nips nips2007 nips2007-128 knowledge-graph by maker-knowledge-mining

128 nips-2007-Message Passing for Max-weight Independent Set

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Author: Sujay Sanghavi, Devavrat Shah, Alan S. Willsky

Abstract: We investigate the use of message-passing algorithms for the problem of ﬁnding the max-weight independent set (MWIS) in a graph. First, we study the performance of loopy max-product belief propagation. We show that, if it converges, the quality of the estimate is closely related to the tightness of an LP relaxation of the MWIS problem. We use this relationship to obtain sufﬁcient conditions for correctness of the estimate. We then develop a modiﬁcation of max-product – one that converges to an optimal solution of the dual of the MWIS problem. We also develop a simple iterative algorithm for estimating the max-weight independent set from this dual solution. We show that the MWIS estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the MWIS is unique. Finally, we show that any problem of MAP estimation for probability distributions over ﬁnite domains can be reduced to an MWIS problem. We believe this reduction will yield new insights and algorithms for MAP estimation. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 First, we study the performance of loopy max-product belief propagation. [sent-7, score-0.091]

2 We show that, if it converges, the quality of the estimate is closely related to the tightness of an LP relaxation of the MWIS problem. [sent-8, score-0.171]

3 We then develop a modiﬁcation of max-product – one that converges to an optimal solution of the dual of the MWIS problem. [sent-10, score-0.221]

4 We also develop a simple iterative algorithm for estimating the max-weight independent set from this dual solution. [sent-11, score-0.204]

5 We show that the MWIS estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the MWIS is unique. [sent-12, score-0.177]

6 1 Introduction The max-weight independent set (MWIS) problem is the following: given a graph with positive weights on the nodes, ﬁnd the heaviest set of mutually non-adjacent nodes. [sent-15, score-0.129]

7 In this paper we investigate the use of message-passing algorithms, like loopy max-product belief propagation, as practical solutions for the MWIS problem. [sent-18, score-0.125]

8 1 we describe one such application: scheduling channel access and transmissions in wireless networks. [sent-24, score-0.168]

9 Message passing algorithms provide a promising alternative to current scheduling algorithms. [sent-25, score-0.123]

10 1 Our contributions In Section 4 we construct a probability distribution whose MAP estimate corresponds to the MWIS of a given graph, and investigate the application of the loopy Max-product algorithm to this distritbuion. [sent-30, score-0.125]

11 We demonstrate that there is an intimate relationship between the max-product ﬁxed-points and the natural LP relaxation of the original independent set problem. [sent-31, score-0.133]

12 The ﬁrst, obtained by a minor modiﬁcation of max-product, calculates the optimal solution to the dual of the LP relaxation of the MWIS problem. [sent-34, score-0.296]

13 The second algorithm uses this optimal dual to produce an estimate of the MWIS. [sent-35, score-0.204]

14 This estimate is correct when the original graph is bipartite. [sent-36, score-0.11]

15 2 Max-weight Independent Set, and its LP Relaxation Consider a graph G = (V, E), with a set V of nodes and a set E of edges. [sent-40, score-0.177]

16 Positive weights wi , i ∈ V are associated with each node. [sent-42, score-0.17]

17 A subset of V will be represented by vector x = (xi ) ∈ {0, 1}|V | , where xi = 1 means i is in the subset xi = 0 means i is not in the subset. [sent-43, score-0.19]

18 A subset x is called an independent set if no two nodes in the subset are connected by an edge: (xi , xj ) = (1, 1) for all (i, j) ∈ E. [sent-44, score-0.173]

19 The linear programing relaxation of IP is obtained by replacing the integrality constraints xi ∈ {0, 1} with the constraints xi ≥ 0. [sent-47, score-0.416]

20 DUAL : xi + xj ≤ 1 for all (i, j) ∈ E, xi ∈ {0, 1}. [sent-52, score-0.226]

21 wi xi , λij , min (i,j)∈E i=1 λij ≥ wi , for all i ∈ V, s. [sent-53, score-0.381]

22 If this is the case, we say that there is no integrality gap between LP and IP or equivalently that the LP relaxation is tight. [sent-57, score-0.27]

23 usIt is well known [3] that the LP relaxation is tight for bipartite graphs. [sent-58, score-0.201]

24 More generally, for non-bipartite graphs, tightness will depend on the node weights. [sent-59, score-0.146]

25 We will use the performance of LP as a benchmark with which to compare the performance of our message passing algorithms. [sent-60, score-0.097]

26 The next lemma states the standard complimentary slackness conditions of linear programming, specialized for LP above, and for the case when there is no integrality gap. [sent-61, score-0.225]

27 1 When there is no integrality gap between IP and LP, there exists a pair of optimal solutions x = (xi ), λ = (λij ) of LP and DUAL respectively, such that: (a) x ∈ {0, 1}n , (b) xi 2. [sent-63, score-0.324]

28 1 j∈N (i) λij − wi = 0 for all i ∈ V , (c) (xi + xj − 1) λij = 0, for all (i, j) ∈ E. [sent-64, score-0.179]

29 Such networks are ubiquitous in the modern world: examples range from sensor networks that lack wired connections to the fusion center, and ad-hoc networks that can be quickly deployed in areas without coverage, to the 802. [sent-68, score-0.112]

30 11 wi-ﬁ networks that currently represent the most widely used method for wireless data access. [sent-69, score-0.109]

31 Fundamentally, any two wireless nodes that transmit at the same time and over the same frequencies will interfere with each other, if they are located close by. [sent-70, score-0.232]

32 Typically in a network only certain pairs 2 of nodes interfere. [sent-72, score-0.106]

33 The scheduling problem is to decide which nodes should transmit at a given time over a given frequency, so that (a) there is no interference, and (b) nodes which have a large amount of data to send are given priority. [sent-73, score-0.344]

34 In particular, it is well known that if each node is given a weight equal to the data it has to transmit, optimal network operation demands scheduling the set of nodes with highest total weight. [sent-74, score-0.363]

35 If a “ conﬂict graph” is made, with an edge between every pair of interfering nodes, the scheduling problem is exactly the problem of ﬁnding the MWIS of the conﬂict graph. [sent-75, score-0.166]

36 The lack of an infrastructure, the fact that nodes often have limited capabilities, and the local nature of communication, all necessitate a lightweight distributed algorithm for solving the MWIS problem. [sent-76, score-0.169]

37 3 MAP Estimation as an MWIS Problem In this section we show that any MAP estimation problem is equivalent to an MWIS problem on a suitably constructed graph with node weights. [sent-77, score-0.216]

38 We now build an auxillary graph G, and assign weights to its nodes, such that the MAP estimation problem above is equivalent to ﬁnding the MWIS of G. [sent-87, score-0.15]

39 There is one node in G for each pair (α, yα ), where yα is an assignment (i. [sent-88, score-0.137]

40 1 2 There is an edge in G between any two nodes δ(α1 , yα1 ) and δ(α2 , yα2 ) if and only if there exists a variable index m such that 1. [sent-92, score-0.215]

41 In other words, we put an edge between all pairs of nodes that correspond to inconsistent assignments. [sent-99, score-0.185]

42 Given this graph G, we now assign weights to the nodes. [sent-100, score-0.121]

43 Assign to each node δ(α, yα ) a weight of c + φα (yα ). [sent-103, score-0.139]

44 (a) If y∗ is a MAP estimate of q, let δ ∗ = ∗ {δ(α, yα ) | α ∈ A} be the set of nodes in G that correspond to each domain being consistent with y∗ . [sent-106, score-0.177]

45 Then, for every ∗ domain α, there is exactly one node δ(α, yα ) included in δ ∗ . [sent-109, score-0.148]

46 The weights on the nodes in G are: θ1 + c on node “1” on the left, θ2 + c for node “1” on the right, θ12 + c for the node “11”, and c for all the other nodes. [sent-115, score-0.481]

47 Now, given an MWIS problem on G = (V, E), associate a binary random variable Xi with each i ∈ V and consider the following joint distribution: for x ∈ {0, 1}n , p (x) = 1 Z exp(wi xi ), 1{xi +xj ≤1} (1) i∈V (i,j)∈E where Z is the normalization constant. [sent-117, score-0.095]

48 It is easy to see that p(x) = Z exp ( i wi xi ) if x is an independent set, and p(x) = 0 otherwise. [sent-119, score-0.269]

49 Thus, any MAP estimate arg maxx p(x) corresponds to a maximum weight independent set of G. [sent-120, score-0.093]

50 At every iteration t each node i sends a message {mt (0), mt (1)} to each neighbor j ∈ N (i). [sent-123, score-0.349]

51 Each node i→j i→j also maintains a belief {bt (0), bt (1)} vector. [sent-124, score-0.233]

52 The message and belief updates, as well as the ﬁnal i i output, are computed as follows. [sent-125, score-0.1]

53 i→j j→i (i) The messages are updated as follows:   mt+1 (0) = max i→j  mt (0) , ewi k→i k=j,k∈N (i) mt+1 (1) i→j k=j,k∈N (i) mt (0). [sent-127, score-0.387]

54 k→i =   mt (1) , k→i  k=j,k∈N (i) (ii) Nodes i ∈ V , compute their beliefs as follows: bt+1 (0) = i mt+1 (0), k→i bt+1 (1) = ewi i k∈N (i) mt+1 (1). [sent-128, score-0.164]

55 independent set x(bt+1 ) as follows: xi (bt+1 ) = 1{bt+1 (1)>bt+1 (0)} . [sent-131, score-0.126]

56 i i i t (iv) Update t = t + 1; repeat from (i) till x(b ) converges and output the converged estimate. [sent-132, score-0.122]

57 t For the purpose of analysis, we ﬁnd it convenient to transform the messages be deﬁning1 γi→j = log mt (0) i→j mt (1) i→j . [sent-133, score-0.319]

58 The estimation of step (iii) of max-product becomes: xi (γ t+1 ) = 1{wi −Pk∈N (i) γk→i >0} . [sent-135, score-0.124]

59 1 If the algorithm starts with all messages being strictly positive, the messages will remain strictly positive over any ﬁnite number of iterations. [sent-138, score-0.188]

60 Then, the following properties hold: (a) Let i be a node with estimate xi (γ) = 1, and let j ∈ N (i) be any neighbor of i. [sent-146, score-0.299]

61 Then, the messages on edge (i, j) satisfy γi→j > γj→i . [sent-147, score-0.152]

62 (b) Let j be a node with xj (γ) = 0, which by deﬁnition means that wj − k∈N (j) γk→j ≤ 0. [sent-149, score-0.226]

63 Suppose now there exists a neighbor i ∈ N (j) whose estimate is xi (γ) = 1. [sent-150, score-0.213]

64 In addition, if there exists a neighbor i1 ∈ N (j1 ) of j1 whose estimate is xi1 (γ) = 1, then it has to be that γj1 →j2 = γj2 →j1 = 0 (and similarly for a neighbor i2 of j2 ). [sent-155, score-0.167]

65 1 reveal striking similarities between the messages γ of ﬁxed points of max-product, and the optimal λ that solves the dual linear program DUAL. [sent-157, score-0.261]

66 In particular, suppose that γ is a ﬁxed point at which the corresponding estimate x(γ) is a maximal independent set: for every j whose estimate xj (γ) = 0 there exists a neighbor i ∈ N (j) whose estimate is xi (γ) = 1. [sent-158, score-0.402]

67 The MWIS, for example, is also maximal (if not, one could add a node to the MWIS and obtain a higher weight). [sent-159, score-0.137]

68 • xi (γ) γi→j + k∈N (i)−j γk→i − wi = 0 for all i, j ∈ V At least semantically, these relations share a close resemblance to the complimentary slackness conditions of Lemma 2. [sent-161, score-0.331]

69 In the following lemma we leverage this resemblance to derive a certiﬁcate of optimality of the max-product ﬁxed point estimate for certain problems. [sent-163, score-0.103]

70 Then, if G′ is acyclic, we have that : (a) x(γ) is a solution to the MWIS for G, and (b) there is no integrality gap between LP and IP, i. [sent-167, score-0.197]

71 For bipartite graph, we know that LP relaxation is tight, i. [sent-181, score-0.169]

72 The following result suggests that if max-product works then it must be that LP relaxation is tight (i. [sent-185, score-0.134]

73 Then, if there exists at least one node i whose estimate xi (γ) = 1, then there is no integrality gap between LP and IP. [sent-190, score-0.448]

74 Next, we present two examples which help us conclude that neither max-product nor LP relaxation dominate the other. [sent-191, score-0.102]

75 In each graph, numbers represent node weights, and an arrow from i to j represents 5 a message value of γi→j = 2. [sent-193, score-0.177]

76 The boxed nodes indicate the ones for which the estimate xi (γ) = 1. [sent-195, score-0.24]

77 In the graph on the right, there is 1 an integrality gap between LP and IP: setting each xi = 2 yields an optimal value of 7. [sent-199, score-0.365]

78 It is based on modifying max-product by drawing upon a dual co-ordinate descent and barrier method. [sent-205, score-0.273]

79 (2) For small enough parameter δ1 , ij use subroutine EST(λε,δ , δ1 ), to produce an estimate for the MWIS as the output of algorithm. [sent-208, score-0.282]

80 Notice the similarity (at least syntactic) between (3) and update of max-product (min-sum) (2): essentially, the dual coordinate descent is a sequential bidirectional version of the max-product algorithm ! [sent-217, score-0.302]

81 It is well known that the coordinate descent always coverges, in terms of cost for linear programs. [sent-218, score-0.143]

82 However, due to constraints of type j∈N (i) λij ≥ wi in DUAL, the algorithm may not 2 A good policy for picking edges is round-robin or uniformly at random 6 converge to an optimal of DUAL. [sent-220, score-0.22]

83 Therefore, a direct adaptation of max-product to mimic dual coordinate descent is not good enough. [sent-221, score-0.277]

84 log  j∈N (i) λij − wi  The following is coordinate descent algorithm for CP(ε). [sent-225, score-0.286]

85 ij ji ◦ Initially, set t = 0 and λ0 = max{wi , wj } for all (i, j) ∈ E. [sent-229, score-0.261]

86 ij (i) In iteration t + 1, update parameters as follows: ◦ Pick an edge (i, j) ∈ E. [sent-230, score-0.269]

87 This edge selection is done so that each edge is chosen inﬁnitely often as t → ∞ (for example, at each t choose an edge uniformly at random. [sent-231, score-0.237]

88 i i′ j ′ ◦ For edge (i, j), nodes i and j exchange messages as follows:     t+1 γi→j = wi − ◦ Update λt+1 ij k=j,k∈N (i) t+1 λt  , γj→i = wj − ki + as follows: with a = λt+1 = ij t+1 γi→j and b = k′ =i,k′ ∈N (j) t+1 γj→i , (a − b)2 + 4ε2 a + b + 2ε + 2 λt ′ j  k . [sent-235, score-0.588]

89 + (4) + (ii) Update t = t + 1 and repeat till algorithm converges within δ for each component. [sent-236, score-0.098]

90 The iterative step (4) can be rewritten as follows: for some β ∈ [1, 2],        λt+1 = βε + max −βε, wi − λt  ,  wj − λt  , ik kj ij   k∈N (i)\j k∈N (j)\i t+1 t+1 where β depends on values of γi→j , γj→i . [sent-238, score-0.325]

91 Thus the updates in DESCENT are obtained by small but important perturbation of dual coordinate descent for DUAL, and making it convergent. [sent-239, score-0.277]

92 2 Subroutine: EST DESCENT yields a good estimate of the optimal solution to DUAL, for small values of ǫ and δ. [sent-242, score-0.099]

93 In general, it is not possible to recover the solution of a linear program from a dual optimal solution. [sent-244, score-0.217]

94 This procedure is likely to extend for general G when LP relaxation is tight and LP has unique solution. [sent-246, score-0.134]

95 (i) Initially, color a node i gray and set xi = 0 if j∈N (i) λij > wi . [sent-249, score-0.392]

96 Color all other nodes with green and leave their values unspeciﬁed. [sent-250, score-0.139]

97 The condition j∈N (i) λij > wi is checked as whether j∈N (i) λij ≥ wi + δ1 or not. [sent-251, score-0.313]

98 (ii) Repeat the following steps (in any order) till no more changes can happen: ◦ if i is green and there exists a gray node j ∈ N (i) with λij > 0, then set xi = 1 and color it orange. [sent-252, score-0.36]

99 ◦ if i is green and some orange node j ∈ N (i), then set xi = 0 and color it gray. [sent-254, score-0.282]

100 (iii) If any node is green, say i, set xi = 1 and color it red. [sent-255, score-0.249]

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same-paper 1 1.0000002 128 nips-2007-Message Passing for Max-weight Independent Set

Author: Sujay Sanghavi, Devavrat Shah, Alan S. Willsky

2 0.27004209 92 nips-2007-Fixing Max-Product: Convergent Message Passing Algorithms for MAP LP-Relaxations

Author: Amir Globerson, Tommi S. Jaakkola

Abstract: We present a novel message passing algorithm for approximating the MAP problem in graphical models. The algorithm is similar in structure to max-product but unlike max-product it always converges, and can be proven to ﬁnd the exact MAP solution in various settings. The algorithm is derived via block coordinate descent in a dual of the LP relaxation of MAP, but does not require any tunable parameters such as step size or tree weights. We also describe a generalization of the method to cluster based potentials. The new method is tested on synthetic and real-world problems, and compares favorably with previous approaches. Graphical models are an effective approach for modeling complex objects via local interactions. In such models, a distribution over a set of variables is assumed to factor according to cliques of a graph with potentials assigned to each clique. Finding the assignment with highest probability in these models is key to using them in practice, and is often referred to as the MAP (maximum aposteriori) assignment problem. In the general case the problem is NP hard, with complexity exponential in the tree-width of the underlying graph. Linear programming (LP) relaxations have proven very useful in approximating the MAP problem, and often yield satisfactory empirical results. These approaches relax the constraint that the solution is integral, and generally yield non-integral solutions. However, when the LP solution is integral, it is guaranteed to be the exact MAP. For some classes of problems the LP relaxation is provably correct. These include the minimum cut problem and maximum weight matching in bi-partite graphs [8]. Although LP relaxations can be solved using standard LP solvers, this may be computationally intensive for large problems [13]. The key problem with generic LP solvers is that they do not use the graph structure explicitly and thus may be sub-optimal in terms of computational efﬁciency. The max-product method [7] is a message passing algorithm that is often used to approximate the MAP problem. In contrast to generic LP solvers, it makes direct use of the graph structure in constructing and passing messages, and is also very simple to implement. The relation between max-product and the LP relaxation has remained largely elusive, although there are some notable exceptions: For tree-structured graphs, max-product and LP both yield the exact MAP. A recent result [1] showed that for maximum weight matching on bi-partite graphs max-product and LP also yield the exact MAP [1]. Finally, Tree-Reweighted max-product (TRMP) algorithms [5, 10] were shown to converge to the LP solution for binary xi variables, as shown in [6]. In this work, we propose the Max Product Linear Programming algorithm (MPLP) - a very simple variation on max-product that is guaranteed to converge, and has several advantageous properties. MPLP is derived from the dual of the LP relaxation, and is equivalent to block coordinate descent in the dual. Although this results in monotone improvement of the dual objective, global convergence is not always guaranteed since coordinate descent may get stuck in suboptimal points. This can be remedied using various approaches, but in practice we have found MPLP to converge to the LP 1 solution in a majority of the cases we studied. To derive MPLP we use a special form of the dual LP, which involves the introduction of redundant primal variables and constraints. We show how the dual variables corresponding to these constraints turn out to be the messages in the algorithm. We evaluate the method on Potts models and protein design problems, and show that it compares favorably with max-product (which often does not converge for these problems) and TRMP. 1 The Max-Product and MPLP Algorithms The max-product algorithm [7] is one of the most often used methods for solving MAP problems. Although it is neither guaranteed to converge to the correct solution, or in fact converge at all, it provides satisfactory results in some cases. Here we present two algorithms: EMPLP (edge based MPLP) and NMPLP (node based MPLP), which are structurally very similar to max-product, but have several key advantages: • After each iteration, the messages yield an upper bound on the MAP value, and the sequence of bounds is monotone decreasing and convergent. The messages also have a limit point that is a ﬁxed point of the update rule. • No additional parameters (e.g., tree weights as in [6]) are required. • If the ﬁxed point beliefs have a unique maximizer then they correspond to the exact MAP. • For binary variables, MPLP can be used to obtain the solution to an LP relaxation of the MAP problem. Thus, when this LP relaxation is exact and variables are binary, MPLP will ﬁnd the MAP solution. Moreover, for any variable whose beliefs are not tied, the MAP assignment can be found (i.e., the solution is partially decodable). Pseudo code for the algorithms (and for max-product) is given in Fig. 1. As we show in the next sections, MPLP is essentially a block coordinate descent algorithm in the dual of a MAP LP relaxation. Every update of the MPLP messages corresponds to exact minimization of a set of dual variables. For EMPLP minimization is over the set of variables corresponding to an edge, and for NMPLP it is over the set of variables corresponding to all the edges a given node appears in (i.e., a star). The properties of MPLP result from its relation to the LP dual. In what follows we describe the derivation of the MPLP algorithms and prove their properties. 2 The MAP Problem and its LP Relaxation We consider functions over n variables x = {x1 , . . . , xn } deﬁned as follows. Given a graph G = (V, E) with n vertices, and potentials θij (xi , xj ) for all edges ij ∈ E, deﬁne the function1 f (x; θ) = θij (xi , xj ) . (1) ij∈E The MAP problem is deﬁned as ﬁnding an assignment xM that maximizes the function f (x; θ). Below we describe the standard LP relaxation for this problem. Denote by {µij (xi , xj )}ij∈E distributions over variables corresponding to edges ij ∈ E and {µi (xi )}i∈V distributions corresponding to nodes i ∈ V . We will use µ to denote a given set of distributions over all edges and nodes. The set ML (G) is deﬁned as the set of µ where pairwise and singleton distributions are consistent x ˆ xi µij (ˆi , xj ) = µj (xj ) , ˆ xj µij (xi , xj ) = µi (xi ) ∀ij ∈ E, xi , xj ˆ ML (G) = µ ≥ 0 ∀i ∈ V xi µi (xi ) = 1 Now consider the following linear program: MAPLPR : µL∗ = arg max µ∈ML (G) µ·θ. (2) where µ·θ is shorthand for µ·θ = ij∈E xi ,xj θij (xi , xj )µij (xi , xj ). It is easy to show (see e.g., [10]) that the optimum of MAPLPR yields an upper bound on the MAP value, i.e. µL∗ ·θ ≥ f (xM ). Furthermore, when the optimal µi (xi ) have only integral values, the assignment that maximizes µi (xi ) yields the correct MAP assignment. In what follows we show how the MPLP algorithms can be derived from the dual of MAPLPR. 1 P We note that some authors also add a term i∈V θi (xi ) to f (x; θ). However, these terms can be included in the pairwise functions θij (xi , xj ), so we ignore them for simplicity. 2 3 The LP Relaxation Dual Since MAPLPR is an LP, it has an equivalent convex dual. In App. A we derive a special dual of MAPLPR using a different representation of ML (G) with redundant variables. The advantage of this dual is that it allows the derivation of simple message passing algorithms. The dual is described in the following proposition. Proposition 1 The following optimization problem is a convex dual of MAPLPR DMAPLPR : min max xi i s.t. max βki (xk , xi ) (3) k∈N (i) xk βji (xj , xi ) + βij (xi , xj ) = θij (xi , xj ) , where the dual variables are βij (xi , xj ) for all ij, ji ∈ E and values of xi and xj . The dual has an intuitive interpretation in terms of re-parameterizations. Consider the star shaped graph Gi consisting of node i and all its neighbors N (i). Assume the potential on edge ki (for k ∈ N (i)) is βki (xk , xi ). The value of the MAP assignment for this model is max max βki (xk , xi ). This is exactly the term in the objective of DMAPLPR. Thus the dual xi k∈N (i) xk corresponds to individually decoding star graphs around all nodes i ∈ V where the potentials on the graph edges should sum to the original potential. It is easy to see that this will always result in an upper bound on the MAP value. The somewhat surprising result of the duality is that there exists a β assignment such that star decoding yields the optimal value of MAPLPR. 4 Block Coordinate Descent in the Dual To obtain a convergent algorithm we use a simple block coordinate descent strategy. At every iteration, ﬁx all variables except a subset, and optimize over this subset. It turns out that this can be done in closed form for the cases we consider. We begin by deriving the EMPLP algorithm. Consider ﬁxing all the β variables except those corresponding to some edge ij ∈ E (i.e., βij and βji ), and minimizing DMAPLPR over the non-ﬁxed variables. Only two terms in the DMAPLPR objective depend on βij and βji . We can write those as f (βij , βji ) = max λ−j (xi ) + max βji (xj , xi ) + max λ−i (xj ) + max βij (xi , xj ) i j xi where we deﬁned λ−j (xi ) = i xj k∈N (i)\j xi xi (4) λki (xi ) and λki (xi ) = maxxk βki (xk , xi ) as in App. A. Note that the function f (βij , βji ) depends on the other β values only through λ−i (xj ) and λ−j (xi ). j i This implies that the optimization can be done solely in terms of λij (xj ) and there is no need to store the β values explicitly. The optimal βij , βji are obtained by minimizing f (βij , βji ) subject to the re-parameterization constraint βji (xj , xi ) + βij (xi , xj ) = θij (xi , xj ). The following proposition characterizes the minimum of f (βij , βji ). In fact, as mentioned above, we do not need to characterize the optimal βij (xi , xj ) itself, but only the new λ values. Proposition 2 Maximizing the function f (βij , βji ) yields the following λji (xi ) (and the equivalent expression for λij (xj )) 1 −j 1 λji (xi ) = − λi (xi ) + max λ−i (xj ) + θij (xi , xj ) j 2 2 xj The proposition is proved in App. B. The λ updates above result in the EMPLP algorithm, described in Fig. 1. Note that since the β optimization affects both λji (xi ) and λij (xj ), both these messages need to be updated simultaneously. We proceed to derive the NMPLP algorithm. For a given node i ∈ V , we consider all its neighbors j ∈ N (i), and wish to optimize over the variables βji (xj , xi ) for ji, ij ∈ E (i.e., all the edges in a star centered on i), while the other variables are ﬁxed. One way of doing so is to use the EMPLP algorithm for the edges in the star, and iterate it until convergence. We now show that the result of 3 Inputs: A graph G = (V, E), potential functions θij (xi , xj ) for each edge ij ∈ E. Initialization: Initialize messages to any value. Algorithm: • Iterate until a stopping criterion is satisﬁed: – Max-product: Iterate over messages and update (cji shifts the max to zero) h i mji (xi )← max m−i (xj ) + θij (xi , xj ) − cji j xj – EMPLP: For each ij ∈ E, update λji (xi ) and λij (xj ) simultaneously (the update for λij (xj ) is the same with i and j exchanged) h i 1 1 λji (xi )← − λ−j (xi ) + max λ−i (xj ) + θij (xi , xj ) j i 2 2 xj – NMPLP: Iterate over nodes i ∈ V and update all γij (xj ) where j ∈ N (i) 2 3 X 2 γij (xj )← max 4θij (xi , xj ) − γji (xi ) + γki (xi )5 xi |N (i)| + 1 k∈N(i) • Calculate node “beliefs”: Set biP i ) to be the sum of incoming messages into node i ∈ V (x (e.g., for NMPLP set bi (xi ) = k∈N(i) γki (xi )). Output: Return assignment x deﬁned as xi = arg maxxi b(ˆi ). x ˆ Figure 1: The max-product, EMPLP and NMPLP algorithms. Max-product, EMPLP and NMPLP use mesP sages mij , λij and γij respectively. We use the notation m−i (xj ) = k∈N(j)\i mkj (xj ). j this optimization can be found in closed form. The assumption about β being ﬁxed outside the star implies that λ−i (xj ) is ﬁxed. Deﬁne: γji (xi ) = maxxj θij (xi , xj ) + λ−i (xj ) . Simple algebra j j yields the following relation between λ−j (xi ) and γki (xi ) for k ∈ N (i) i 2 λ−j (xi ) = −γji (xi ) + γki (xi ) (5) i |N (i)| + 1 k∈N (i) Plugging this into the deﬁnition of γji (xi ) we obtain the NMPLP update in Fig. 1. The messages for both algorithms can be initialized to any value since it can be shown that after one iteration they will correspond to valid β values. 5 Convergence Properties The MPLP algorithm decreases the dual objective (i.e., an upper bound on the MAP value) at every iteration, and thus its dual objective values form a convergent sequence. Using arguments similar to [5] it can be shown that MPLP has a limit point that is a ﬁxed point of its updates. This in itself does not guarantee convergence to the dual optimum since coordinate descent algorithms may get stuck at a point that is not a global optimum. There are ways of overcoming this difﬁculty, for example by smoothing the objective [4] or using techniques as in [2] (see p. 636). We leave such extensions for further work. In this section we provide several results about the properties of the MPLP ﬁxed points and their relation to the corresponding LP. First, we claim that if all beliefs have unique maxima then the exact MAP assignment is obtained. Proposition 3 If the ﬁxed point of MPLP has bi (xi ) such that for all i the function bi (xi ) has a unique maximizer x∗ , then x∗ is the solution to the MAP problem and the LP relaxation is exact. i Since the dual objective is always greater than or equal to the MAP value, it sufﬁces to show that there exists a dual feasible point whose objective value is f (x∗ ). Denote by β ∗ , λ∗ the value of the corresponding dual parameters at the ﬁxed point of MPLP. Then the dual objective satisﬁes λ∗ (xi ) = ki max i xi k∈N (i) ∗ max βki (xk , x∗ ) = i i k∈N (i) xk ∗ βki (x∗ , x∗ ) = f (x∗ ) k i i 4 k∈N (i) To see why the second equality holds, note that bi (x∗ ) = maxxi ,xj λ−j (xi ) + βji (xj , xi ) and i i bj (x∗ ) = maxxi ,xj λ−i (xj ) + βij (xi , xj ). By the equalization property in Eq. 9 the arguments of j j the two max operations are equal. From the unique maximum assumption it follows that x∗ , x∗ are i j the unique maximizers of the above. It follows that βji , βij are also maximized by x∗ , x∗ . i j In the general case, the MPLP ﬁxed point may not correspond to a primal optimum because of the local optima problem with coordinate descent. However, when the variables are binary, ﬁxed points do correspond to primal solutions, as the following proposition states. Proposition 4 When xi are binary, the MPLP ﬁxed point can be used to obtain the primal optimum. The claim can be shown by constructing a primal optimal solution µ∗ . For tied bi , set µ∗ (xi ) to 0.5 i and for untied bi , set µ∗ (x∗ ) to 1. If bi , bj are not tied we set µ∗ (x∗ , x∗ ) = 1. If bi is not tied but bj i i ij i j is, we set µ∗ (x∗ , xj ) = 0.5. If bi , bj are tied then βji , βij can be shown to be maximized at either ij i x∗ , x∗ = (0, 0), (1, 1) or x∗ , x∗ = (0, 1), (1, 0). We then set µ∗ to be 0.5 at one of these assignment i j i j ij ∗ pairs. The resulting µ∗ is clearly primal feasible. Setting δi = b∗ we obtain that the dual variables i (δ ∗ , λ∗ , β ∗ ) and primal µ∗ satisfy complementary slackness for the LP in Eq. 7 and therefore µ∗ is primal optimal. The binary optimality result implies partial decodability, since [6] shows that the LP is partially decodable for binary variables. 6 Beyond pairwise potentials: Generalized MPLP In the previous sections we considered maximizing functions which factor according to the edges of the graph. A more general setting considers clusters c1 , . . . , ck ⊂ {1, . . . , n} (the set of clusters is denoted by C), and a function f (x; θ) = c θc (xc ) deﬁned via potentials over clusters θc (xc ). The MAP problem in this case also has an LP relaxation (see e.g. [11]). To deﬁne the LP we introduce the following deﬁnitions: S = {c ∩ c : c, c ∈ C, c ∩ c = ∅} is the set of intersection between clusters ˆ ˆ ˆ and S(c) = {s ∈ S : s ⊆ c} is the set of overlap sets for cluster c.We now consider marginals over the variables in c ∈ C and s ∈ S and require that cluster marginals agree on their overlap. Denote this set by ML (C). The LP relaxation is then to maximize µ · θ subject to µ ∈ ML (C). As in Sec. 4, we can derive message passing updates that result in monotone decrease of the dual LP of the above relaxation. The derivation is similar and we omit the details. The key observation is that one needs to introduce |S(c)| copies of each marginal µc (xc ) (instead of the two copies in the pairwise case). Next, as in the EMPLP derivation we assume all β are ﬁxed except those corresponding to some cluster c. The resulting messages are λc→s (xs ) from a cluster c to all of its intersection sets s ∈ S(c). The update on these messages turns out to be:   1 1 λ−c (xs ) + max  λ−c (xs ) + θc (xc ) λc→s (xs ) = − 1 − ˆ s s ˆ |S(c)| |S(c)| xc\s s∈S(c)\s ˆ where for a given c ∈ C all λc→s should be updated simultaneously for s ∈ S(c), and λ−c (xs ) is s deﬁned as the sum of messages into s that are not from c. We refer to this algorithm as Generalized EMPLP (GEMPLP). It is possible to derive an algorithm similar to NMPLP that updates several clusters simultaneously, but its structure is more involved and we do not address it here. 7 Related Work Weiss et al. [11] recently studied the ﬁxed points of a class of max-product like algorithms. Their analysis focused on properties of ﬁxed points rather than convergence guarantees. Speciﬁcally, they showed that if the counting numbers used in a generalized max-product algorithm satisfy certain properties, then its ﬁxed points will be the exact MAP if the beliefs have unique maxima, and for binary variables the solution can be partially decodable. Both these properties are obtained for the MPLP ﬁxed points, and in fact we can show that MPLP satisﬁes the conditions in [11], so that we obtain these properties as corollaries of [11]. We stress however, that [11] does not address convergence of algorithms, but rather properties of their ﬁxed points, if they converge. MPLP is similar in some aspects to Kolmogorov’s TRW-S algorithm [5]. TRW-S is also a monotone coordinate descent method in a dual of the LP relaxation and its ﬁxed points also have similar 5 guarantees to those of MPLP [6]. Furthermore, convergence to a local optimum may occur, as it does for MPLP. One advantage of MPLP lies in the simplicity of its updates and the fact that it is parameter free. The other is its simple generalization to potentials over clusters of nodes (Sec. 6). Recently, several new dual LP algorithms have been introduced, which are more closely related to our formalism. Werner [12] presented a class of algorithms which also improve the dual LP at every iteration. The simplest of those is the max-sum-diffusion algorithm, which is similar to our EMPLP algorithm, although the updates are different from ours. Independently, Johnson et al. [4] presented a class of algorithms that improve duals of the MAP-LP using coordinate descent. They decompose the model into tractable parts by replicating variables and enforce replication constraints within the Lagrangian dual. Our basic formulation in Eq. 3 could be derived from their perspective. However, the updates in the algorithm and the analysis differ. Johnson et al. also presented a method for overcoming the local optimum problem, by smoothing the objective so that it is strictly convex. Such an approach could also be used within our algorithms. Vontobel and Koetter [9] recently introduced a coordinate descent algorithm for decoding LDPC codes. Their method is speciﬁcally tailored for this case, and uses updates that are similar to our edge based updates. Finally, the concept of dual coordinate descent may be used in approximating marginals as well. In [3] we use such an approach to optimize a variational bound on the partition function. The derivation uses some of the ideas used in the MPLP dual, but importantly does not ﬁnd the minimum for each coordinate. Instead, a gradient like step is taken at every iteration to decrease the dual objective. 8 Experiments We compared NMPLP to three other message passing algorithms:2 Tree-Reweighted max-product (TRMP) [10],3 standard max-product (MP), and GEMPLP. For MP and TRMP we used the standard approach of damping messages using a factor of α = 0.5. We ran all algorithms for a maximum of 2000 iterations, and used the hit-time measure to compare their speed of convergence. This measure is deﬁned as follows: At every iteration the beliefs can be used to obtain an assignment x with value f (x). We deﬁne the hit-time as the ﬁrst iteration at which the maximum value of f (x) is achieved.4 We ﬁrst experimented with a 10 × 10 grid graph, with 5 values per state. The function f (x) was 5 a Potts model: f (x) = The values for θij and θi (xi ) ij∈E θij I(xi = xj ) + i∈V θi (xi ). were randomly drawn from [−cI , cI ] and [−cF , cF ] respectively, and we used values of cI and cF in the range range [0.1, 2.35] (with intervals of 0.25), resulting in 100 different models. The clusters for GEMPLP were the faces of the graph [14]. To see if NMPLP converges to the LP solution we also used an LP solver to solve the LP relaxation. We found that the the normalized difference between NMPLP and LP objective was at most 10−3 (median 10−7 ), suggesting that NMPLP typically converged to the LP solution. Fig. 2 (top row) shows the results for the three algorithms. It can be seen that while all non-cluster based algorithms obtain similar f (x) values, NMPLP has better hit-time (in the median) than TRMP and MP, and MP does not converge in many cases (see caption). GEMPLP converges more slowly than NMPLP, but obtains much better f (x) values. In fact, in 99% of the cases the normalized difference between the GEMPLP objective and the f (x) value was less than 10−5 , suggesting that the exact MAP solution was found. We next applied the algorithms to the real world problems of protein design. In [13], Yanover et al. show how these problems can be formalized in terms of ﬁnding a MAP in an appropriately constructed graphical model.6 We used all algorithms except GNMPLP (since there is no natural choice for clusters in this case) to approximate the MAP solution on the 97 models used in [13]. In these models the number of states per variable is 2 − 158, and there are up to 180 variables per model. Fig. 2 (bottom) shows results for all the design problems. In this case only 11% of the MP runs converged, and NMPLP was better than TRMP in terms of hit-time and comparable in f (x) value. The performance of MP was good on the runs where it converged. 2 As expected, NMPLP was faster than EMPLP so only NMPLP results are given. The edge weights for TRMP corresponded to a uniform distribution over all spanning trees. 4 This is clearly a post-hoc measure since it can only be obtained after the algorithm has exceeded its maximum number of iterations. However, it is a reasonable algorithm-independent measure of convergence. 5 The potential θi (xi ) may be folded into the pairwise potential to yield a model as in Eq. 1. 6 Data available from http://jmlr.csail.mit.edu/papers/volume7/yanover06a/Rosetta Design Dataset.tgz 3 6 (a) (b) (c) 100 (d) 0.6 2000 0.04 0.4 0.02 −50 0 −0.02 −0.04 ∆(Value) 0 1000 ∆(Hit Time) ∆(Value) ∆(Hit Time) 50 0 MP TRMP GMPLP 0 −0.2 −1000 −0.4 −0.06 −100 0.2 MP TRMP GMPLP MP TRMP MP TRMP Figure 2: Evaluation of message passing algorithms on Potts models and protein design problems. (a,c): Convergence time results for the Potts models (a) and protein design problems (c). The box-plots (horiz. red line indicates median) show the difference between the hit-time for the other algorithms and NMPLP. (b,d): Value of integer solutions for the Potts models (b) and protein design problems (d). The box-plots show the normalized difference between the value of f (x) for NMPLP and the other algorithms. All ﬁgures are such that better MPLP performance yields positive Y axis values. Max-product converged on 58% of the cases for the Potts models, and on 11% of the protein problems. Only convergent max-product runs are shown. 9 Conclusion We have presented a convergent algorithm for MAP approximation that is based on block coordinate descent of the MAP-LP relaxation dual. The algorithm can also be extended to cluster based functions, which result empirically in improved MAP estimates. This is in line with the observations in [14] that generalized belief propagation algorithms can result in signiﬁcant performance improvements. However generalized max-product algorithms [14] are not guaranteed to converge whereas GMPLP is. Furthermore, the GMPLP algorithm does not require a region graph and only involves intersection between pairs of clusters. In conclusion, MPLP has the advantage of resolving the convergence problems of max-product while retaining its simplicity, and offering the theoretical guarantees of LP relaxations. We thus believe it should be useful in a wide array of applications. A Derivation of the dual Before deriving the dual, we ﬁrst express the constraint set ML (G) in a slightly different way. The deﬁnition of ML (G) in Sec. 2 uses a single distribution µij (xi , xj ) for every ij ∈ E. In what follows, we use two copies of this pairwise distribution for every edge, which we denote µij (xi , xj ) ¯ and µji (xj , xi ), and we add the constraint that these two copies both equal the original µij (xi , xj ). ¯ For this extended set of pairwise marginals, we consider the following set of constraints which is clearly equivalent to ML (G). On the rightmost column we give the dual variables that will correspond to each constraint (we omit non-negativity constraints). µij (xi , xj ) = µij (xi , xj ) ¯ µji (xj , xi ) = µij (xi , xj ) ¯ x xi µij (ˆi , xj ) = µj (xj ) ˆ ¯ µji (ˆj , xi ) = µi (xi ) ¯ x xj ˆ xi µi (xi ) = 1 ∀ij ∈ E, xi , xj ∀ij ∈ E, xi , xj ∀ij ∈ E, xj ∀ji ∈ E, xi ∀i ∈ V βij (xi , xj ) βji (xj , xi ) λij (xj ) λji (xi ) δi (6) ¯ We denote the set of (µ, µ) satisfying these constraints by ML (G). We can now state an LP that ¯ is equivalent to MAPLPR, only with an extended set of variables and constraints. The equivalent ¯ problem is to maximize µ · θ subject to (µ, µ) ∈ ML (G) (note that the objective uses the original ¯ µ copy). LP duality transformation of the extended problem yields the following LP min i δi s.t. λij (xj ) − βij (xi , xj ) ≥ 0 βij (xi , xj ) + βji (xj , xi ) = θij (xi , xj ) − k∈N (i) λki (xi ) + δi ≥ 0 ∀ij, ji ∈ E, xi , xj ∀ij ∈ E, xi , xj ∀i ∈ V, xi (7) We next simplify the above LP by eliminating some of its constraints and variables. Since each variable δi appears in only one constraint, and the objective minimizes δi it follows that δi = maxxi k∈N (i) λki (xi ) and the constraints with δi can be discarded. Similarly, since λij (xj ) appears in a single constraint, we have that for all ij ∈ E, ji ∈ E, xi , xj λij (xj ) = maxxi βij (xi , xj ) and the constraints with λij (xj ), λji (xi ) can also be discarded. Using the eliminated δi and λji (xi ) 7 variables, we obtain that the LP in Eq. 7 is equivalent to that in Eq. 3. Note that the objective in Eq. 3 is convex since it is a sum of point-wise maxima of convex functions. B Proof of Proposition 2 We wish to minimize f in Eq. 4 subject to the constraint that βij + βji = θij . Rewrite f as f (βij , βji ) = max λ−j (xi ) + βji (xj , xi ) + max λ−i (xj ) + βij (xi , xj ) j i xi ,xj xi ,xj (8) The sum of the two arguments in the max is λ−j (xi ) + λ−i (xj ) + θij (xi , xj ) i j (because of the constraints on β). Thus the minimum must be greater than −j −i 1 2 maxxi ,xj λi (xi ) + λj (xj ) + θij (xi , xj ) . One assignment to β that achieves this minimum is obtained by requiring an equalization condition:7 λ−i (xj ) + βij (xi , xj ) = λ−j (xi ) + βji (xj , xi ) = j i 1 θij (xi , xj ) + λ−j (xi ) + λ−i (xj ) i j 2 (9) which implies βij (xi , xj ) = 1 θij (xi , xj ) + λ−j (xi ) − λ−i (xj ) and a similar expression for βji . i j 2 The resulting λij (xj ) = maxxi βij (xi , xj ) are then the ones in Prop. 2. Acknowledgments The authors acknowledge support from the Defense Advanced Research Projects Agency (Transfer Learning program). Amir Globerson was also supported by the Rothschild Yad-Hanadiv fellowship. References [1] M. Bayati, D. Shah, and M. Sharma. Maximum weight matching via max-product belief propagation. IEEE Trans. on Information Theory (to appear), 2007. [2] D. P. Bertsekas, editor. Nonlinear Programming. Athena Scientiﬁc, Belmont, MA, 1995. [3] A. Globerson and T. Jaakkola. Convergent propagation algorithms via oriented trees. In UAI. 2007. [4] J.K. Johnson, D.M. Malioutov, and A.S. Willsky. Lagrangian relaxation for map estimation in graphical models. In Allerton Conf. Communication, Control and Computing, 2007. [5] V. Kolmogorov. Convergent tree-reweighted message passing for energy minimization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(10):1568–1583, 2006. [6] V. Kolmogorov and M. Wainwright. On the optimality of tree-reweighted max-product message passing. In 21st Conference on Uncertainty in Artiﬁcial Intelligence (UAI). 2005. [7] J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, 1988. [8] B. Taskar, S. Lacoste-Julien, and M. Jordan. Structured prediction, dual extragradient and bregman projections. Journal of Machine Learning Research, pages 1627–1653, 2006. [9] P.O. Vontobel and R. Koetter. Towards low-complexity linear-programming decoding. In Proc. 4th Int. Symposium on Turbo Codes and Related Topics, 2006. [10] M. J. Wainwright, T. Jaakkola, and A. S. Willsky. Map estimation via agreement on trees: messagepassing and linear programming. IEEE Trans. on Information Theory, 51(11):1120–1146, 2005. [11] Y. Weiss, C. Yanover, and T. Meltzer. Map estimation, linear programming and belief propagation with convex free energies. In UAI. 2007. [12] T. Werner. A linear programming approach to max-sum, a review. IEEE Trans. on PAMI, 2007. [13] C. Yanover, T. Meltzer, and Y. Weiss. Linear programming relaxations and belief propagation – an empirical study. Jourmal of Machine Learning Research, 7:1887–1907, 2006. [14] J.S. Yedidia, W.T. W.T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. on Information Theory, 51(7):2282–2312, 2005. 7 Other solutions are possible but may not yield some of the properties of MPLP. 8

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