighbors. Dynamic networks are commonoccurrences in opinion dynamics [19], [20], [21], [22] but,to our knowledge, somewhat new in the context of sociallearning.Our working model in this paper posits a Gaussian setting:the likelihoods and initial priors of the agents are normaldistributions. During the learning process, signals are gener-ated as noisy measurements of agents’ beliefs and the noiseis assumed normal and unbiased. Thus all beliefs remainGaussian at all times [11], [23].Our main result is that, under the assumption that eachagent hears a noisy signal from the truth at a frequencybounded away from zero, the system reaches truthful con-sensus almost surely with a convergence rate polynomial inexpectation. Specifically, we show that, as long as each agentreceives a signal from the truth at least once every /γ steps,the convergence rate is O ( t − γ/ d ) , where d is the maximumnode outdegree.Somewhat paradoxically, high outdegree can slow downlearning. The reason is that signals from peer agents areimperfect conveyors of the truth and can, on occasion,contaminate the network with erroneous information; thisfinding is in line with a similar phenomenon uncoveredby Harel et al. [24], in which social learning system withtwo Bayesian agents is found to be hindered by increasedinteraction between the agents. We note that our lower-boundassumption on the truth-hearing frequency is necessary: eveninfinitely frequent access to the truth is not enough to achievetruthful consensus in the limit. Further background.
Researchers have conducted empir-ical evaluations of both Bayesian and non-Bayesian mod-els [25], [26], [27], [28]. In [29], Mossel et al. analyzed aBayesian learning system in which each agent gets signalsfrom the truth only once at the beginning and then inter-act with other agents using Gaussian estimators. In [30],Moscarini et al. considered social learning in a model wherethe truth is not fixed but is, instead, supplied by a Markovchain. In the different but related realm of iterated learning,gents learn from ancestors and teach descendants. The goalis to pass on the truth through generations while seeking toprevent information loss [31], [32].
Organization.
Section II introduces the model and thebasic formulas for single-step belief updates. Section IIIinvestigates the dynamics of the beliefs in expectation andderive the polynomial upper bound on the convergence rateunder the assumption that each agent hears a signal fromthe truth at a frequency bounded away from zero. Wedemonstrate the necessity of this assumption in Section IVand prove that the convergence occurs almost surely.II. P
RELIMINARIES
A. The Model
We choose the real line R as the state space and we denotethe agents by , , . . . , n ; for convenience, we add an extraagent, labeled , whose belief is a fixed number, unknownto others, called the truth . At time t = 0 , , . . . , the belief ofagent i is a probability distribution over the state space R ,which is denoted by µ t,i . We assume that the initial belief µ ,i of agent i is Gaussian: µ ,i ∼ N ( x ,i , σ ,i ) . Withoutloss of generality, we assume the truth is a constant (single-point distribution: µ t, = 0 ; σ t, = 0 for all t ) and thestandard deviation is the same for all other agents, ie, σ ,i = σ > for i > .The interactions between agents are modeled by an infinitesequence ( G t ) t ≥ , where each G t is a directed graph overthe node set { , . . . , n } . An edge pointing from i to j in G t indicates that i receives data from j at time t . Typically, thesequence of graphs is specified ahead of time or it is chosenrandomly: the only condition that matters is that it should beindependent of the randomness used in the learning process;specifically, taking expectations and variances of the randomvariables that govern the dynamics will assume a fixed graphsequence (possibly random). Because agent holds the truth,no edge points away from it. The adjacency matrix of G t isdenoted by A t : it is an ( n + 1) × ( n + 1) matrix whose firstrow is (1 , , . . . , . B. Information Transfer
At time t ≥ , each agent i > samples a state θ t,i ∈ R consistent with her own belief: θ t,i ∼ µ t,i . A noisymeasurement a t,i = θ t,i + ε t,i is then sent to each agent j such that ( A t ) ji = 1 . All the noise terms ε t,i are sampled iid from N (0 , σ ) . An equivalent formulation is to say that thelikelihood function l ( a | θ ) is drawn from N ( θ, σ ) . In oursetting, agent i sends the same data to all of her neighbors;this is done for notational convenience and the same resultswould still hold if we were to resample independently foreach neighbor. Except for the omission of explicit utilitiesand actions, our setting is easily identified as a variant of theBWR model [11]. C. Updating Beliefs
A single-step update for agent i > consists of setting µ t +1 ,i as the posterior P [ µ t,i | d ] ∝ P [ d | µ t,i ] P [ µ t,i ] , where d isthe data from the neighbors of i received at time t . Pluggingin the corresponding Gaussians gives us the classical updaterules from Bayesian inference [23]. Updated beliefs remainGaussian so we can use the notation µ t,i ∼ N ( x t,i , τ − t,i ) ,where τ t,i denotes the precision σ − t,i . Writing τ = σ − andletting d t,i denote the outdegree of i in G t , for any i > and t ≥ , ( x t +1 ,i = ( τ t,i x t,i + τ a + · · · + τ a d t,i ) / ( τ t,i + d t,i τ ); τ t +1 ,i = τ t,i + d t,i τ, (1)where a , . . . , a d t,i are the signals received by agent i fromits neighbors at time t . D. Expressing the Dynamics in Matrix Form
Let D t and P t denote the ( n + 1) -by- ( n + 1) diagonalmatrices diag ( d t,i ) and ( τ /τ ) I + P t − k =0 D k , respectively,where I is the identity matrix and the sum is for t = 0 . Itfollows from (1) that µ t,i ∼ N ( x t,i , ( τ P t ) − ii ) for i > . Re-grouping the means in vector form, x t := ( x t, , . . . , x t,n ) T ,where x t, = 0 and x , , . . . , x ,n are given as inputs, wehave x t +1 = ( P t + D t ) − ( P t x t + A t ( x t + u t + ε t )) , (2)where u t is such that u t, ∼ N (0 , and, for i > , u t,i ∼ N ( , ( τ ( P t ) ii ) − ) ; and ε t is such that ε t, ∼ N (0 , and, for i > , ε t,i ∼ N ( , /τ ) . We refer to the vectors x t and y t := E x t as the mean process and the expectedmean process , respectively. Taking expectations on both sidesof (2) with respect to the random vectors u t and ε t yieldsthe update rule for the expected mean process: y = x and,for t > , y t +1 = ( P t + D t ) − ( P t + A t ) y t . (3)A key observation is that ( P t + D t ) − ( P t + A t ) is astochastic matrix, so the expected mean process y t forms adiffusive influence system [22]: the vector evolves by takingconvex combinations of its own coordinates. What makesthe analysis different from standard multiagent agreementsystems is that the weights vary over time. In fact, someweights typically tend to 0, which violates one of the cardinalassumptions used in the analysis of averaging systems [22],[33]. This leads us to the use of arguments, such as fourth-order moment bounds, that are not commonly encounteredin this area. E. Our Results
The belief vector µ t is Gaussian with mean x t andcovariance matrix Σ t formed by zeroing out the top-leftelement of ( τ P t ) − . We say that the system reaches truthfulconsensus if both the mean process x t and the covariancematrix tend to zero as t goes to infinity. This indicates that allthe agents’ beliefs share a common mean equal to the truthand the “error bars” vanish over time. In view of (1), theovariance matrix indeed tends to as long as the degreesare nonzero infinitely often, a trivial condition. To establishtruthful consensus, therefore, boils down to studying themean process x t . We do this in two parts: first, we showthat the expected mean process converges to the truth; thenwe prove that fluctuations around it eventually vanish almostsurely. Truth-hearing assumption : Given any interval of length κ := ⌊ /γ ⌋ , every agent i > has an edge ( i, in G t for at leastone value of t in that interval.T HEOREM O ( t − γ/ d ) , where d is the maximum outdegreeover all the networks.We prove the theorem in the next two sections. It willfollow directly from Lemmas 3.1 and 4.1 below. The con-vergence rate can be improved to the order of t − (1 − ε ) γ/d ,for arbitrarily small ε > . The inverse dependency on γ is not surprising: the more access to the truth the strongerthe attraction to it. On the other hand, it might seemcounterintuitive that a larger outdegree should slow downconvergence. This illustrates the risk of groupthink. It paysto follow the crowds when the crowds are right. When theyare not, however, this distracts from the lonely voice thathappens to be right.How essential is the truth-hearing assumption? We showthat it is necessary. Simply having access to the truthinfinitely often is not enough to achieve truthful consensus. F. Useful Matrix Inequalities
We highlight certain matrix inequalities to be usedthroughout. We use the standard element-wise notation R ≤ S to indicate that R ij ≤ S ij for all i, j . The infinity norm k R k ∞ = max i P j | r ij | is submultiplicative: k RS k ∞ ≤k R k ∞ k S k ∞ , for any matching rectangular matrices. On theother hand, the max-norm k R k max := max i,j | r ij | is not,but it is transpose-invariant and also satisfies: k RS k max ≤k R k ∞ k S k max . It follows that k RSR T k max ≤ k R k ∞ k SR T k max = k R k ∞ k RS T k max ≤ k R k ∞ k S T k max = k R k ∞ k S k max . (4)III. T HE E XPECTED M EAN P ROCESS D YNAMICS
We analyze the convergence of the mean process inexpectation. The expected mean y t = E x t evolves throughan averaging process entirely determined by the initial value y = (0 , x , , . . . , x ,n ) T and the graph sequence G t . Intu-itively, if an agent communicates repeatedly with a holderof the truth, the weight of the latter should accumulate and The Kullback-Leibler divergence [12] is not suitable here because theestimator is Gaussian, hence continuous, whereas the truth is a single-pointdistribution. increasingy influence the belief of the agent in question. Ourgoal in this section is to prove the following result:L
EMMA y t converges to the truth asymptotically.If, at each step, no agent receives information from more than d agents, then the convergence rate is bounded by Ct − γ/ d ,where C is a constant that depends on x , γ, d, σ /σ . Proof.
We define B t as the matrix formed by removingthe first row and the first column from the stochastic P − t +1 ( P t + A t ) . If we write y t as (0 , z t ) then, by (3), (cid:18) z t +1 (cid:19) = (cid:18) α t B t (cid:19) (cid:18) z t (cid:19) , (5)where α t,i = ( P − t +1 ) ii if there is an edge ( i, at time t and α t,i = 0 otherwise. This further simplifies to z t +1 = B t z t . (6)Let be the all-one column vector of length n . Since P − t +1 ( P t + A t ) is stochastic, α t + B t = (7)In matrix terms, the truth-hearing assumption means that, forany t ≥ , α t + α t +1 + · · · + α t + κ − ≥ Q − t + κ , (8)where Q t is the matrix derived from P t by removing the firstrow and the last column; the inequality relies on the fact that P t is monotonically nondecreasing. For any t > s ≥ , wedefine the product matrix B t : s defined as B t : s := B t − B t − . . . B s , (9)with B t : t = I . By (6), for any t > s ≥ , z t = B t : s z s . (10)To bound the infinity norm of B t :0 , we observe that, for any ≤ l < κ − , the i -th diagonal element of B s + κ : s + l +1 islower-bounded by κ − Y j = l +1 ( B s + j ) ii = κ − Y j = l +1 ( P s + j + A s + j ) ii ( P s + j +1 ) ii (11) ≥ κ − Y j = l +1 ( P s + j ) ii ( P s + j +1 ) ii = ( P s + l +1 ) ii ( P s + κ ) ii ≥ ( P s ) ii ( P s + κ ) ii . The inequalities follow from the nonnegativity of the entriesand the monotonicity of ( P t ) ii . Note that (11) also holds for l = κ − since ( B s + κ : s + κ ) ii = 1 .Since P − t +1 ( P t + A t ) is stochastic, the row-sum of B t does not exceed 1; therefore, by premultiplymultiplying B s +1 , B s +1 , . . . on both sides of (7), we obtain: B s + κ : s ≤ − κ − X l =0 B s + κ : s + l +1 α s + l . (12)oting that k B t k ∞ = k B t k ∞ for any t , as B t is non-negative, we combine (8), (11), and (12) together to derive: k B s + κ : s k ∞ ≤ − min i> ( P s ) ii ( P s + κ ) ii . (13)Let d := max t ≥ max ≤ i ≤ n d t,i denote the maximum outde-gree in all the networks, and define δ = min { τ /τ, } . Forany i > and s ≥ κ , sδκ ≤ ( P s ) ii ≤ ds + τ τ ; (14)hence, max i ( P s + κ ) ii ≤ d ( s + κ ) + τ τ . (15)It follows that ( P s + κ ) ii − ( P s ) ii ( P s + κ ) ii = P κ − l =0 d s + l,i ( P s + κ ) ii ≤ dκ δ − s + κ . (16)Thus, we have min i> ( P s ) ii ( P s + κ ) ii = 1 − max i> ( P s + κ ) ii − ( P s ) ii ( P s + κ ) ii ≥ − dκ δ − s + κ . (17)We can replace the upper bound of (13) by − i> ( P s + κ ) ii min i> ( P s ) ii ( P s + κ ) ii , which, together with (15) and (17) gives us k B s + κ : s k ∞ ≤ − d ( s + κ ) + τ /τ (cid:18) − dκ δ − s + κ (cid:19) ≤ − dκ ( m + 2) . (18)The latter inequality holds as long as s = mκ > and m ≥ m ∗ := 2 dκδ + τ dκτ . It follows that, for m ≥ m ∗ , k B ( m + m ) κ : m κ k ∞ ≤ m +1 Y j =2 (cid:18) − dκ ( m + j ) (cid:19) ≤ exp − dκ m +1 X j =2 m + j . (19)The matrices B t are sub-stochastic so that k B t z k ∞ ≤ k B t k ∞ k z k ∞ ≤ k z k ∞ . By (10), for any t ≥ ( m + m ) κ , z t = B t :( m + m ) κ B ( m + m ) κ : m κ z m , so that, by using standard bounds for the harmonic series, ln( k + 1) < + · · · + k ≤ k , we find that k z t k ∞ ≤ k B ( m + m ) κ : m κ z m k ∞ ≤ k B ( m + m ) κ : m κ k ∞ k z k ∞ ≤ Ct − / (2 dκ ) , where C > depends on z , κ, d, τ /τ . We note that theconvergence rate can be improved to the order of t − (1 − ε ) γ/d ,for arbitrarily small ε > , by working a little harderwith (18). (cid:3) IV. T HE M EAN P ROCESS D YNAMICS
Recall that µ t,i ∼ N ( x t,i , τ − t,i ) , where τ t,i denotes theprecision σ − t,i . A key observation about the updating rulein (1) is that the precision τ t,i is entirely determined bythe graph sequence G t and is independent of the actualdynamics. Adding to this the connectivity property impliedby the truth-hearing assumption, we find immediately that τ t,i → ∞ for any agent i . This ensures that the covariancematrix Σ t tends to as t goes to infinity, which satisfies thesecond criterion for truthful consensus. The first criterionrequires that the mean process x t should converge to thetruth . Take the vector x t − y t and remove the firstcoordinate ( x t − y t ) to form the vector ∆ t ∈ R n . Underthe truth-hearing assumption, we have seen that y t → (Lemma 3.1), so it suffices to prove the following:L EMMA ∆ t vanishes almost surely. Proof.
We use a fourth-moment argument. The justificationfor the high order is technical: it is necessary to make acertain “deviation power” series converge. By (2), x t is alinear combination of independent Gaussian random vectors u s and ε s for ≤ s ≤ t − , and thus x t itself is a Gaussianrandom vector. Therefore ∆ t is also Gaussian and its meanis zero. From Markov’s inequality, for any c > , X t ≥ P [ | ∆ t,i | ≥ c ] ≤ X t ≥ E ∆ t,i c . (20)If we are able to show the right hand side of (20) is finitefor any c > , then, by the Borel-Cantelli lemma, withprobability one, the event | ∆ t,i | ≥ c occurs only a finitenumber of times, and so ∆ t,i goes to zero almost surely.Therefore, we only need to analyze the order of the fourthmoment E ∆ t,i . By subtracting (3) from (2), we have: ∆ t +1 = B t ∆ t + M t v t , (21)where v t := u t + ε t and M t := P − t +1 A t ; actually, fordimensions to match, we remove the top coordinate of v t and the first row and first column of M t (see previous sectionfor definition of B t ). Transforming the previous identity intoa telescoping sum, it follows from ∆ = x − y = andthe definition B t : s = B t − B t − . . . B s that ∆ t = t − X s =0 B t : s +1 M s v s = t − X s =0 R t,s v s , (22)where R t,s := B t : s +1 M s . We denote by C , C , . . . suit-ably large constants (possibly depending on κ, d, n, τ, τ ).y (14), k M s k ∞ ≤ C / ( s + 1) and, by (19), for sufficientlylarge s , k B t : s +1 k ∞ ≤ C ( s + 1) β ( t + 1) − β , where β = 1 / dκ < . Combining the above inequalities,we obtain the following estimate of R t,s as k R t,s k ∞ ≤ C ( s + 1) − β ( t + 1) − β . (23)In the remainder of the proof, the power of a vector isunderstood element-wise. We use the fact that v s and v s ′ are independent if s = s ′ and that the expectation of an oddpower of an unbiased Gaussian is always zero. By Cauchy-Schwarz and Jensen’s inequalities, E ∆ t = t − X s =0 R t,s v s ! = t − X s =0 E ( R t,s v s ) + X ≤ s = s ′ We now show why the truth-hearing assumption is nec-essary. We describe a sequence of graphs G t that allowsevery agent infinite access to the truth and yet does not leadto truthful consensus. For this, it suffices to ensure that theexpected mean process y t does not converge. Consider asystem with two learning agents with priors µ , and µ , from the same distribution N (2 , . We have x , = x , = y , = y , = 2 and, as usual, the truth is assumed to be0; the noise variance is σ = 1 . The graph sequence isdefined as follows: set t = 0 ; for k = 1 , , . . . , agent 1links to the truth agent at time t k and to agent 2 at times t k + 1 , . . . , s k − ; then at time s k , agent 2 links to the truthagent, and then to agent 1 at times s k + 1 , . . . , t k +1 − . Thetime points s k and t k are defined recursively to ensure that y s k , ≥ − k +1 and y t k , ≥ − k . (27)In this way, the expected mean processes of the two agentsalternate while possibly sliding down toward 1 but neverlower. The existence of these time points can be proved byinduction. Since y , = 2 , the inequality y t k , ≥ − k holds for k = 1 , so let’s assume it holds up to k > . Thekey to the proof is that, by (3), as agent 1 repeatedly linksto agent 2, she is pulled arbitrarily close to it. 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