nips nips2007 nips2007-120 knowledge-graph by maker-knowledge-mining

120 nips-2007-Linear programming analysis of loopy belief propagation for weighted matching

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Author: Sujay Sanghavi, Dmitry Malioutov, Alan S. Willsky

Abstract: Loopy belief propagation has been employed in a wide variety of applications with great empirical success, but it comes with few theoretical guarantees. In this paper we investigate the use of the max-product form of belief propagation for weighted matching problems on general graphs. We show that max-product converges to the correct answer if the linear programming (LP) relaxation of the weighted matching problem is tight and does not converge if the LP relaxation is loose. This provides an exact characterization of max-product performance and reveals connections to the widely used optimization technique of LP relaxation. In addition, we demonstrate that max-product is effective in solving practical weighted matching problems in a distributed fashion by applying it to the problem of self-organization in sensor networks. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 In this paper we investigate the use of the max-product form of belief propagation for weighted matching problems on general graphs. [sent-5, score-0.495]

2 We show that max-product converges to the correct answer if the linear programming (LP) relaxation of the weighted matching problem is tight and does not converge if the LP relaxation is loose. [sent-6, score-0.943]

3 In addition, we demonstrate that max-product is effective in solving practical weighted matching problems in a distributed fashion by applying it to the problem of self-organization in sensor networks. [sent-8, score-0.46]

4 In this paper we study the application of the widely used max-product form of LBP (or simply max-product (MP) algorithm), to the weighted matching problem. [sent-14, score-0.396]

5 Given a graph G = (V, E) with non-negative weights we on its edges e ∈ E, the weighted matching problem is to ﬁnd the heaviest set of mutually disjoint edges (i. [sent-15, score-0.933]

6 a set of edges such that no two edges share a node). [sent-17, score-0.386]

7 Weighted matching is a classic problem that has played a central role in computer science and combinatorial optimization, with applications in resource allocation, scheduling in communications networks [10], and machine learning [5]. [sent-18, score-0.313]

8 Weighted matching thus naturally suggests itself as a good candidate for the study of convergence and correctness of algorithms like max-product. [sent-20, score-0.288]

9 Weighted matching can be naturally formulated as an integer program. [sent-21, score-0.318]

10 The technique of linear programming (LP) relaxation involves replacing the integer constraints with linear inequality constraints. [sent-22, score-0.248]

11 This relaxation is tight if the resulting linear program has an integral optimum. [sent-23, score-0.311]

12 LP relaxation is not always tight for the weighted matching problem. [sent-24, score-0.653]

13 The primary contribution of this paper is an exact characterization of max-product performance for the matching problem, which also establishes a link to LP relaxation. [sent-25, score-0.336]

14 We show that (i) if the LP relaxation is tight then max-product 1 converges to the correct answer, and (ii) if the LP relaxation is not tight then max-product does not converge. [sent-26, score-0.557]

15 Weighted matching is a special case of the weighted b-matching problem, where there can be up to bi edges touching node i (setting all bi = 1 reduces to simple matching). [sent-27, score-0.871]

16 However, in the interests of clarity, we provide proofs only for the conceptually easier case of simple matchings where bi = 1. [sent-29, score-0.199]

17 al [2] established that max-product converges for weighted matching in bipartite graphs, and [5] extended this result to b-matching. [sent-32, score-0.478]

18 These results are implied by our result1 , as for bipartite graphs, the LP relaxation is always tight. [sent-33, score-0.232]

19 In Section 2 we set up the weighted matching problem and its LP relaxation. [sent-34, score-0.396]

20 We describe the maxproduct algorithm for weighted matching in Section 3. [sent-35, score-0.396]

21 2 Weighted Matching and its LP Relaxation Suppose that we are given a graph G with weights we , we also positive integers bi for each node i ∈ V . [sent-38, score-0.271]

22 A b-matching is any set of edges such that the total number of edges in the set incident to any node i is at most bi . [sent-39, score-0.618]

23 Weighted b-matching can be naturally formulated as the following integer program (setting all bi = 1 gives an integer program for simple matching): IP : max we xe , e∈E xe ≤ bi for all i ∈ V, s. [sent-41, score-1.168]

24 e∈Ei xe ∈ {0, 1} for all e ∈ E Here Ei is the set of edges incident to node i. [sent-43, score-0.753]

25 The linear programming (LP) relaxation of the above problem is to replace the constraint xe ∈ {0, 1} with the constraint xe ∈ [0, 1], for each e ∈ E. [sent-44, score-1.074]

26 The LP relaxation is said to be tight if the unique optimum is integral (i. [sent-47, score-0.317]

27 Note that the LP relaxation is not tight in general. [sent-50, score-0.257]

28 Apart from the bipartite case, the tightness of LP relaxation is a function of both the weights and the graph structure2 . [sent-51, score-0.321]

29 Associate a binary variable xe ∈ {0, 1} with each edge e ∈ E, and consider the following probability distribution: p(x) ∝ ψ(xEi ) i∈V exp(we xe ), (1) e∈E which contains a factor ψ(xEi ) for each node i ∈ V , the value of which is ψ(xEi ) = 1 if e∈Ei xe ≤ bi , and 0 otherwise. [sent-54, score-1.61]

30 Note that we use i to refer both to the nodes of G and factors of p, and e to refer both to the edges of G and variables of p. [sent-55, score-0.268]

31 The factor ψ(xEi ) enforces the constraint that at most one edge incident to node i can be assigned the value “1”. [sent-56, score-0.276]

32 It is easy to see that, for any x, p(x) = exp( e we xe ) if the set of edges {e|xe = 1} constitute a b-matching in G, and p(x) = 0 otherwise. [sent-57, score-0.621]

33 If all we = 1, LP relaxation is loose: setting each xe = 1 is the optimum. [sent-61, score-0.621]

34 However, if instead the weights are {1, 1, 3}, then LP relaxation is tight. [sent-62, score-0.247]

35 However, for our problem the special form of ψ makes mt [1] this step easy even for large degrees: for each edge e ∈ Ei compute ae = max 1, me→i [0] . [sent-72, score-0.469]

36 Then, mt+1 [1] = i→e mt →i [0] e ae e ∈Ge , e ∈Ei −e mt+1 [0] = i→e mt →i [0] e ae e ∈Fe e ∈Ei −e These updates are linear in the degree |Ei |. [sent-74, score-0.702]

37 The Computation Tree for Weighted Matching Our proofs rely on the computation tree interpretation [12, 11] of loopy max-product beliefs, which we now describe for the special case of simple matching (bi = 1). [sent-75, score-0.449]

38 Recall the variables of p correspond to edges in G, and nodes in G correspond to factors. [sent-76, score-0.244]

39 For any edge e, the computation tree Te (1) at time 1 is just the edge e, the root of the tree. [sent-77, score-0.344]

40 The weights of the edges in Te are copied from the corresponding edges in G. [sent-80, score-0.44]

41 Suppose M is a matching on the original graph G, and Te is a computation tree. [sent-81, score-0.323]

42 Then, the image of M in Te is the set of edges in Te whose corresponding copy in G is a member of M . [sent-82, score-0.229]

43 G appears on the left, the numbers are the edge weights and the letters are node labels. [sent-85, score-0.254]

44 The max-weight matching M ∗ = {(a, b), (c, d)} is depicted in bold. [sent-86, score-0.288]

45 In the center plot we show T(a,b) (4), the computation tree at time t = 4 rooted at edge (a, b). [sent-87, score-0.207]

46 The bold edges in the middle tree depict the matching which is the image of M ∗ onto T(a,b) (4). [sent-89, score-0.593]

47 6, and it is easy to see that any matching on T(a,b) (4) that includes the root edge will have weight at most 6. [sent-91, score-0.454]

48 On the right we depict M , the max-weight matching on the tree T(a,b) (4). [sent-93, score-0.372]

49 In this example we see that even though (a, b) is in the unique optimal matching in G, the beliefs at the root are such that n4 [0] > n4 [1]. [sent-96, score-0.436]

50 Note also that the dotted edges are not an image of any matching in the (a,b) (a,b) original graph G. [sent-97, score-0.516]

51 Theorem 1 Let G = (V, E) be a graph with nonnegative real weights we on the edges e ∈ E. [sent-102, score-0.282]

52 Assume the linear programming relaxation LP has a unique optimal solution. [sent-103, score-0.277]

53 The above theorem implies that LP relaxation and Max-product will both succeed, or both fail, on the same problem instances, and thus are equally powerful for the weighted matching problem. [sent-112, score-0.589]

54 wij ≤ zi + zj for all (i, j) ∈ E, zi ≥ 0 for all i ∈ V The following lemma states that the standard linear programming properties of complimentary slackness hold in the strict sense for the weighted matching problem (this is a special case of [3, ex. [sent-120, score-0.75]

55 Lemma 1 (strict complimentary slackness) If the solution to LPis unique and integral, and M ∗ is the optimal matching, then there exists an optimal dual solution z to DUAL such that 1. [sent-123, score-0.178]

56 There exists > 0 such that for all (i, j) ∈ M ∗ we have wij ≤ zi + zj − / 3. [sent-125, score-0.193]

57 if no edge in M ∗ is incident on node i, then zi = 0 4. [sent-126, score-0.323]

58 zi ≤ maxe we for all i Let t ≥ 2wmax , where wmax = maxe we is the weight of the heaviest edge, and is as in part 2 of Lemma 1 above. [sent-127, score-0.176]

59 Suppose now that there exists an edge e ∈ M ∗ for which the belief at time t is / incorrect, i. [sent-128, score-0.197]

60 e e 4 Recall that nt [1] > nt [0] means that there is a matching M in Te (t) such that (a) the root e ∈ M , and e e ∗ (b) M is a max-weight matching on Te (t). [sent-131, score-0.792]

61 Note that this will be a path, ∗ because M and MT are matchings and so any node in Te (t) can have at most one edge adjacent to it in each of the two matchings. [sent-135, score-0.271]

62 It is easy to see that this path alternates between edges in M and MT . [sent-139, score-0.289]

63 Then, from parts 1 and 3 of Lemma 1 it follows that ∗ w(P ∩ MT ) = wij = zi + zj = zi . [sent-143, score-0.236]

64 ∗ (i,j)∈P ∩MT i∈P ∗ (i,j)∈P ∩MT On the other hand, from part 2 of Lemma 1 it follows that w(P ∩ M ) = wij ≤ (i,j)∈P ∩M zi + zj − zi = (i,j)∈P ∩M − |P ∩ M |. [sent-144, score-0.236]

65 We now show that M cannot be an optimal matching in Te (t). [sent-148, score-0.315]

66 We do so by “ﬂipping” the edges in ∗ P to obtain a matching with higher weight. [sent-149, score-0.481]

67 Speciﬁcally, let M = M − (P ∩ M ) + (P ∩ MT ) be the matching containing all edges in M except the ones in P , which are replaced by the edges in ∗ P ∩ MT . [sent-150, score-0.674]

68 It is easy to see that M is a matching in Te (t), and that w(M ) > w(M ). [sent-151, score-0.288]

69 This contradicts the choice of M , and shows that for e ∈ M ∗ the beliefs satisfy nt [1] ≤ nt [0] for all t large enough. [sent-152, score-0.209]

70 For the weighted matching problem this local optimality implies global optimality, because the symmetric difference of any two matchings is a union of disjoint paths and cycles. [sent-161, score-0.56]

71 We ﬁrst provide a combinatorial characterization of when the LP relaxation is loose. [sent-164, score-0.218]

72 An alternating path in G is a path in which every alternate edge is in M ∗ , and each node appears at most once. [sent-166, score-0.459]

73 A blossom is an alternating path that wraps onto itself, such that the result is a single odd cycle C and a path R leading out of that cycle3 . [sent-167, score-0.564]

74 The importance of blossoms for matching problems is well-known [4]. [sent-168, score-0.341]

75 A bad blossom is a blossom in which the edge weights satisfy w(C ∩ M ∗ ) + 2w(R ∩ M ∗ ) < w(C − M ∗ ) + 2w(R − M ∗ ). [sent-169, score-0.638]

76 3 The path may be of zero length, in which case the blossom is just the odd cycle. [sent-170, score-0.307]

77 5 b 3 Example: On the right is a bad blossom: bold edges are in M ∗ , numbers are edge weights and alphabets are node labels. [sent-171, score-0.557]

78 5 1 f 1 c g i 3 u 3 d A dumbbell is an alternating path that wraps onto itself twice, such that the result is two disjoint odd cycles C1 and C2 and an alternating path R connecting the two cycles. [sent-174, score-0.485]

79 In a bad dumbbell the edge weights satisfy w(C1 ∩ M ∗ ) + w(C2 ∩ M ∗ ) + 2w(R ∩ M ∗ ) < w(C1 − M ∗ ) + w(C2 − M ∗ ) + 2w(R − M ∗ ). [sent-175, score-0.318]

80 3 3 a 3 c 3 3 u 3 d 1 h f 3 3 j 3 i Proposition 1 If LP relaxation is loose, then in G there exists either a bad blossom, or a bad dumbbell. [sent-178, score-0.443]

81 Suppose now that max-product converges to M ∗ by iteration τ , and suppose also there exists a bad blossom B1 in G. [sent-182, score-0.392]

82 For an edge e ∈ B1 ∩ M ∗ consider the computation tree Te (τ + |V |) for e at time τ + |V |. [sent-183, score-0.178]

83 From the deﬁnition of convergence, it follows that near the root e, M will be the image of M ∗ onto Te : for any edge e in the tree at distance less than |V | from the root, e ∈ M if and only if its copy in G is in M ∗ . [sent-185, score-0.29]

84 This means that the copies in Te of the edges in B1 will contain an alternating path P in Te : every alternate edge of P will be in M . [sent-186, score-0.474]

85 For the bad blossom example above, the alternating path is ihgf cbaudcf ghi (it will go once around the cycle and twice around the path of the blossom). [sent-187, score-0.605]

86 Make a new matching M on Te (τ + |V |) by “switching” the edges in this path: M = M − (M ∩ P ) + (P − M ). [sent-188, score-0.481]

87 It follows from Proposition 1 that if LP relaxation is loose, then max-product cannot converge to M ∗ . [sent-193, score-0.193]

88 Hence, sparser networks with fewer edges are needed [7]. [sent-200, score-0.218]

89 The throughput of a link drops fast with distance, so the sparse network should mostly contain short edges. [sent-201, score-0.176]

90 An interesting set of sparse subgraph constructions with various tradeoffs addressing power efﬁciency in wireless networks is proposed in [7]. [sent-204, score-0.197]

91 984 0 0 10 20 30 40 50 40 60 60 80 100 120 140 160 180 200 (a) (b) Figure 2: (a) Histogram of node degrees versus node density. [sent-256, score-0.19]

92 (b) Average fraction of the LP upper bound on optimal cost obtained using LP relaxation and max-product. [sent-257, score-0.22]

93 We assign edge weights to be proportional to the throughput of the link. [sent-259, score-0.25]

94 odd-cycles where the weights on the edges are quite similar. [sent-272, score-0.247]

95 We show in simulations that bad blossoms appear rarely for the random graphs and weights in our construction, and LP-relaxation and MP produce nearly optimal b-matchings. [sent-273, score-0.278]

96 For the cases where LP relaxation has fractional edges, and MP has oscillating (or non-converged) edges, we erase them from the ﬁnal matching and ensure that LP and MP solutions are valid matchings. [sent-274, score-0.551]

97 An important performance metric for sensor network self-organization is the power-stretch factor5 , which compares the weights of shortest paths in G to weights of shortest paths in the sparse subgraph. [sent-290, score-0.317]

98 We run MP for a ﬁxed number of iterations, remove oscillating edges to get a valid matching, and plot the average fraction of the LP upper bound that the solution gets. [sent-299, score-0.25]

99 Sharma, “Maximum weight matching via max-product belief propagation,” in ISIT, Sept. [sent-316, score-0.337]

100 5 To compute the power-stretch the edges are weighted by d3 , i. [sent-387, score-0.301]

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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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