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    "text": "Consensus ranking under the exponential model∗ Marina Meil˘a Kapil Phadnis Arthur Patterson JeffBilmes\nDepartment of Statistics University of Washington Department of Electrical Engineering\nUniversity of Washington Seattle, WA 98195 University of Washington\nSeattle, WA 98195 Seattle, WA 98195",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "text": "Abstract Various measures of \"agreement\" have been proposed\n(for a good overview, see [Critchlow, 1985]). Of\nthese, Kendall's metric [Fligner and Verducci, 1986]\nWe analyze the generalized Mallows model, has been the measure of choice in many rea popular exponential model over rankings. cent applications centered on the analysis of\nEstimating the central (or consensus) rank- ranked data [Ailon et al., 2005, Cohen et al., 1999,\ning from data is NP-hard. We obtain the fol- Lebanon and Lafferty, 2002].",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "text": "The Kendall distance is\nlowing new results: (1) We show that search defined as:\nmethods can estimate both the central rank-\n1[j≺π0l] (1) ing π0 and the model parameters θ exactly. dK(π, π0) = X The search is n! in the worst case, but is l≺πj\ntractable when the true distribution is concentrated around its mode; (2) We show that In the above, π, π0 represent permutations and l ≺π\nthe generalized Mallows model is jointly ex- j (l ≺π0 j) mean that l precedes j (i.e is preferred to j)\nponential in (π0, θ), and introduce the con- in permutation π (π0). Hence dK is the total number\njugate prior for this model class; (3) The of pairwise disagreements between π and π0.\nsufficient statistics are the pairwise marginal\nThis distance was further generalized to a family\nprobabilities that item i is preferred to item\nof parametrized distances dθ(π, π0) depending on a\nj. Preliminary experiments confirm the theparameter vector θ by [Fligner and Verducci, 1986].\noretical predictions and compare the new alBased on these distances, defining probabilistic models\ngorithm and existing heuristics. of the form P(π) ∝e−dθ(π,π0) is immediate. Estimating π0 by e.g Maximum Likelihood (ML) is equivalent\nto finding the consensus ranking. In fact, various vot-\n1 Introduction ing rules have been studied in the context of statistical estimation [Conitzer and Sandholm, 2005]. Such\nWe are given a set of N rankings, or permutations1 estimation problems for generalizations of the Kendall\non n objects. These rankings might represent individ- distance are the focus of the present paper.\nual preferences of a panel of N judges, each presented\nwith the same set of n candidates. Alternatively, they 2 Background: Generalized Mallows\nmay represent the ranking votes of a population of N models\nvoters. The problem of rank aggregation, or of finding\na consensus ranking, is to find a single ranking π0 that\nThis section follows the excellent paper of\nbest \"agrees\" with all the N rankings.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "text": "This process\n[Fligner and Verducci, 1986] which should be concan also be seen as a voting rule, where the N voters'\nsulted for more details. Let π denote a permutation\npreferences are aggregated in an election to produce\nover the set [n] = {1, 2, 3 . . . n}, where π(l) is the rank\na consensus order over the candidates, the top ranked of element l in π and π−1(j) is the element at rank\nbeing the winner.\nj. One can uniquely determine any π by the n −1\n∗This material is based upon work supported by the integers V1(π), V2(π), . . . Vn−1(π) defined as\nNational Science Foundation under grant IIS-0535100 and\nby an ONR MURI grant N000140510388. Vj(π) = X 1[l≺πj] (2) 1We use permutation and ranking interchangeably. l>j In other words, Vj is the number of elements in j+1 : n nential models, one for each Vj and that\nwhich are ranked before j by π. It follows from the\nn−1 n−1\nabove that Vj takes values in {0, . . .n −j}. We note 1 −e−(n−j+1)θj\nψ(θ) = ψj(θj) = (5) that while the values π(l) are dependent, the values Y Y 1 −e−θj j=1 j=1\nVj may be chosen independently in specifying a π. In\n[Feller, 1968] it is moreover shown that the numbers e−θjr\nVj are uniformly distributed if π is sampled uniformly. P[Vj(ππ−10 ) = r] = (6)\nψj(θj)\nWe say that a distance between permutations d(π, π0)\nThe above models are well defined for any real val- is right-invariant if d(ππ, π0π) = d(π, π0) for any perues of the parameters θ. However, we are interested mutation π, where ππ(l) = π(π(l)).",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
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    "text": "Requiring that a\nonly in the values θj ≥0, for which the probability distance is right invariant means that we want it to be\ndistribution has a maximum at V ≡0. This case cor- indifferent to the relabeling of the n objects, which is\nresponds to a distribution over orderings where all the a standard assumption. For any right-invariant d, we\nhigh probability instances are small perturbations of have d(π, π0) = d(ππ−10 , id) ≡D(ππ−10 ) and therefore\nthe central permutation. For θ ≡0, Pθ ≡P0 is the the distance is completely determined by the function\ndef uniform distribution. For θ1 = θ2 = . . . = θn−1 (4) is\nD(π) = d(π, id) where id denotes the identity per- the Mallows model [Mallows, 1957]. The size of the θ\nmutation id = (1, 2, . . . n). parameters controls the concentration of the distribution around its mode π0; smaller values make the dis-\n2.1 Generalized Kendall distance tribution closer to uniform, while larger values make\nit more concentrated. From (1) and (2) it is easy to see that the Kendall distance has a simple expression DK(π) = j=1 Vj(π). Pn−1 3 The ML estimation problem Therefore, [Fligner and Verducci, 1986] proposed the\nparametrized generalization of the Kendall distance\n3.1 Parameter estimation. defined by n−1 Assume an independent sample π1:N of size N has been\nDθ(π) = X θjVj(π), θj ≥0 (3) obtained from model (4). Then the data log-likelihood j=1 can be written as where θ = (θ1:n−1) is a parameter vector. The Kendall l(θ, π0) = ln P(π1:N; θ, π0) (7)\ndistance is a metric [Mallows, 1957]. The generaliza- n−1 i=1 Vj(πiπ−10 ) −ln ψj(θj) # (8) \" θj PN tion (3) may be asymmetric unless θj is constant for = −N X N\nall j. Therefore, dθ is not in general a metric. j=1\nn−1\nDθ is a versatile and convenient measure of divergence\n= −N X θj ¯Vj −ln ψj(θj) (9) between rankings. By choosing the θ parameters to e.g j=1\ndecrease with j we can emphasize the greater importance of ranking the first items in π0 correctly relative In the above ¯V is the sample expectation of Vj(ππ−10 ).\nto the correct ranking of items with low ranks in π0.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
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    "text": "It is easy to see that for any fixed π0, the model (4)\nVariations of this model where the \"emphasized ranks\" is an exponential family [DeGroot, 1975] with paramj can be selected at will are also possible. eters θ. Moreover, because the random variables Vj\nare independent, each Vj is distributed according to\n2.2 Generalized Mallows models an exponential model with one parameter θj. This is\nreflected in equation (9) where the log-likelihood l deThe following family of exponential models based on composes into a sum of terms, each depending on a\nthe divergence (3) is called the generalized Mallows single θj.\nmodel [Fligner and Verducci, 1986]\nMaximizing the log-likelihood to estimate θ when π0 is\n0 ) known is therefore immediate. It amounts to solving e−dθ(π,π0) e−Dθ(ππ−1\nPθ,π0(π) = = (4) the implicit equation in one variable obtained by tak-\nψ(θ) ψ(θ)\ning the partial derivative w.r.t. θj in equation (9). As\nin [Fligner and Verducci, 1986] this equation is rewrit- In the above, ψ(θ) is a normalization constant\nten that does not depend on π0. It was shown in\n[Fligner and Verducci, 1986] that the model (4) fac- 1 n −j + 1\ntors into a product of independent univariate expo- ¯Vj = − j = 1 : n −1 (10) eθj −1 e(n−j+1)θj −1, Note that l(θ, π0) is log-concave in θ. Hence equa- In [Cohen et al., 1999] a greedy heuristic (the CSS\ntion (10) has a unique solution for any j and any greedy algorithm) based on graph operations is intro-\n¯Vj ∈[0, n −j] (see e.g [Fligner and Verducci, 1986]). duced and tested. The heuristic works under slightly\nThis solution has in general no closed form expres- more general conditions, as it assumes that not all\nsion, but can be obtained numerically by standard of the n items are ranked under all permutations πi.\niterative algorithms for convex/concave optimization A good discussion of the sources of difficulty for this\n[Bertsekas, 1999]. problem is also given. This greedy heuristic achieves\na factor 2 approximation. We will return to the CSS\n3.2 The centroid estimation problem heuristic in sections 7 and 8. Interestingly enough, none of the above tie the conIn the following we study the combinatorial problem of\ncentration of the distribution to the hardness of the\nestimating the unknown mode π0. Before addressing\nproblem (recent work that explores the effect of a\nthis, however, we introduce a summary statistic that\nform of concentration includes [Conitzer et al., 2006,\nwill prove pivotal to our findings. This is the matrix\nDavenport and Kalagnanam, 2004]). Intuitively, howQ(π1:N) defined as\never, the problem should not be difficult if Pθ,π0 is\nN concentrated around π0.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
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    "text": "It is also intuitive that if the\nQjl(π1:N) = N X 1[j≺πil] for j, l = 1 : n (11) distribution is uniform, then any permutation will be i=1 equally qualified to be the mode. The next section\nexploits exactly this observation.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
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    "text": "In other words, Qjl(π1:N) is the probability that j precedes l in the sample. In the rest of the paper, when\n4 Exact ML estimation for π0no confusion is possible, we will denote Q(π1:N) simply\nby Q. Also, Q(π) denotes the Q matrix corresponding\n4.1 Estimation of π0 for θ known.to a single permutation π. The elements of Q(π) are\n{0, 1} valued while the elements of Q ≡Q(π1:N) are Maximizing the log-likelihood (9) w.r.t π0 is the same\nrational numbers for any finite N.\nas minimizing j=1 θj ¯Vj. The following key obser- Pn−1One of the most effective mode estimation procedures vation allows us to do so. Let us denote for simis the FV heuristic [Fligner and Verducci, 1988] and plicity ˜Vj(π) = Vj(ππ−10 ). It can be verified that\nj=1 Qjl. if π−10 (1) = r, then ˜V1(π) is the number of elementscan be described in terms of Q. Let ¯ql = PnNote that ¯ql is one less than the average rank of l which come before r in π. ¯V1, the expectation of ˜V1 unin the data. Let ¯π0 denote the permutation given der the sampling distribution, is the expected number\nby sorting the ¯ql values in increasing order.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
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      "Marina Meila",
      "Kapil Phadnis",
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      "Jeff A. Bilmes"
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    "text": "In of elements before r. Therefore we have:\n[Fligner and Verducci, 1988] it is argued that this per-\n¯V1 = Qjr whenever π−10 (1) = r (13)mutation is an unbiased estimator of π0. X\nj̸=r\nThe FV heuristic starts with this permutation, plus\nHence, to estimate the first element of π0, we can comthe set of all its neighbors at dK = 1; for each of these pute all column sums of Q and then choose π−10 (1) ∈candidates, the parameters θ are estimated and the\nargmin l̸=r Qlr.data likelihood computed. The most likely π0 of the r P\nset is then chosen. This idea can be generalized by induction to all subFor θ1 = θ2 = . . . = θn−1 ≥0 (the Mallows model) sequent j's. Assume π−10 = r1 fixed and denote by\nthe optimal π0 does not depend on θ and the problem π|−r1 the permutation over n −1 elements resulting\nbecomes one of finding from π if r1 was removed. Again, a simple verification\nshows that ˜V2(π) represents the number of items that\nN precede π−10 (2) in π|−r1. By averaging, it follows that\nπ0 ∈argmin (12) π′ X dK(πi, π′)\n0 (1 : 2) = (r1, r2). (14) i=1 ¯V2 = X Qlr2 when π−1\nl̸=r1,r2\nThis is precisely the consensus ranking problem. It is known that this problem is NP-hard By induction, we obtain\n[Bartholdi et al., 1989], and solving it approximately\n¯Vj = Qlrj when π−10 (1 : j) = (r1, r2, . . . , rj).has been addressed in the literature. The approxima- X\nl̸=r1,r2,...rj\ntion algorithm that guarantees best theoretical bounds\n(15)\nis that of [Ailon et al., 2005]; this is a randomized algorithm that achieves a factor 11/7 approximation in Therefore, we have in Q the information necessary\nminimizing the r.h.s of (12). to find the π0 maximizing the likelihood and that an exhaustive search over all the possible permutaTable 1: The SearchPi algorithm with an admissitions can obtain it. One can represent this as a\nble heuristic A.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
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      "Jeff A. Bilmes"
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    "text": "Node ρ stores: ρ = r1, . . . , rj, j =\nsearch tree, whose nodes represent partial orderings |ρ|, Vj(ρ), θj, C(ρ), L(ρ); S is the priority queue holding\nρ = (r1, . . . rj). Denote by |ρ| the length of the se- the nodes to be expanded.\nquence ρ. A node ρ with |ρ| = j has n −j children,\nrepresented by the sequences ρ′ = ρ|rj+1 where the Algorithm SearchPi\nsymbol | stands for concatenation of sequences and Initialize\nrj+1 ranges in [n] \\ ρ the set complement of ρ in [n]. S = {ρ∅}, ρ∅ =the empty sequence, j =\nAny path of length n through the search tree start- 0, C(ρ∅) = V (ρ∅) = L(ρ∅) = 0\ning from the root represents a permutation. A node\nRepeat (r1, . . . rj) at level j < n can be thought of as the set\nof all permutations that start with r1, . . . , rj. remove ρ ∈argmin L(ρ) from S\nρ∈S\nWe define the variables of a search algorithm. First,\nif |ρ| = n (Return)\nOutput ρ, L(ρ) = C(ρ) and Stop. Vj(r1, r2, . . . rj) = X Qlrj. (16)\nl̸∈{r1,r2,...rj}\nelse (Expand ρ)\nThe cost at node ρ = (r1, . . . rj) is given by for rj+1 ∈[n] \\ ρ\nj create node ρ′ = ρ|rj+1\nC(r1, . . . rj) = X θlVl(r1, . . . rl) (17) Vj+1(ρ′)min= Vj(r1:j−1, rj+1) −Qrjrj+1\nl=1 compute V = rj+1∈[n]\\ρVj+1(ρ|rj+1)min\nThis cost can be computed recursively on the tree by calculate A(ρ)\nfor rj+1 ∈[n] \\ ρ\nC(r1, . . . rj) = C(r1, . . . rj−1) + θjVj(r1, . . . rj) (18)\nθj+1 = tn−j−1(Vj+1(ρ′))\nC(ρ′) = C(ρ) + θj+1Vj+1(ρ′) The tree nodes can be expanded according to any\nL(ρ′) = C(ρ′) + A(ρ) standard search procedure, such as A∗. To direct the\nstore node (ρ′, j + 1, Vj+1, θj+1, C(ρ′),\nsearch, one also needs a lower bound A(r1, . . . rj) on L(ρ′)) in S\nthe cost to go from the current partial solution. We\nwill describe possible bounds in the next section. The\nsum L(ρ) = C(ρ) + A(ρ) represents a lower bound for\nany permutation in the set prefixed by ρ. In such a be computed off-line once and then used for any data\ntree, search can finish with the optimal solution be- with n up to a preset maximum value.\nfore the whole tree is expanded. Table 1 provides an\nA∗Best-First (BF) search algorithm. 5 Computational aspects 4.2 Simultaneous estimation of π0 and θ. 5.1 Admissible heuristics\nAlgorithm SearchPi can immediately be extended We now describe possible functions A(ρ) to be used in\nto the more interesting case when both the centroid place of the cost to go. Such a function needs to satisfy\nπ0 and the parameters θ are unknown. Recall, for two conditions: to be easily computable, and to lower\nany fixed π0 the model (4) is an exponential family bound the true cost to go.",
    "paper_id": "1206.5265",
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    "text": "The simplest heuristic is\nmodel and thus the parameter estimates depend only evidently A(ρ) = 0.\non the sufficient statistics ¯V1:n−1. Moreover, the estimate θMLj depends only on ¯Vj. Hence, any time a node Admissible heuristic for V with known θ. If the\nparameters θ are known, then we only need to find ρ in the search tree is created, θML|ρ| can be readily com- lower bounds on Vj′ for j′ > j. When node ρ is exputed at the node by solving (10) with ¯Vj = V|ρ|(ρ). panded, after computing Vj+1 for all children, we find\nAs mentioned before, this equation does not generally the minimum over these values as\nhave a closed form solution. However, the values θ\ncan be tabulated as a function of ¯V . The value of V min = min Vj+1(ρ|r). (19) r∈[n]\\ρ\nθMLj in (10) depends only on ¯Vj and n −j. Therefore,\nthe curve ¯Vn−j(θ), and consequently its inverse which For j + 1 < j′ < n −1, the best Vj′ on the current\nwe denote tn−j( ¯V ) depend only on n −j and not on branch are column sums of sub-matrices of Q. This set of curves, one for each value of n −j can (rj+1, rj+2, . . . , rj′) be any length j′ −j continuation, we get: by the path of the greedy search strategy, is n + (n −\n1) + . . . + 2 = n(n + 1)/2 −1. The number of nodes\nVj′(ρ|(rj+1, . . . , rj′))\nexpanded by the greedy strategy is one node in each\n= X Qirj′ − X Qirj′ level, i.e a total of n −1 nodes.\ni∈[n]\\ρ i∈{rj+1...rj′ }\nA qualitative examination of the cost (17) shows that\n≥max[V min−(j′ −j)Qmax, 0] = aj′(ρ) the larger the value of θj, the greater the advantage of\nwhere Qmax = maxjl Qjl is computed offline. Then the best rj w.r.t the second best.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
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      "Kapil Phadnis",
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    "text": "Hence, large values\nof θj imply that the chance of a non-optimal subtree\nA can be computed as A(ρ) = j′=j+1 θj′aj′(ρ). Pn−1 at level j to contain the optimal solution is small. In\nAdmissible heuristic for V with constant θ. For other words, when the values of the parameters are\nthe special case of consensus ranking, when θj ≡1, an large, which corresponds to a distribution Pθ that deeven better heuristic can be used. Sort the off-diagonal cays fast away from the mode π0, then the number\nvalues of Qlr in increasing order, denoting the resulting of nodes explored will be small. For any admissible\nsequence by heuristic A, there are parameters θML for which the\nBF algorithm will explore exactly the same nodes as\nq(1) ≤q(2) ≤. . . ≤q(n(n−1)/2) (20)\nthe greedy algorithm and no more. The cost to go in consensus ranking is independent of θ\nAt the other end of the spectrum, if θj ≈0 for all j,and equal to Vj+1(ρ′j+1)+Vj+2(ρ′j+2)+. . . Vn−1(ρ′n−1)\nthe search is likely to be intractable. In this case areon some (unknown) path from the current node to the\ndata sets sampled from an almost uniform distribution,bottom of the tree.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "text": "Since each Vj is the sum of n −j\nwhich will have all values Qlr ≈0.5. Data sampledoff-diagonal Qlr's, this cost to go is equal to the sum of\nfrom multi-modal distributions can also fall under this(n−j−1)(n−j−1+1)/2 distinct off-diagonal elements\ncategory2. For multi-modal distributions, individual\nof Q. Hence A(ρ) = l=1 q(l) is always lower P(n−j−1)(n−j)/2 Qlr values can take extreme values near 0 or 1, butbounding the cost to go. This heuristic depends only\nbecause no consensus exists, the average Qlr along a\non the level j and can be entirely computed before the\ncolumn or sub-column will be near 0.5 as well.\nsearch. In this latter case, the algorithm can be stopped any\nAdmissible heuristics for unknown θ. If the patime, and it will provide the best solution it was able\nrameters θj are estimated simultaneously with the cento find so far. For this case, practical optimization\ntral permutation π0, then lower bounding the cost to\nusually involves inadmissible heuristics (e.g. beam\ngo requires us to find lower bounds on the parameters\nθj′, with j′ > j = |ρ| the current level. search). We leave this avenue open for further research.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "text": "Any non-zero lower bound on θj′ can then be combined\nwith the lower bounds on Vj′ described above pro- 5.3 Number of operations per node.\nduce an admissible A. The derivation of possible lower\nbounds for the parameters is in [Meil˘a et al., 2007]. In Upon creating node ρ′ = ρ|rj+1 = r1, . . . , rj+1, the\nthis case too, the bounds will be computed off-line and SearchPi algorithm needs to compute the value of\nl∈[n]\\ρ Qlrj+1. Computing this sum explicitlywill depend only on the tree level j. ¯Vρ′ = P\ntakes O(n −j) operations, which makes the time of\n5.2 Number of node expansions exploring one vertical path to the terminal of a tree be\nO(n(n−1)+(n−1)(n−2)+. . .+2 = O(n3). However,\nLet us further analyze the algorithm SearchPi from by better organizing the data we can obtain a constant\na computational point of view. BF algorithms with\ncomputation time per node.\nadmissible heuristics are guaranteed to find the optimal solution given enough time. The stopping condiVj+1(r1, . . . , rj+1) = Qlrj+1−Qrjrj+1tion is met when the most promising node is a terminal X\nl̸=r1:j−1,rj+1\nnode. This condition can be met before all nodes in the\nsearch tree are expanded. Hence, an important perfor- = Vj(r1, . . . , rj−1, rj+1) −Qrjrj+1 (21)\nmance parameter for a BF algorithm is the number of\nnodes that it visits before it finds the optimum. This The node (r1, . . . , rj−1, rj+1) is a sibling of\nnumber clearly depends on the quality of the heuristic (r1, . . . , rj+1)'s parent (hence an \"uncle\"). In\n– the better a lower bound is A on the cost to go, the our algorithm, and in any search algorithm that\nmore nodes can be pruned from the search tree. creates all children of a node at once, this node will In our case, the worst case running time will be n!. 2In this case, since there is no true parameter θ, we refer\nThe lower limit on the number of nodes created, given to the estimated θML have been created and its V value available by the is not only convenient, it is also necessary to ensure\ntime we need to compute Vj+1(r1, . . . , rj+1). that the model is identifiable. A model Pθ,π0 with\nθ > 0 is strongly unimodal; in such a model the\nTo use this value, we must only make sure that no\nprobability of any inversion w.r.t π0 is less than 0.5\nnodes are deleted from memory while their V values\n[Fligner and Verducci, 1988].\nare still needed. This can be achieved with a counter\nvariable associated with each Vj(ρ) which signals when While almost3 each Q ∈Q defines uniquely a pair\nthe value is no longer needed.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "text": "Another possible solu- θML, π0ML, the converse is not true. There are\ntion is to pass the V values alone, as tables, down the an infinity of matrices Q which produce the same\ntree. This way any node can be deleted independently θML, π0ML.\nof the rest of the tree. Keeping a table at a node adds\na storage of n −j per node. 6.2 The conjugate prior Selecting the next node can be done efficiently if all\nThe existence of finite sufficient statistics implies that\nthe nodes are kept in a priority queue sorted by L(ρ). Pθ,π0(π) is an exponential family model jointly in\nFibonacci heaps can attain constant access time, while\n(θ, π0). As such, it will have a conjugate prior, whose\nour STL based implementation uses a binomial heap\nform is given below.\nwith access time logarithmic in the length of the queue. Proposition 2 Let Γ ∈Q, ν > 0; denote Γ∞=\n6 Identifiability and conjugate prior Q(id) ∈Q, Θ = diag(θ, 0) ∈Rn×n and Π0 the permutation matrix associated to permutation π0. Then\n6.1 Identifiability\nΓ∞Π0ΓΠT0 Θ+ln ψ(θ)] P(π0, θ ; ν, Γ) ∝e−ν[trace (23)\nThe matrix Q represents the sufficient statistics for the\nparameters π0, θ. Because by definition is a conjugate prior for the parameters (θ, π0) of model\n(4). Qlj + Qjl = 1 for l ̸= j, Qjj = 0 (22)\nProof. Vj(ππ−10 ) can be written as element (j, j) the number r of free parameters in Q is at most n(n −\nof Γ∞Π0Q(π)ΠT0 and consequently ln Pθ,π0(π) = 1)/2.\ntrace Γ∞Π0Q(π)ΠT0 Θ + ln ψ(θ). Moreover, NQ =\nThe set Q = {Q} of matrices satisfying (22) is a i=1 Q(πi). Hence, convex polytope, with n! extreme points given by PN\nQ(π) = [1[l≺πj]]lj. By the Caratheodory theorem P(π0, θ |π1:N) ∝ P(π1:N|π0, θ)P(π0, θ ; ν, Γ)\nΠT0 Θ+ln ψ(θ)] [Rockafellar, 1970], any Q in the polytope can be rep- Γ∞Π0 NQ+νΓN+ν ∝ e−(N+ν)[trace resented by a convex combination of at most n(n −\nNQ + νΓ\n1)/2+1 extreme points.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "text": "This implies that Q can be ap- = P(π0, θ; N + ν, ) (24)\nN + ν proximated arbitrarily closely by finite data sets with\nN large enough. So, asymptotically, any Q ∈Q can We have shown that the distribution in (23) is closed\nrepresent a set of sufficient statistics. under sampling, in other words it is a conjugate prior\nNote also that for any Q ∈Q and for any permuta- [DeGroot, 1975]. It remains to show that the prior is\ntion π0, there is a unique parameter vector θML(π0) ∈ integrable on θj ≥0, j = 1 : n −1. This is straightforargmax Pθ,π0(Q) (because equation (10) has a unique ward and left to the reader. 2\nsolution). The following result says that for any data We note that the general form of a conset there is a non-negative θ estimate. jugate prior family is P(π0, θ ; ν, Γ) ∝\nh(θ, π0)e−ν[trace Γ∞Π0ΓΠT 0 Θ+ln ψ(θ)]a where h(θ, π0)\nProposition 1 For any Q ∈Q there exists a permu- is a function that renders the prior integrable and\ntation π0, so that θMLj (π0) ≥0. doesn't depend on ν, Γ. Our proposition extends\nimmediately to this case as well.\njl Qjl = Proof. Since Qjl = 1 −Qlj we have P n(n−1)/2; therefore there is at least one column r for The prior above is defined up to a normalization conl Qlr ≤(n −1)/2. For this column, equation stant. At present we do not have a closed form formula which P (10) with j = 1 will have a non-negative solution θ1. for this constant. We also stress that the sufficient\nWe now delete column and row r from Q and proceed statistics Q for the model (4) are not minimal and the\nrecursively for j = 2 : n −1. 2 model itself, in the above parametrization, is not a\nminimal exponential model. This proposition justifies our focusing on the domain\nof non-negative θ. It shows that such a restriction 3Except for those Q for which there are ties in π0. It is interesting, nevertheless, to interpret the prior's 106 θ= 1\nparameters. The ν parameter's role as \"equivalent θ=1.5θ= 2 1.3\nsample size\" is obvious; let us now look at the matrix 104 θ= 3\n1.2parameter Γ. If Γ represents an expectation matrix expanded",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
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    "text": "102 1.1Eθ∗,π∗[Q(π)] under model (4) the conjugate prior is nodesequivalent to having seen ν samples from a distribu- Cost/Cost(BB−CSS)\n0tion centered at π∗with spread θ∗. If one uses another 10 0 10 20 30 40 50 60 8 10 14 20 25\nΓ in the prior, that corresponds to having seen ν sam- n n\nples from a distribution not in the class represented by a b\n(4). Figure 1: (a) The average number of nodes expanded by\nThe matrix Γ0 obtained from θ∗≡0 has (Γ0)ij = 0.5 the SearchPi with heuristic A = 0 for various values of n\nin each off-diagonal entry. This matrix corresponds and θ. The error bars mark the minimum and maximum\nto an non-informative prior w.r.t π0, as θ∗≡0 rep- values over niter = 10 replications. (b) The cost of the\ngreedy CSS heuristics as a fraction of the BF-CSS cost.resents the uniform distribution. Using a conjugate\nThe BF-CSS heuristic is in effect the exact BF algorithm for\nprior with Γ0 implements a smoothing over the pa- n ≤14. The data are Q matrices with independent random\nrameters while being non-informative w.r.t the central entries. The boxplots are over niter = 10 replications.\npermutation.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
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    "text": "It can be easily verified that any other\nΓ ∈Q is informative w.r.t both θ and π0. Hence, in\nthe conjugate prior framework it is impossible to ex- Although theoretically the search time should not depress ignorance w.r.t to the central distribution, while pend on the true π0 in all our experiments we select\nexpressing knowledge about the parameters θ. a random π0 every time in order to average out any\nFrom an algorithmic standpoint, working with the con- artifacts of the implementation (for example, having\njugate prior is, as expected, straightforward. The full the first branch always be the optimal one could make\nposterior, up to the normalization constant, is ob- the algorithm faster). We also mention that our impletained as a summation of sufficient statistics and prior mentation of the SearchPi is a pilot implementation\nparameters.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
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    "text": "This allows one to compare the posteriors not optimized w.r.t running time.\nof any two models. If one is interested in the Maxi- The other algorithms we compared were the FV heurismum A-Posteriori (MAP) estimate, this can be readily tic of [Fligner and Verducci, 1988], the Greedyobtained by algorithm SearchPi with Q replaced by Order algorithm of [Cohen et al., 1999] (here de-\n(NQ + νΓ)/(N + ν). noted CSS) and the algorithm of [Ailon et al., 2005]\n(denoted ACN here). Our implementation of the FV\nheuristic omits the search around ¯π and therefore has a\n7 Experiments run-time complexity of O(n2). The ACN algorithm is\nalso O(n2) while the greedy algorithm is O(n3). In our\nThe experiments in this section evaluate various ex- experiments, these algorithms ran very fast (fractions\nisting algorithms on the consensus ranking problem of of a second) in all the experiments performed.\nestimating π0. Since estimating θ adds only a small\nExperiments with concentrated distributions\nconstant time per search step, we consider that this\nAs mentioned in section 2, the consensus ranking probcase embodies the core difficulties of the estimation\nlem has two regimes. In the asymptotic regime the\nproblem. Exception would make the cases when θj has distribution is concentrated around its mode (θML is\ncomparatively large values at large j's, signifying that\nlarge), and N is large enough that πML coincides with\nthe most important stages of the ranking are among\nthe true π0. This is an easy case for the BF search,\nthe last ones, while getting the highly ranked elements\nbut it is also an easy case for all heuristic algorithms\nof π0 is less important. This case is rather unrealistic\nmentioned in section 2.\nin practice. We have confirmed this experimentally, on samplesWe implemented the SearchPi in C++ with the\nwith N = 5000 from distribution Pθ,π0 with random π0\nheuristics mentioned in section 5. This algorithm\nand with θ ≡1, 1.5, 2, 3. Each experiment was repliis denoted in the experiments as BF. We also imcated niter = 10 times.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
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    "text": "In all cases, all the heuristics\nplemented a sub-optimal search algorithm that runs\nreturned the optimal permutation. For this experi-the SearchPi for a predefined amount of time (5\nment, Figure 1,a shows the number of nodes expanded\nminutes) then continues with greedy search from the\nby the BF algorithm as a function of θ and n.\nlargest level j attained in the BF search. This algorithm is denoted BF-CSS (the greedy search is denoted We also ran a comparison of the heuristics FV, ACN,\nby CSS as described below). CSS on samples of size N = 5000 from a distribution 1.01 1.01 Cost/Cost(BB−CSS) 1 1 Cost/Cost(CSS) 0.99\n0.99 8 10 14 20 25\n0.98\n8 10 20 25 30 35 40 45 50 100 Figure 3: The cost (from left to right) of the FV, ACN,\nn CSS, and SearchPi (only for n = 8, 10) algorithms, as\nfractions of the BF-CSS cost.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
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    "text": "The data are as in Figure 2. Figure 2: The cost (from left to right) of the true π0, the The boxplots are over niter = 50 replications. FV and the ACN heuristics, as fractions of the CSS cost. The data are N = 100 permutations from P0.03,π0 with\nrandom π0. The boxplots are over niter = 500 replications.\ntics, the optimal BF (for n = 8, 10 only) and the approximate search BF-CSS. The costs are plotted as\nfractions of the cost BF-CSS. Therefore, the optimal\nwith θ = 0.3 (only moderately concentrated) and with BF cost always appears below or equal to 1.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "text": "The exn = 10, . . .50. Each experiment was replicated niter = periments also show that in a large number of cases,\n100 times. For up to n = 40, all the heuristics returned the suboptimal BF-CSS outperforms all the other algothe true permutation π0. For these experiments, the rithms and improves on the closely related CSS greedy\noptimal permutation was not known except for n ≤15 algorithm.\nbut the large N ensures that with high probability the\nWe do not claim the BF-CSS to be the ultimate apoptimum coincides with the true π0.\nproximate search heuristic. Better and faster subExperiments with almost uniform distributions optimal searches (e.g beam-search) could be impleAt the other end of the spectrum is the combinato- mented. We only demonstrate by BF-CSS that the\nrial regime, where the observed permutations are dis- search tree approach is effective in improving the cost,\ntributed almost uniformly (θ ≈0) and N is relatively or alternatively, in getting closer to a consensus, over\nsmall so that the true π0 is different from π0ML. We the traditional heuristics.\nhave simulated this case by generating N = 100 samExperiments with no consensus and large range\nples from a model with θ = 0.003. The distribution\nof Q. In this set of experiments, the data consists\nbeing practically indistinguishable from uniform, and\nof a matrix Q with elements randomly sampled from\nthe Qij values being very close to 0.5, the differences\n[0, 1] subject to the constraint Qij + Qji = 1 and 0\nin cost between various solutions are minute, and they\ndiagonal. This simulates the case of a multi-modal\nare presented only as surrogates of a quality of the\ndistribution, where the permutations exhibit no consearch, since the optimal π0 is not known. For the\nsensus, but are also non-uniform.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
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    "text": "Such a setting was same reason, all algorithms except SearchPi have\nexamined experimentally by [Cohen et al., 1999]. In\nbeen compared on niter = 500 replicated experiments.\nthis problem, because the cost C can vary significantly\nThe comparison between the heuristic algorithms is with the choice of π0, finding a central permutation π0\npresented in Figure 2. The greedy CSS heuristic is minimizing this cost is a legitimate practical question.\nconsistently the best at all scales. Its advantage over For instance, this task is a subtask of learning to rank\nthe randomized algorithm of ACN is increasing with in [Cohen et al., 1999].\nlarger n. The true model π0 is never optimal for this\nThe experimental setting is identical to the previdata distribution, while its estimate by the FV heurisous, except that the experiments are now replicated\ntic is better but loses to the other algorithms. The\nniter = 10 times.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
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    "text": "Figure 1,b shows the costs, as a frac-\n\"shrinking towards\" 1 effect observed for larger n retion of the cost of BF-CSS. Similarly to 3, the BF alflects the fact that a the larger number of values in Q\ngorithm improves on all heuristics for small n and the\nare near 0.5 when n is large. This in turn shrinks the\nsuboptimal BF-CSS improves by a few percent over\nrange between the maximum and minimum cost.\nthe greedy algorithm (the best contender of the other\nFigure 3 shows comparisons between the three heuris- heuristics) for larger values of n. In the interest of fairness, we stress once more regime.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
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    "text": "If this is of interest, then our experiments have\nthat the FV algorithm could be improved by local shown that the existing heuristics differ and that the\nsearch like in [Fligner and Verducci, 1988] and that SearchPi outperforms the other contenders when it's\nthe CSS algorithm can also be improved by first find- tractable. We are currently implementing faster and\ning the strongly connected components as described in non-admissible versions of SearchPi, with the expec-\n[Cohen et al., 1999]. tation that, even if exact optimization is not tractable,\nusing a search like in SearchPi for a pre-specified\ntime can improve over greedy search.\n8 Related work and Discussion\nWe can show (proof omitted) that the GreedyOrder algorithm of [Cohen et al., 1999] is the greedyThis work builds on [Fligner and Verducci, 1986] and\ncounterpart of the SearchPi algorithm. In this sense,[Fligner and Verducci, 1990] who introduced the genthe good results of the CSS heuristic for larger n sug-eralized Mallows model and exploited the fact that it\ngest that adding an amount of search to this alreadyis an exponential family model in θ alone. As such,\ngood heuristic is worthwhile.they use a conjugate prior on θ with a uniform prior\non π0. We have shown in section 6 such a prior is We conclude by pointing out that with real ranking\nnot the conjugate prior for θ, π0 jointly. The normal- data we expect to encounter few unimodal distribuization constant for their posterior is not computable tions. We plan to continue this work toward the\nin closed form, and it has strong similarities with the more ambitious goal of estimating parametric and nonnormalization constant of (23), suggesting that the lat- parametric mixtures over the space of rankings.\nter may not be computable in closed for either. Another notable spin-offof [Fligner and Verducci, 1990] References\nis [Lebanon and Lafferty, 2002] where the posterior\nof [Fligner and Verducci, 1990] is used as a con- [Ailon et al., 2005] Ailon, N., Charikar, M., and Newman,\nditional probability model over permutations, to A. (2005). Aggregating inconsistent information: Rankbe estimated from data by a MCMC algorithm. ing and clustering. In The 37-th ACM Symposium on the\nTheory of Computing (STOC). Association for ComputOther exhaustive procedures for computing consen- ing Machinery.\nsus rankings have been developed as well. In\n[Davenport and Kalagnanam, 2004], a greedy heuris- [Bartholdi et al., 1989] Bartholdi, J., Tovey, C. A., and\nTrick, M. (1989). Voting schemes for which it can betic and branch-and-bound procedure is developed for\ndifficult to tell who won. Social Choice and Welfare,\ncomputing the consensus ranking based on the pair- 6(2):157–165.\nwise winner-looser graph.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "chunk_id": "ed4939b4-6e57-4e18-a606-7c3f070e35cb",
    "text": "This procedure was extended in [Conitzer et al., 2006], which utilizes not [Bertsekas, 1999] Bertsekas, D. Nonlinear programming. Athena Scientific, Cambridge, MA, 2 edition.only graphs but also linear programming approximations leading to better bounds. These papers empir- [Cohen et al., 1999] Cohen, W. S., and\nically explore the effect of concentration based on a Singer, Y. (1999). Learning to order things. Journal of\nsingle probability of a deviation from pairwise pref- Artificial Intelligence Research, 10:243–270.\nerences in π0. They also find that as concentration [Conitzer et al., 2006] Conitzer, V., Davenport, A., and\nincreases, compute-time decreases. Kalagnanam, J. (2006). Improved bounds for computing Kemeny rankings. In Proceedings of The 21st NaWe have presented a new algorithm and a compari- tional Conference on Artificial Intelligence, AAAI-2006,\nson of algorithms from various fields on the estimation Boston, MA.\nof the consensus ranking. Our approach to concen-\n[Conitzer and Sandholm, 2005] Conitzer, V. and Sandtration is based on the parameters of an exponential holm, T. (2005). Common voting rules as maximum\nmodel. While our algorithm is certainly optimal, it is likelihood estimators. In Uncertainty in Artificial Intelalso by far the slowest.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
    "chunk_index": 25,
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    "text": "Experiments have highlighted ligence: Proceedings of the Twentieth Conference (UAIthe existing trade-offs: in the asymptotic regime, all 2005), pages 145–152, Edinburgh, Scotland, UK. Morheuristics work well; using SearchPi is also efficient. gan Kaufmann Publishers. In the combinatorial case, if we are interested in the [Critchlow, 1985] Critchlow, D. Metric methcost only, then the differences in cost are so minute ods for analyzing partially ranked data. Number 34 in\nthat almost any heuristic (even no optimization) will Lecture notes in statistics. Springer-Verlag, Berlin Heibe acceptable.",
    "paper_id": "1206.5265",
    "title": "Consensus ranking under the exponential model",
    "authors": [
      "Marina Meila",
      "Kapil Phadnis",
      "Arthur Patterson",
      "Jeff A. Bilmes"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1206.5265v1",
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    "text": "In other words, while the problem of delberg New York Tokyo.\nconsensus ranking is theoretically NP hard, minimiz- [Davenport and Kalagnanam, 2004] Davenport, A. and\ning the cost (approximately) is practically easy. Kalagnanam, J. (2004). A computational study of the\nKemeny rule for preference aggregation. In Proceedings\nWhat is hard is finding the individual permutation of The 19th National Conference on Artificial Intellithat achieves best consensus in the combinatorial gence, AAAI-2004, pages 697–702, San Jose, CA. [DeGroot, 1975] DeGroot, M. Probability and\nStatistics. [Feller, 1968] Feller, W. (1968). An introduction to probability theory and its applications, volume 1. Wiley, New\nYork, third edition. [Fligner and Verducci, 1986] Fligner, M.",
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    "paper_id": "1206.5265",
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    "authors": [
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      "Arthur Patterson",
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    "paper_id": "1206.5265",
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      "Marina Meila",
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