A start of Variational Methods for ERGM Ranran Wang, UW

Transcription

1 A start of Variational Methods for ERGM Ranran Wang, UW MURI-UCI April 24, 2009

2 Outline A start of Variational Methods for ERGM [1] Introduction to ERGM Current methods of parameter estimation: MCMCMLE: Markov chain Monte-Carlo estimation MPLE: Maximum pseudo-likelihood estimation Variational methods: Exponential families and variational inference Approximation of intractable families Application on ERGM Simulation study

3 Introduction to ERGM A start of Variational Methods for ERGM [2] Network Notations m actors; n = m(m 1) 2 dyads Sociomatrix (adjacency matrix) Y : {y i,j } i,j=1,,n Edge set {(i, j) :y i,j =1}. Undirected network: {y i,j = y j,i =1}

4 ERGM A start of Variational Methods for ERGM [3] Exponential Family Random Graph Model (Frank and Strauss, 1986; Wasserman and Pattison, 1996; Handcock, Hunter, Butts, Goodreau and Morris, 2008): log[p (Y = y obs ; η)] = η T φ(y obs ) κ(η, Y), y Y where Y is the random matrix η Ω R q is the vector of model parameters φ(y) is a q-vector of statistics κ(η, Y) = log P z Y exp{ηt φ(z)} is the normalizing factor, which is difficult to calculate. R package: statnet

5 Current estimation approaches for ERGM A start of Variational Methods for ERGM [4] MCMC-MLE (Geyer and Thompson 1992, Snijders, 2002; Hunter, Handcock, Butts, Goodreau and Morris, 2008): 1. Set an initial value η 0, for parameter η. 2. Generate MCMC samples of size m from P η0 by Metropolis algorithm. 3. Iterate to obtain a maximizer η of the approximate log-likelihood ratio: h 1 (η η 0 ) T φ(y obs ) log m mx exp (η η 0 ) T φ(y i ) i i=1 4. If the estimated variance of the approximate log-likelihood ratio is too large in comparison to the estimated log-likelihood for η, return to step 2 with η 0 = η. 5. Return η as MCMCMLE.

6 A start of Variational Methods for ERGM [5] MPLE (Besag, 1975; Strauss and Ikeda, 1990): Conditional formulation: logit[p (Y ij =1 Y C ij = yc ij )] = ηt δ(y C ij ). where δ(y C ij )=φ(y+ ij ) φ(y ij ), the change in φ(y) when y ij changes from 0 to 1 while the rest of network remains y C ij.

7 A start of Variational Methods for ERGM [6] Comparison Simulation study: van Duijn, Gile and Handcock (2008) MCMC-MLE Slow-mixing Highly depends on initial values Be able to model various network characteristics together. MPLE Deterministic model; computation is fast Unstable Dyadic-independent model; could not capture high-order network characteristics.

8 Variational method A start of Variational Methods for ERGM [7] Exponential families and variational representations Basics of exponential family: log[p(x; θ)] = θ, φ(x) κ(θ). Sufficient statistics: φ(x). Log-partition function: κ(θ) = log P x X exp θ, φ(x). Mean value parametrization: µ R q := E(φ(x)) Mean value space (convex hull): M = µ R q p( ) s.t. X X φ(x)p(x) =µ.

9 A start of Variational Methods for ERGM [8] The log-partition function is smooth and convex in terms of θ. Suppose θ =(θ α,θ β, ) and φ(x) =(φ α (x),φ β (x), ): κ θ α (θ) =E[φ α (x)] := X x X φ α (x)p(x; θ). (1) κ θ α θ β (θ) =E[φ α (x)φ β (x)] E[φ α (x)]e[φ β (x)]. (2) So, µ(θ) can be reexpressed as µ(θ) = κ θ (θ) and it has gradient 2 κ θ T θ (θ). (Barndorff-Nielson, 1978; Handcock, 2003; Wainwright and Jordan, 2003)

10 A start of Variational Methods for ERGM [9] Exp: Ising model on graph G(V, E) log p(x, θ) ={ X s V θ s x s + X θ st x s x t κ(θ)}, (3) (s,t) E where: x s, associated with s V is a Bernoulli random variable; components x s and x t are allowed to interact directly only if s and t are joined by an edge in the graph. The relevant mean parameters in this representation are as follows: µ s = E θ [x s ]=p(x s = 1; θ), µ st = E θ [x s x t ]=p(x s =1,x t = 1; θ). For each edge (s, t), the triplet {µ s,µ t,µ st } uniquely determines a joint marginal p(x s,x t ; µ) as follows: p(x s,x t ; µ) =» (1 + µst µ s µ t ) (µ t µ st ) (µ s µ st ) µ st.

11 A start of Variational Methods for ERGM [10] To ensure the joint marginal, we impose non-negativity constraints on all four entries, as follows: 1+µ st µ s µ t 0 µ st 0 µ s(/t) µ st 0 The inequalities above define M.

12 A start of Variational Methods for ERGM [11] Variational inference and mean value estimation For any µ rim (ri: relative interior), we have following lower bound: κ(θ) = sup θ, µ κ (µ) (4) µ M κ(θ) = log X x X exp{ θ, φ(x) } p(x; θ) p(x; θ) X x X log `exp{ θ, φ(x) } p(x; θ) p(x; θ) = X x X θ, φ(x) p(x; θ) X x X log(p(x; θ))p(x; θ) = E θ, φ(x) E[log(p(x; θ))] = θ, µ κ (µ). The inequality follows from Jensen s inequality, and the last equality follows from E(φ(x)) = µ and κ (µ) =E[log(p(x; θ(µ)))], the negative entropy of distribution p(x; θ).

13 Why variational method? A start of Variational Methods for ERGM [12] Variational representation turns the problem of calculating intractable summation/integrals to optimization problem (finding lower bound of κ over M). The problem of computing mean parameters can be solved simultaneously. Two main difficulties: The constraint set M of realizable mean parameters is difficult to characterize in an explicit manner. κ (µ) is lack of explicit form and needs proper approximation.

14 Mean value estimation A start of Variational Methods for ERGM [13] µ is obtained by solving the optimization problem in (4). However, the dual function κ lacks an explicit form in many cases. We restrict the choice of µ to a tractable subset M t (H) of M(G), where H is the tractable subgraph of G. The lower bound in (4) will then be computable. The solution of the optimization problem specifies optimal approximation µ t of µ. sup { µ, θ κ H (µ)} µ M t (H) The optimal µ t, in fact, minimizes the Kullback-Leibler divergence between the tractable M t and the target constraint M, and KL divergence between their natural parameter spaces as well.

15 A start of Variational Methods for ERGM [14] Ising model on Graph: Approximation of κ Exp: Ising model on Graph: Approximation of κ Assume the tractable graph H 0 is fully disconnected, then the mean value parameter set is M 0 (H 0 )={(µ s,µ st ) 0 µ s 1,µ st = µ s µ t } Here, µ s = p(x s = 1) and µ st = p(x s =1,x t = 1) = µ s µ t. So, the distribution on H 0 is fully factorizable. Deriving from Bernoulli distribution, κ H 0 (µ) = X s V [µ s log µ s + (1 µ s ) log(1 µ s )]. By (4), κ(θ) = max {µs} [0,1] n X s V θ s µ s + X (s,t) E θ st µ s µ t X s V [µ s log µ s +(1 µ s ) log(1 µ s )]. (5)

16 A start of Variational Methods for ERGM [15] After taking gradient and setting it to zero, we have following updates for µ: logit(µ s ) θ s + X θ st µ t. (6) t N (s) Apply (6) iteratively (coordinate ascent) to each node until convergence is reached.

17 Applications to ERGM A start of Variational Methods for ERGM [16] Dependence Graph G Y is a graph with m actors and n = m(m 1) 2 dyads Construct a dependence graph D Y to describe the dependence structure of G Y : D Y = G(V (D),E(D)). Each dyad (i, j), i < j on G is an actor on D. Each actor (ij) V (D) has a binary variable y ij. Each edge on D exists if (ij) and (kl) as actors on D Y share a common value, i.e (ij) and (kl) as dyads on G Y share a node. Frank and Strauss, Dependence Graph: D Original Graph: G Figure 1: Dependence Graph D

18 A start of Variational Methods for ERGM [17] Exp: Erdos-Renyi Model: For an undirected random graph Y = {Y ij }, all dyads are mutually independent, so the dependency graph D is fully disconnected. Each y ij, (ij) D(V ) is a Bernoulli random variable. The model can be written as log[p θ (Y = y)] = X i<j θ ij y ij κ(θ, Y), y Y. Calculating entropy of Bernoulli distribution, we have κ (µ) = X i<j [µ ij log(µ ij ) + (1 µ ij ) log(1 µ ij )], (7) where µ ij = P (Y ij = 1). Then, κ(θ) = sup µ M{ θ, µ κ (µ)} = X i<j log(1 + exp(θ ij )), when θ ij = log( µ ij 1 µ ij ).

19 2-star ERGM model A start of Variational Methods for ERGM [18] Analogous to Ising model, on dependence graph D = G(V (D),E(D)), log P (Y, θ) = X θ s y s + X θ st y s y t κ(θ), s:(ij) V (G). s V (D) (s,t) E(D) If θ s = η 1,s V and θ st = η 2, (s, t) E, X X X log P (Y, η) ={η 1 y ij + η 2 y ij y ik κ(η)}, i<j i j,k>i which corresponds to the canonical 2-star model.

20 A start of Variational Methods for ERGM [19] Given a graph G Y with 6 actors and its dependency graph D Y with 15 nodes. For Ising model log p(x, θ) ={ X θ s y s + s V D X (s,t) E D θ st y s y t κ(θ)},

21 A start of Variational Methods for ERGM [20] Compare µ var obtained from naive mean field algorithm to µ mcmc obtained from MCMC samples for fixed θ s. θ st = 0.2, s,t (ij):s θ s µ mcmc s µ var s

22 A start of Variational Methods for ERGM [21] For 2-star model, let θ s = η 1 [ 2, 2] and θ st = η 2 [ 2, 2]. µ = P (x s = 1), s. Compare µ var (η 1,η 2 ) with µ mcmc. MU_var MU_mcmc Figure 2: µ MCMC vs. µ var

23 A start of Variational Methods for ERGM [22] Parameter estimation by variational inference 1. Start with θ (0) 2. Estimate eµ(θ) from naive mean field algorithm 3. Calculate κ(θ) = θ, eµ κ (eµ) and log-likelihood l(θ, y). Also, calculate κ(θ) =E θ (φ(x)) and l(θ, y) =φ(x) E θ (φ(x)). 4. Update θ by gradient ascent: eθ (n+1) = e θ (n) + γ l(θ (n),y),γ Iterate until e θ converges.

24 Simulation study A start of Variational Methods for ERGM [23] Figure 3: A sample graph with 6 edges and 12 2-stars 2-star ERGM η 1 η 2 MLE MCMC-MLE MPLE Var-MLE

25 A start of Variational Methods for ERGM [24] eta_1!2.1!1.9!1.7! n.iter eta_ n.iter Figure 4: Convergence of Var-MLE

26 Discussion and Future work A start of Variational Methods for ERGM [25] Future work: Better approximation of A : Structured mean field algorithm Bethe entropy approximation Clustered variational method Extension to general ERGM: clustering structure of dependence graph; constraint space Continuous graph: Gaussian random field Curved-exponential family Hybrid of MCMC and variational methods

27 Thanks for your attention! A start of Variational Methods for ERGM [26]