nips nips2013 nips2013-104 knowledge-graph by maker-knowledge-mining

104 nips-2013-Efficient Online Inference for Bayesian Nonparametric Relational Models

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Author: Dae Il Kim, Prem Gopalan, David Blei, Erik Sudderth

Abstract: Stochastic block models characterize observed network relationships via latent community memberships. In large social networks, we expect entities to participate in multiple communities, and the number of communities to grow with the network size. We introduce a new model for these phenomena, the hierarchical Dirichlet process relational model, which allows nodes to have mixed membership in an unbounded set of communities. To allow scalable learning, we derive an online stochastic variational inference algorithm. Focusing on assortative models of undirected networks, we also propose an efﬁcient structured mean ﬁeld variational bound, and online methods for automatically pruning unused communities. Compared to state-of-the-art online learning methods for parametric relational models, we show signiﬁcantly improved perplexity and link prediction accuracy for sparse networks with tens of thousands of nodes. We also showcase an analysis of LittleSis, a large network of who-knows-who at the heights of business and government. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Stochastic block models characterize observed network relationships via latent community memberships. [sent-7, score-0.244]

2 In large social networks, we expect entities to participate in multiple communities, and the number of communities to grow with the network size. [sent-8, score-0.241]

3 We introduce a new model for these phenomena, the hierarchical Dirichlet process relational model, which allows nodes to have mixed membership in an unbounded set of communities. [sent-9, score-0.29]

4 To allow scalable learning, we derive an online stochastic variational inference algorithm. [sent-10, score-0.219]

5 Focusing on assortative models of undirected networks, we also propose an efﬁcient structured mean ﬁeld variational bound, and online methods for automatically pruning unused communities. [sent-11, score-0.486]

6 Compared to state-of-the-art online learning methods for parametric relational models, we show signiﬁcantly improved perplexity and link prediction accuracy for sparse networks with tens of thousands of nodes. [sent-12, score-0.387]

7 1 Introduction A wide range of statistical models have been proposed for the discovery of hidden communities within observed networks. [sent-14, score-0.231]

8 The simplest stochastic block models [20] create communities by clustering nodes, aiming to identify demographic similarities in social networks, or proteins with related functional interactions. [sent-15, score-0.235]

9 We propose a novel hierarchical Dirichlet process relational (HDPR) model, which allows mixed membership in an unbounded collection of latent communities. [sent-18, score-0.261]

10 By adapting the HDP [18], we allow data-driven inference of the number of communities underlying a given network, and growth in the community structure as additional nodes are observed. [sent-19, score-0.483]

11 The inﬁnite relational model (IRM) [10] previously adapted the Dirichlet process to deﬁne a nonparametric relational model, but restrictively associates each node with only one community. [sent-20, score-0.286]

12 The more ﬂexible nonparametric latent feature model (NLFM) [14] uses an Indian buffet process (IBP) [7] to associate nodes with a subset of latent communities. [sent-21, score-0.21]

13 The inﬁnite multiple membership relational model (IMRM) [15] also uses an IBP to allow multiple memberships, but uses a non-conjugate observation model to allow more scalable inference for sparse networks. [sent-22, score-0.229]

14 The nonparametric metadata dependent relational (NMDR) model [11] employs a logistic stick-breaking prior on the node-speciﬁc community frequencies, and thereby models relationships between communities and metadata. [sent-23, score-0.534]

15 In contrast, the conditionally conjugate structure of our HDPR model allows us to easily develop a stochastic variational inference algorithm [17, 2, 9]. [sent-25, score-0.18]

16 Its online structure, which incrementally updates global community parameters based on random subsets of the full graph, is highly scalable; our experiments consider social networks with tens of thousands of nodes. [sent-26, score-0.264]

17 1 While the HDPR is more broadly applicable, our focus in this paper is on assortative models for undirected networks, which assume that the probability of linking distinct communities is small. [sent-27, score-0.321]

18 Our work builds on stochastic variational inference methods developed for the assortative MMSB (aMMSB) [6], but makes three key technical innovations. [sent-29, score-0.266]

19 First, adapting work on HDP topic models [19], we develop a nested family of variational bounds which assign positive probability to dynamically varying subsets of the unbounded collection of global communities. [sent-30, score-0.22]

20 Finally, we derive a structured mean ﬁeld variational bound which models dependence among the pair of community assignments associated with each edge. [sent-32, score-0.336]

21 In this paper, we use our assortative HDPR (aHDPR) model to recover latent communities in social networks previously examined with the aMMSB [6], and demonstrate substantially improved perplexity scores and link prediction accuracy. [sent-34, score-0.633]

22 We also use our learned community structure to visualize business and governmental relationships extracted from the LittleSis database [13]. [sent-35, score-0.16]

23 2 Assortative Hierarchical Dirichlet Process Relational Models We introduce the assortative HDP relational (aHDPR) model, a nonparametric generalization of the aMMSB for discovering shared memberships in an unbounded collection of latent communities. [sent-36, score-0.319]

24 We focus on undirected binary graphs with N nodes and E = N (N − 1)/2 possible edges, and let yij = yji = 1 if there is an edge between nodes i and j. [sent-37, score-0.407]

25 For some experiments, we assume the yij variables are only partially observed to compare the predictive performance of different models. [sent-38, score-0.227]

26 Letting βk denote the expected frequency of community k, and γ > 0 the concentration, we deﬁne a stick-breaking representation of β: k−1 β k = vk (1 − v ), vk ∼ Beta(1, γ), k = 1, 2, . [sent-41, score-0.282]

27 (1) =1 Adapting a two-layer hierarchical DP [18], the mixed community memberships for each node i are then drawn from DP with base measure β, πi ∼ DP(αβ). [sent-44, score-0.253]

28 To generate a possible edge yij between nodes i and j, we ﬁrst sample a pair of indicator variables from their corresponding community membership distributions, sij ∼ Cat(πi ), rij ∼ Cat(πj ). [sent-46, score-0.705]

29 We then determine edge presence as follows: p(yij = 1 | sij = rij = k) = wk , p(yij = 1 | sij = rij ) = . [sent-47, score-0.446]

30 (2) For our assortative aHDPR model, each community has its own self-connection probability wk ∼ Beta(τa , τb ). [sent-48, score-0.332]

31 Our HDPR model could easily be generalized to more ﬂexible likelihoods in which each pair of communities k, have their own interaction probability [1], but motivated by work on the aMMSB [6], we do not pursue this generalization here. [sent-50, score-0.214]

32 3 Scalable Variational Inference Previous applications of the MMSB associate a pair of community assignments, sij and rij , with each potential edge yij . [sent-51, score-0.593]

33 In assortative models these variables are strongly dependent, since present edges only have non-negligible probability for consistent community assignments. [sent-52, score-0.302]

34 To improve accuracy and reduce local optima, we thus develop a structured variational method based on joint conﬁgurations of these assignment pairs, which we denote by eij = (sij , rij ). [sent-53, score-0.339]

35 Left: Conventional representation, in which source sij and receiver rij community assignments are independently sampled. [sent-56, score-0.354]

36 Right: Blocked representation in which eij = (sij , rij ) denotes the pair of community assignments underlying yij . [sent-57, score-0.57]

37 (3) For the nonparametric aHDPR model, the number of latent community parameters wk , βk , and the dimensions of the community membership distributions πi , are both inﬁnite. [sent-60, score-0.55]

38 1 Variational Bounds via Nested Truncations We begin by deﬁning categorical edge assignment distributions q(eij | φij ) = Cat(eij | φij ), where φijk = q(eij = (k, )) = q(sij = k, rij = ). [sent-63, score-0.149]

39 For the global community parameters, we deﬁne an untruncated factorized variational distribution: q(β, w | v ∗ , λ) = ∞ k−1 ∗ δvk (vk )Beta(wk | λka , λkb ), ∗ βk (v ∗ ) = vk k=1 (1 − v ∗ ). [sent-68, score-0.366]

40 (4) =1 Our later derivations show that for communities k > K above the truncation level, the optimal variational parameters equal the prior: λka = τa , λkb = τb . [sent-69, score-0.37]

41 Matched to this, we associate a (K + 1)-dimensional community membership distribution πi to each node, where the ﬁnal component contains the sum of all mass not assigned to the ﬁrst K. [sent-72, score-0.27]

43 k (5) Unlike truncations of the global stick-breaking process [4], our variational bounds are nested, so that lower-order approximations are special cases of higher-order ones with some zeroed parameters. [sent-75, score-0.164]

44 2 Structured Variational Inference with Linear Time and Storage Complexity Conventional, coordinate ascent variational inference algorithms iteratively optimize each parameter given ﬁxed values for all others. [sent-77, score-0.159]

45 Community membership and interaction parameters are updated as λka = τa + E ij K k=1 φijkk yij , θik = αβk + λkb = τb + (i,j)∈E 3 K =1 E ij φijk . [sent-78, score-0.341]

46 K k=1 φijkk (1 − yij ), (6) (7) Here, the ﬁnal summation is over all potential edges (i, j) linked to node i. [sent-79, score-0.313]

47 Updates for assignment distributions depend on expectations of log community assignment probabilities: Eq [log(wk )] = ψ(λka ) − ψ(λka + λkb ), πik ˜ Eq [log(1 − wk )] = ψ(λkb ) − ψ(λka + λkb ), exp{Eq [log(πik )]} = exp{ψ(θik ) − ψ( K+1 =1 θi )}, πi ˜ K k=1 πik . [sent-80, score-0.321]

48 ˜ (8) (9) Given these sufﬁcient statistics, the assignment distributions can be updated as follows: φijkk ∝ πik πjk f (wk , yij ), ˜ ˜ φijk ∝ πik πj f ( , yij ), ˜ ˜ = k. [sent-81, score-0.449]

49 (10) (11) Here, f (wk , yij ) = exp{yij Eq [log(wk )] + (1 − yij )Eq [log(1 − wk )]}. [sent-82, score-0.506]

50 (7) can be expanded as follows: θik = αβk + (i,j)∈E φijkk + = αβk + (i,j)∈E φijkk + 1 ˜ ˜ =k πik πj f ( , yij ) Zij 1 ˜ π ˜ Zij πik f ( , yij )(˜j − πjk ). [sent-89, score-0.42]

51 The normalization constant Zij , ˜ which is deﬁned so that φij is a valid categorical distribution, can also be computed in linear time: Zij = πi πj f ( , yij ) + ˜ ˜ K k=1 πik πjk (f (wk , yij ) − f ( , yij )). [sent-91, score-0.63]

52 3 Stochastic Variational Inference Standard variational batch updates become computationally intractable when N becomes very large. [sent-97, score-0.146]

53 Recent advancements in applying stochastic optimization techniques within variational inference [8] showed that if our variational mean-ﬁeld family of distributions are members of the exponential family, we can derive a simple stochastic natural gradient update for our global parameters λ, θ, v. [sent-98, score-0.346]

54 By taking natural gradients with respect to our new variational objective for our global variables λ, θ, we have λ∗ = ka ∗ θik = 1 g(i,j) φijkk yij 1 g(i,j) + τa − λka ; K =1 (i,j)∈E φijk + αβk − θik , (14) (15) where the natural gradient for λ∗ is symmetric to λ∗ and where yij in Eq. [sent-102, score-0.676]

55 4 Restricted Stratiﬁed Node Sampling Stochastic variational inference provides us with the ability to choose a sampling scheme that allows us to better exploit the sparsity of real world networks. [sent-119, score-0.159]

56 We show how performing this simple modiﬁcation results in signiﬁcant improvements in both perplexity and link prediction scores. [sent-129, score-0.235]

57 5 Pruning Moves Our nested truncation requires setting an initial number of communities K. [sent-131, score-0.277]

58 To remedy this, we deﬁne a set of pruning moves aimed at improving inference by removing communities that have very small posterior mass. [sent-134, score-0.51]

59 Figure 2 provides an example illustrating how pruning occurs in our model. [sent-136, score-0.194]

60 To determine communities which are good candidates for pruning, for each community k we ﬁrst N N K compute Θk = ( i=1 θik )/( i=1 k=1 θik ). [sent-137, score-0.374]

61 Any community for which Θk < (log K)/N for ∗ t = N/2 consecutive iterations is then evaluated in more depth. [sent-138, score-0.16]

62 To estimate an approximate but still informative ELBO for the pruned model, we must associate a set of relevant observations to each pruning candidate. [sent-140, score-0.255]

63 In particular, we approximate the pruned ELBO L(q prune ) by considering observations yij among pairs of nodes with signiﬁcant mass in the pruned community. [sent-141, score-0.42]

64 We then compare these two values to accept or reject the pruning of the low-weight community. [sent-143, score-0.234]

65 We show signiﬁcant gains in AUC and perplexity scores by using the restricted form of 5 New Model Adjacency Matrix ✓ ✓⇤ Y N nodes K communities Prune k=3 v⇤ , ⇤ ✓ , ⇥k < (log K)/N ⇥k = ( L(qold) L(qprune) PN i=1 ✓ik )/( PN PK i=1 k=1 ✓ik ) Uniformly redistribute mass. [sent-145, score-0.54]

66 Select nodes relevant to pruned topic and its corresponding subgraph (red box) to generate a new ELBO: L(qprune) ⇤ , ⇤ ⇤ ij2S v, , ✓, ij2S If L(qprune) > L(qold), accept or else reject and continue inference with old model Figure 2: Pruning extraneous communities. [sent-147, score-0.225]

67 Suppose that community k = 3 is considered for removal. [sent-148, score-0.16]

68 We specify a new model by redistributing its mass N θi3 uniformly across the remaining communities i=1 θi , = 3. [sent-149, score-0.233]

69 To accurately estimate the true a b ∗ change in ELBO for this pruning, we select the n = 10 nodes with greatest participation θi3 in community 3. [sent-151, score-0.229]

70 ij∈S for a model in which community k = 3 is pruned, and a corresponding ELBO L(q Using the data from the same sub-graph, but the old un-pruned model parameters, we estimate an alternative ELBO L(q old ). [sent-154, score-0.238]

71 ij∈S stratiﬁed node sampling, a quick K-means initialization1 for θ, and our efﬁcient structured meanﬁeld approach combined with pruning moves. [sent-157, score-0.265]

72 We perform a detailed comparison on a synthetic toy dataset, as well as the real-world relativity collaboration network, using a variety of metrics to show the beneﬁts of each contribution. [sent-158, score-0.168]

73 We then show signiﬁcant improvements over the baseline aMMSB model in both AUC and perplexity metrics on several real-world datasets previously analyzed by [6]. [sent-159, score-0.214]

74 Finally, we perform a qualitative analysis on the LittleSis network and demonstrate the usefulness of using our learned latent community structure to create visualizations of large networks. [sent-160, score-0.227]

75 1 Synthetic and Collaboration Networks The synthetic network we use for testing is generated from the standards and software outlined in [12] to produce realistic synthetic networks with overlapping communities and power-law degree distributions. [sent-164, score-0.338]

76 On this network the true number of latent communities was found to be K = 56. [sent-166, score-0.281]

77 To focus on the more challenging problem of identifying overlapping community structure, we take the largest connected component of each graph for analysis. [sent-170, score-0.178]

78 3 compare different aHDPR inference algorithms, and the perplexity scores achieved on various networks. [sent-173, score-0.279]

79 Using a combination of both modiﬁcations, we achieve the best perplexity scores on these datasets. [sent-176, score-0.239]

80 The naive mean-ﬁeld approach is the aHDPR model where the community indicator assignments are split into sij and rij . [sent-180, score-0.411]

81 The aMMSB in some 1 Our K-means initialization views the rows of the adjacency matrix as distinct data points and produces a single community assignment zi for each node. [sent-183, score-0.208]

82 To initialize community membership distributions based on these assignments, we set θizi = N − 1 and θi\zi = α. [sent-184, score-0.227]

83 The lower left shows various perplexity scores for the synthetic and relativity networks with the best performing model (aHDPR-Pruning) scoring an average AUC of 0. [sent-194, score-0.37]

84 The lower right shows the pruning process for the toy data and the ﬁnal K communities discovered on our real-world networks. [sent-199, score-0.442]

85 Pruning moves were applied every N/2 iterations with a maximum of K/10 communities removed per move. [sent-203, score-0.255]

86 If the number of prune candidates was greater than K/10, then K/10 communities with the lowest mass were chosen. [sent-204, score-0.281]

87 3 shows that our pruning moves can learn close to the true underlying number of clusters (K=56) on a synthetic network even when signiﬁcantly altering its initial K. [sent-206, score-0.285]

88 Across several real world networks, there was low variance between runs with respect to the ﬁnal K communities discovered, suggesting a degree of robustness. [sent-207, score-0.214]

89 Furthermore, pruning moves improved perplexity and AUC scores across every dataset as well as reducing computational costs during inference. [sent-208, score-0.474]

90 The ﬁgures above show that the aHDPR with pruning moves has the best performance, in terms of both perplexity (top) and AUC (bottom) scores. [sent-247, score-0.449]

91 For this analysis, we ran the aHDPR with pruning on the entire dataset using the same settings from our previous experiments. [sent-261, score-0.194]

92 We then took the top 200 degree nodes and generated weighted edges based off of a variational distance between their learned expected variational |E [π ]−E [π ]| posteriors such that dij = 1 − q i 2 q j . [sent-262, score-0.363]

93 Larger nodes have greater posterior bridgeness while node colors represent its dominant community membership. [sent-266, score-0.321]

94 Our learned latent communities can drive these types of visualizations that otherwise might not have been possible given the raw subgraph (see ‡A. [sent-267, score-0.254]

95 5 Discussion Our model represents the ﬁrst Bayesian nonparametric relational model to use a stochastic variational approach for efﬁcient inference. [sent-269, score-0.278]

96 Our pruning moves allow us to save computation and improve inference in a principled manner while our efﬁcient structured mean-ﬁeld inference procedure helps us escape local optima. [sent-270, score-0.339]

97 Future extensions of interest could entail advanced split-merge moves that can grow the number of communities as well as extending these scalable inference algorithms to more sophisticated relational models. [sent-271, score-0.417]

98 Truly nonparametric online variational inference for hierarchical dirichlet processes. [sent-299, score-0.254]

99 Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities. [sent-354, score-0.178]

100 Fuzzy communities and the concept of bridgeness in complex networks. [sent-381, score-0.238]

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