nips nips2011 nips2011-47 knowledge-graph by maker-knowledge-mining

47 nips-2011-Beyond Spectral Clustering - Tight Relaxations of Balanced Graph Cuts

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Author: Matthias Hein, Simon Setzer

Abstract: Spectral clustering is based on the spectral relaxation of the normalized/ratio graph cut criterion. While the spectral relaxation is known to be loose, it has been shown recently that a non-linear eigenproblem yields a tight relaxation of the Cheeger cut. In this paper, we extend this result considerably by providing a characterization of all balanced graph cuts which allow for a tight relaxation. Although the resulting optimization problems are non-convex and non-smooth, we provide an efﬁcient ﬁrst-order scheme which scales to large graphs. Moreover, our approach comes with the quality guarantee that given any partition as initialization the algorithm either outputs a better partition or it stops immediately. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 de Abstract Spectral clustering is based on the spectral relaxation of the normalized/ratio graph cut criterion. [sent-5, score-0.792]

2 While the spectral relaxation is known to be loose, it has been shown recently that a non-linear eigenproblem yields a tight relaxation of the Cheeger cut. [sent-6, score-0.328]

3 In this paper, we extend this result considerably by providing a characterization of all balanced graph cuts which allow for a tight relaxation. [sent-7, score-0.42]

4 1 Introduction The problem of ﬁnding the best balanced cut of a graph is an important problem in computer science [9, 24, 13]. [sent-10, score-0.802]

5 In particular, in machine learning spectral clustering is one of the most popular graph-based clustering methods as it can be applied to any graph-based data or to data where similarity information is available so that one can build a neighborhood graph. [sent-12, score-0.215]

6 Spectral clustering is originally based on a relaxation of the combinatorial normalized/ratio graph cut problem, see [28]. [sent-13, score-0.713]

7 In contrast, the computation of eigenvectors of a sparse graph scales easily to large graphs. [sent-16, score-0.127]

8 In a line of recent work [6, 26, 14] it has been shown that relaxation based on the nonlinear graph p-Laplacian lead to similar runtime performance while providing much better cuts. [sent-17, score-0.214]

9 In particular, for p = 1 one obtains a tight relaxation of the Cheeger cut, see [8, 26, 14]. [sent-18, score-0.127]

10 Namely, we provide for almost any balanced graph cut problem a tight relaxation into a continuous problem. [sent-20, score-0.947]

11 This allows ﬂexible modeling of different graph cut criteria. [sent-21, score-0.571]

12 The resulting non-convex, non-smooth continuous optimization problem can be efﬁciently solved by our new method for the minimization of ratios of differences of convex functions, called RatioDCA. [sent-22, score-0.202]

13 Moreover, compared to [14], we also provide a more efﬁcient way how to solve the resulting convex inner problems by transferring recent methods from total variation denoising, cf. [sent-23, score-0.14]

14 In ﬁrst experiments, we illustrate the effect of different balancing terms and show improved clustering results of USPS and MNIST compared to [14]. [sent-25, score-0.176]

15 We work on weighted, undirected graphs G = (V, W ) with vertex set V and 1 a symmetric, non-negative weight matrix W . [sent-28, score-0.139]

16 ≤ fn and deﬁne Ci = {j ∈ V | fj > fi } where C0 = V . [sent-38, score-0.151]

17 Then S : RV → R given by n n−1 ˆ ˆ fi S(Ci−1 ) − S(Ci ) S(f ) = ˆ ˆ S(Ci )(fi+1 − fi ) + f1 S(V ) = i=1 i=1 ˆ ˆ is called the Lovasz extension of S. [sent-39, score-0.278]

18 A particular interesting class of set functions are the submodular set functions implies S(V as their Lovasz extension is convex. [sent-42, score-0.363]

19 2 A set function, F : 2V → R is submodular if for all A, B ⊂ V , ˆ ˆ ˆ ˆ F (A ∪ B) + F (A ∩ B) ≤ F (A) + F (B). [sent-44, score-0.203]

20 ˆ F is called strictly submodular if the inequality is strict whenever A B or B A. [sent-45, score-0.203]

21 Note that symmetric submodular set functions are always non-negative as for all A ⊂ V , ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2F (A) = F (A) + F (A) ≥ F (A ∪ A) + F (A ∩ A) = F (V ) + F (∅) = 0. [sent-46, score-0.3]

22 If F : A → g(s(A)) is submodular for all s ∈ R+ , then g is concave. [sent-50, score-0.203]

23 ˆ • S is submodular if and only if S is convex, • S is positively one-homogeneous, ˆ ˆ • S(f ) ≥ 0, ∀ f ∈ RV and S(1) = 0 if and only if S(A) ≥ 0, ∀ A ⊂ V and S(V ) = 0. [sent-54, score-0.258]

24 One might wonder if the Lovasz extension of all submodular set functions generates the set of all positively one-homogeneous convex functions. [sent-56, score-0.455]

25 In the next section we will be interested in the class of positively onehomogeneous, even, convex functions S with S(f + α1) = S(f ) for all f ∈ RV . [sent-58, score-0.176]

26 From the above proposition we deduce that these properties are fulﬁlled for the Lovasz extension of any symmetric, submodular set function. [sent-59, score-0.328]

27 every f ∈ C 2 (Rn ) can be written as a difference of convex functions. [sent-66, score-0.128]

28 As submodular functions correspond to convex functions in the sense of the Lovasz extension, one can ask if the same result holds for set functions: Is every set function a difference of submodular set functions ? [sent-67, score-0.633]

29 3 Every set function S : 2V → R can be written as the difference of two submodular functions. [sent-71, score-0.252]

30 The corresponding Lovasz extension S : RV → R can be written as a difference of convex functions. [sent-72, score-0.204]

31 Thus we can always ﬁnd the decomposition of the set function into a difference of two submodular functions and thus also the decomposition of its Lovasz extension into a difference of convex functions. [sent-75, score-0.444]

32 Direct minimization of the cut leads typically to very unbalanced partitions, where often just a single vertex is split off. [sent-77, score-0.564]

33 Therefore one has to introduce a balancing term which biases the criterion towards balanced partitions. [sent-78, score-0.383]

34 Two popular balanced graph cut criterion are the Cheeger cut RCC(A, A) and the ratio cut RCut(A, A) 1 cut(A, A) cut(A, A) 1 + , RCut(A, A) = |V | = cut(A, A) . [sent-79, score-1.791]

35 Spectral clustering is derived as relaxation of the ratio cut criterion based on the second eigenvector of the graph Laplacian. [sent-81, score-0.829]

36 In particular there exist graphs where the spectral relaxation is as bad [12] as the isoperimetric inequality suggests [1]. [sent-83, score-0.272]

37 In a recent line of work [6, 26, 14] it has been shown that a tight relaxation for the Cheeger cut can be achieved by moving from the linear eigenproblem to a nonlinear eigenproblem associated to the nonlinear graph 1-Laplacian [14]. [sent-84, score-0.85]

38 1 that a tight relaxation exists for every balanced graph cut measure which is of the form cut divided by balancing term. [sent-86, score-1.496]

39 Then a balanced graph V cut criterion φ : 2 → R+ of a partition (A, A) has the form, φ(A) := cut(A, A) . [sent-88, score-0.889]

40 ˆ S(A) (1) As we consider undirected graphs, the cut is a symmetric set function and thus φ(A) = φ(A). [sent-89, score-0.541]

41 In ˆ order to get a balanced graph cut, S is typically chosen as a function of |A| (or some other type of volume) which is monotonically increasing on [0, |V |/2]. [sent-90, score-0.343]

42 The ﬁrst part of the theorem showing the equivalence of combinatorial and continuous problem is motivated by a result derived by Rothaus in [25] in the context of isoperimetric inequalities on Riemannian manifolds. [sent-91, score-0.16]

43 1 A function S : V → R is positively one-homogeneous, even, convex and S(f + α1) = S(f ) for all f ∈ RV , α ∈ R if and only if S(f ) = supu∈U u, f where U ⊂ Rn is a closed symmetric convex set and u, 1 = 0 for any u ∈ U . [sent-95, score-0.268]

44 1 Let G = (V, E) be a ﬁnite, weighted undirected graph and S : RV → R and let ˆ ˆ S : 2V → R be symmetric with S(∅) = 0, then inf 1 2 f ∈RV n i,j=1 wij |fi − fj | S(f ) = inf A⊂V cut(A, A) , ˆ S(A) if either one of the following two conditions holds 1. [sent-97, score-0.357]

45 S is positively one-homogeneous, even, convex and S(f + α1) = S(f ) for all f ∈ RV , ˆ ˆ α ∈ R and S is deﬁned as S(A) := S(1A ) for all A ⊂ V . [sent-98, score-0.134]

46 S is the Lovasz extension of the non-negative, symmetric set function S with S(∅) = 0. [sent-100, score-0.131]

47 Let f ∈ RV and denote by Ct := {i ∈ V | fi > t}, then it holds under both conditions, mint∈R cut(Ct , Ct ) ≤ ˆ S(Ct ) 1 2 n i,j=1 wij |fi − fj | S(f ) . [sent-101, score-0.228]

48 1 can be generalized by replacing the cut with an arbitrary other set function. [sent-103, score-0.475]

49 However, the emphasis of this paper is to use the new degree of freedom for balanced graph clustering. [sent-104, score-0.358]

50 ˆ ˆ ˆ Moreover, S is non-negative with S(∅) = S(V ) = 0 as S is even, convex and positively onehomogeneous. [sent-108, score-0.134]

51 3 the Lovasz extension of any set function can be written as a difference of convex (d. [sent-110, score-0.204]

52 ˆ We would like to point out a related line of work for the case where the balancing term S is submodular and the balanced graph cut measure is directly optimized using submodular minimization techniques. [sent-117, score-1.344]

53 In [23] this idea is proposed for the ratio cut and subsequently generalized [22, 17] so ˆ that every submodular balancing function S can be used. [sent-118, score-0.813]

54 1 opens up new modeling possibilities for clustering based on balanced graph cuts. [sent-124, score-0.411]

55 We discuss in the experiments differences and properties of the individual balancing terms. [sent-125, score-0.129]

56 However, for a given random graph model it might be possible to suggest a suitable balancing term given one knows how cut and volume behave. [sent-128, score-0.7]

57 A ﬁrst step in this direction has been done in [20] where the limit of cut and volume has been discussed for different neighborhood graph types. [sent-129, score-0.592]

58 In the following we assume that we work with graphs which have non-negative edge weights W = (wij ) and non-negative vertex weights e : V → R+ . [sent-130, score-0.16]

59 The volume reduces to the cardinality if ei = 1 for all i ∈ V (unnormalized case) or to the volume considered in the normalized cut, vol(A) = i∈A di for ei = di for all i ∈ V (normalized case), where di is the degree of vertex i. [sent-132, score-0.416]

60 Using general vertex weights allows us to present the unnormalized and normalized case in a uniﬁed framework. [sent-137, score-0.172]

61 one can give two different vertices very large vertex weights and so implicitly enforce that they will be in different partitions. [sent-140, score-0.129]

62 1, 0, |V | f − median(f )1 1 − gmax, K−1 (f ) − gmin, K−1 (f ) |V | 1− 1 p 1 1 p − gmax, K−1 (f ) − gmin, K−1 (f ) |V | Hard Cheeger cut e,f |p vol(V ) gmax,α (f ) − gmin,α (f ) Trunc. [sent-143, score-0.459]

63 Cheeger cut 1 1 i=1  0, if min{|A|, |A|} < K,  min{|A|, |A|}  −(K − 1), else. [sent-144, score-0.459]

64 Table 1: Examples of balancing set functions and their continuous counterpart. [sent-145, score-0.168]

65 For the hard balanced and hard Cheeger cut we have unit vertex weights, that is ei ≡ 1. [sent-146, score-0.976]

66 For that we deﬁne the functions gmax,α and gmin,α as: n gmax,α (f ) = max ρ, f 0 ≤ ρi ≤ ei , ∀ i = 1, . [sent-148, score-0.135]

67 , n, ρi = α vol(V ) i=1 n and the weighted p-mean wmeanp (f ) is deﬁned as wmeanp (f ) = inf a∈R i=1 ei |fi − a|p . [sent-154, score-0.207]

68 Both functions can be easily evaluated by sorting the componentwise product ei fi . [sent-156, score-0.262]

69 For the Cheeger and Normalized p-cut family and the truncated Cheeger cut the functions S are convex ˆ and not necessarily the Lovasz extension of the induced set functions S (ﬁrst case in Theorem 3. [sent-164, score-0.73]

70 However, in In the case of hard balanced and hard Cheeger cut the set function S ˆ both cases we know an explicit decomposition of the set function S into a difference of submodular functions and thus their Lovasz extension S can be written as a difference of the convex functions. [sent-167, score-1.277]

71 4 Minimization of Ratios of Non-negative Differences of Convex Functions In [14], the problem of computing the optimal Cheeger cut partition is formulated as a nonlinear eigenproblem. [sent-169, score-0.53]

72 We propose a general scheme called RatioDCA for minimizing ratios of non-negative differences of convex functions and thus generalizes Algorithm 1 of [14] which could handle only ratios of convex functions. [sent-175, score-0.322]

73 However, we will show that for every balanced graph cut criterion our algorithm improves a given partition or it terminates directly. [sent-177, score-0.907]

74 Note that such types of algorithms have been considered for speciﬁc graph cut criteria [23, 22, 2]. [sent-178, score-0.591]

75 5 Figure 1: Left: Illustration of different balancing functions (rescaled so that they attain value |V |/2 at |V |/2). [sent-179, score-0.15]

76 1 has the form minf ∈RV 1 2 n i,j=1 wij |fi − fj | , (2) S(f ) where S is one-homogeneous and either convex or the Lovasz extension of a non-negative symmetric set function. [sent-185, score-0.337]

77 3 the Lovasz extension of any set function can be written as a difference of one-homogeneous convex functions. [sent-187, score-0.204]

78 We refer to the convex optimization problem 2 which has to be solved at each step in RatioDCA (line 4) as the inner problem. [sent-195, score-0.139]

79 In the balanced graph cut problem (2) we minimize implicitly over non-constant functions. [sent-202, score-0.802]

80 1 For every balanced graph cut problem, the RatioDCA converges to a non-constant f ∗ given that the initial vector f 0 is non-constant. [sent-205, score-0.817]

81 Now we are ready to state the following key property of our balanced graph clustering algorithm. [sent-206, score-0.411]

82 ˆ ˆ S(B) S(A) The above “improvement theorem” implies that we can use the result of any other graph partitioning method as initialization. [sent-212, score-0.147]

83 Let us restrict our attention to the case where R(f ) = R1 (f ) = 2 i,j=1 wij |fi − fj | and S2 = 0. [sent-217, score-0.127]

84 , for the tight relaxations of the Cheeger cut, normalized cut and truncated Cheeger cut families. [sent-220, score-1.07]

85 Hence, the inner problem of the RatioDCA algorithm (line 4) has the form n f k+1 = arg min{ u 2 ≤1 1 wij |fi − fj | − λk u, s1 (f k ) }. [sent-221, score-0.173]

86 In the supplementary material, we also consider the inner problem of RatioDCA for the tight relaxation of the hard balanced cut. [sent-236, score-0.462]

87 Although, in this case we have to deal with S2 = 0 in the inner problem of RatioDCA, we can derive a similar PDHG method since the objective function is still the a sum of convex terms. [sent-237, score-0.125]

88 7 5 Experiments In a ﬁrst experiment, we study the inﬂuence of the different balancing criteria on the obtained clustering. [sent-238, score-0.128]

89 For better interpretation, we report Figure 2: From left to right: Cheeger 1-cut, Normalized 1-cut, truncated Cheeger cut (TCC), hard balanced cut (HBC), hard Cheeger cut (HCC). [sent-242, score-1.756]

90 all resulting partitions with respect to all balanced graph cut criteria, cut and the size of the largest component in the following table. [sent-246, score-1.28]

91 The parameter for truncated, hard Cheeger cut and hard balanced cut is set to K = 200. [sent-247, score-1.265]

92 One observes that the normalized 1-cut results in a less balanced partition but with a much smaller cut than the Cheeger 1-cut, which is itself less balanced than the hard Cheeger cut. [sent-248, score-1.064]

93 The latter is fully balanced but has an even higher cut. [sent-249, score-0.231]

94 The truncated Cheeger cut has a smaller cut than the hard balanced cut but its partition is not feasible. [sent-250, score-1.741]

95 Note that the hard balanced cut is similar to the normalized 1-cut but achieves smaller cut at the prize of a larger maximal component. [sent-251, score-1.249]

96 9298, 70000 and 630000 points) in the same setting as in [14] with no vertex weights, that is ei = 1, ∀i ∈ V . [sent-290, score-0.17]

97 As clustering criterion for multi-partitioning we use the M cut(Ci ,Ci ) multicut version of the normalized 1-cut, given as RCut(C1 , . [sent-291, score-0.154]

98 Fast spectral methods for ratio cut partitioning and clustering. [sent-429, score-0.6]

99 An inverse power method for nonlinear eigenproblems with applications in 1u spectral clustering and sparse pca. [sent-437, score-0.175]

100 Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. [sent-442, score-0.147]

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We argue that online submodular set cover, as we have deﬁned it, is an interesting and useful model for ranking and repeated active learning problems. In a search result ranking problem, after presenting search results to a user we can obtain implicit feedback from this user (e.g., clicks, time spent viewing each result) to determine which results were actually relevant. We can then construct an objective F t (S) such that F t (S) ≥ 1 iff S covers or summarizes the relevant results. Alternatively, we can avoid explicitly constructing an objective by considering the bandit version of the problem t where we only observe the values F t (Si ). For example, if the user clicked on k total results then t we can let F (Si ) ci /k where ci ≤ k is the number of results in the subset Si which were clicked. Note that the user may click an arbitrary set of results in an arbitrary order, and the user’s decision whether or not to click a result may depend on previously viewed and clicked results. All that we assume is that there is some unknown submodular function explaining the click counts. If the user clicks on a small number of very early results, then coverage is achieved quickly and the ordering is desirable. This coverage objective makes sense if we assume that the set of results the user clicked are of roughly equal importance and together summarize the results of interest to the user. In the medical diagnosis application, we can deﬁne F t (S) to be proportional to the number of candidate diseases which are eliminated after performing the set of tests S on patient t. If we assume that a particular test result always eliminates a ﬁxed set of candidate diseases, then this function is submodular. Speciﬁcally, this objective is the reduction in the size of the version space [5, 6]. Other active learning problems can also be phrased in terms of satisfying a submodular coverage constraint including problems that allow for noise [7]. Note that, as in the search result ranking problem, F t is not initially known but can be inferred after we have chosen S t and suffered loss (F t , S t ). 2 Background and Related Work Recently, Azar and Gamzu [8] extended the O(ln 1/ ) greedy approximation algorithm for submodular set cover to the more general problem of minimizing the average cover time of a set of objectives. Here is the smallest non-zero gain of all the objectives. Azar and Gamzu [8] call this problem ranking with submodular valuations. More formally, we have a known set of functions F1 , F2 , . . . , Fm each with an associated weight wi . The goal is then to choose a permutation S of m the ground set V to minimize i=1 wi (Fi , S). The ofﬂine approximation algorithm for ranking with submodular valuations will be a crucial tool in our analysis of online submodular set cover. In particular, this ofﬂine algorithm can viewed as constructing the best single permutation S for a sequence of objectives F 1 , F 2 . . . F T in hindsight (i.e., after all the objectives are known). Recently the ranking with submodular valuations problem was extended to metric costs [9]. Online learning is a well-studied problem [10]. In one standard setting, the online learning algorithm has a collection of actions A, and at each time step t the algorithm picks an action S t ∈ A. The learning algorithm then receives a loss function t , and the algorithm incurs the loss value for the action it chose t (S t ). We assume t (S t ) ∈ [0, 1] but make no other assumptions about the form of loss. The performance of an online learning algorithm is often measured in terms of regret, the difference between the loss incurred by the algorithm and the loss of the best single ﬁxed action T T in hindsight: R = t=1 t (S t ) − minS∈A t=1 t (S). There are randomized algorithms which guarantee E[R] ≤ T ln |A| for adversarial sequences of loss functions [11]. Note that because 2 E[R] = o(T ) the per round regret approaches zero. In the bandit version of this problem the learning algorithm only observes t (S t ) [12]. Our problem ﬁts in this standard setting with A chosen to be the set of all ground set permutations (v1 , v2 , . . . vn ) and t (S t ) (F t , S t )/n. However, in this case A is very large so standard online learning algorithms which keep weight vectors of size |A| cannot be directly applied. Furthermore, our problem generalizes an NP-hard ofﬂine problem which has no polynomial time approximation scheme, so it is not likely that we will be able to derive any efﬁcient algorithm with o(T ln |A|) regret. We therefore instead consider α-regret, the loss incurred by the algorithm as compared to α T T times the best ﬁxed prediction. Rα = t=1 t (S t ) − α minS∈A t=1 t (S). α-regret is a standard notion of regret for online versions of NP-hard problems. If we can show Rα grows sub linearly with T then we have shown loss converges to that of an ofﬂine approximation with ratio α. Streeter and Golovin [13] give online algorithms for the closely related problems of submodular function maximization and min-sum submodular set cover. In online submodular function maximization, the learning algorithm selects a set S t with |S t | ≤ k before F t is revealed, and the goal is to maximize t F t (S t ). This problem differs from ours in that our problem is a loss minimization problem as opposed to an objective maximization problem. Online min-sum submodular set cover is similar to online submodular set cover except the loss is not cover time but rather n ˆ(F t , S t ) t max(1 − F t (Si ), 0). (2) i=0 t t Min-sum submodular set cover penalizes 1 − F t (Si ) where submodular set cover uses I(F t (Si ) < 1). We claim that for certain applications the hard threshold makes more sense. For example, in repeated active learning problems minimizing t (F t , S t ) naturally corresponds to minimizing the number of questions asked. Minimizing t ˆ(F t , S t ) does not have this interpretation as it charges less for questions asked when F t is closer to 1. One might hope that minimizing could be reduced to or shown equivalent to minimizing ˆ. This is not likely to be the case, as the approximation algorithm of Streeter and Golovin [13] does not carry over to online submodular set cover. Their online algorithm is based on approximating an ofﬂine algorithm which greedily maximizes t min(F t (S), 1). Azar and Gamzu [8] show that this ofﬂine algorithm, which they call the cumulative greedy algorithm, does not achieve a good approximation ratio for average cover time. Radlinski et al. [14] consider a special case of online submodular function maximization applied to search result ranking. In their problem the objective function is assumed to be a binary valued submodular function with 1 indicating the user clicked on at least one document. The goal is then to maximize the number of queries which receive at least one click. For binary valued functions ˆ and are the same, so in this setting minimizing the number of documents a user must view before clicking on a result is a min-sum submodular set cover problem. Our results generalize this problem to minimizing the number of documents a user must view before some possibly non-binary submodular objective is met. With non-binary objectives we can incorporate richer implicit feedback such as multiple clicks and time spent viewing results. Slivkins et al. [15] generalize the results of Radlinski et al. [14] to a metric space bandit setting. Our work differs from the online set cover problem of Alon et al. [16]; this problem is a single set cover problem in which the items that need to be covered are revealed one at a time. Kakade et al. [17] analyze general online optimization problems with linear loss. If we assume that the functions F t are all taken from a known ﬁnite set of functions F then we have linear loss over a |F| dimensional space. However, this approach gives poor dependence on |F|. 3 Ofﬂine Analysis In this work we present an algorithm for online submodular set cover which extends the ofﬂine algorithm of Azar and Gamzu [8] for the ranking with submodular valuations problem. Algorithm 1 shows this ofﬂine algorithm, called the adaptive residual updates algorithm. Here we use T to denote the number of objective functions and superscript t to index the set of objectives. This notation is chosen to make the connection to the proceeding online algorithm clear: our online algorithm will approximately implement Algorithm 1 in an online setting, and in this case the set of objectives in 3 Algorithm 1 Ofﬂine Adaptive Residual Input: Objectives F 1 , F 2 , . . . F T Output: Sequence S1 ⊂ S2 ⊂ . . . Sn S0 ← ∅ for i ← 1 . . . n do v ← argmax t δ(F t , Si−1 , v) v∈V Si ← Si−1 + v end for Figure 1: Histograms used in ofﬂine analysis the ofﬂine algorithm will be the sequence of objectives in the online problem. The algorithm is a greedy algorithm similar to the standard algorithm for submodular set cover. The crucial difference is that instead of a normal gain term of F t (S + v) − F t (S) it uses a relative gain term δ(F t , S, v) min( F 0 t (S+v)−F t (S) , 1) 1−F t (S) if F (S) < 1 otherwise The intuition is that (1) a small gain for F t matters more if F t is close to being covered (F t (S) close to 1) and (2) gains for F t with F t (S) ≥ 1 do not matter as these functions are already covered. The main result of Azar and Gamzu [8] is that Algorithm 1 is approximately optimal. t Theorem 1 ([8]). The loss t (F , S) of the sequence produced by Algorithm 1 is within 4(ln(1/ ) + 2) of that of any other sequence. We note Azar and Gamzu [8] allow for weights for each F t . We omit weights for simplicity. Also, Azar and Gamzu [8] do not allow the sequence S to contain duplicates while we do–selecting a ground set element twice has no beneﬁt but allowing them will be convenient for the online analysis. The proof of Theorem 1 involves representing solutions to the submodular ranking problem as histograms. Each histogram is deﬁned such that the area of the histogram is equal to the loss of the corresponding solution. The approximate optimality of Algorithm 1 is shown by proving that the histogram for the solution it ﬁnds is approximately contained within the histogram for the optimal solution. In order to convert Algorithm 1 into an online algorithm, we will need a stronger version of Theorem 1. Speciﬁcally, we will need to show that when there is some additive error in the greedy selection rule Algorithm 1 is still approximately optimal. For the optimal solution S ∗ = argminS∈V n t (F t , S) (V n is the set of all length n sequences of ground set elements), deﬁne a histogram h∗ with T columns, one for each function F t . Let the tth column have with width 1 and height equal to (F t , S ∗ ). Assume that the columns are ordered by increasing cover time so that the histogram is monotone non-decreasing. Note that the area of this histogram is exactly the loss of S ∗ . For a sequence of sets ∅ = S0 ⊆ S1 ⊆ . . . Sn (e.g., those found by Algorithm 1) deﬁne a corresponding sequence of truncated objectives ˆ Fit (S) min( F 1 t (S∪Si−1 )−F t (Si−1 ) , 1) 1−F t (Si−1 ) if F t (Si−1 ) < 1 otherwise ˆ Fit (S) is essentially F t except with (1) Si−1 given “for free”, and (2) values rescaled to range ˆ ˆ between 0 and 1. We note that Fit is submodular and that if F t (S) ≥ 1 then Fit (S) ≥ 1. In this ˆ t is an easier objective than F t . Also, for any v, F t ({v}) − F t (∅) = δ(F t , Si−1 , v). In ˆ ˆ sense Fi i i ˆ other words, the gain of Fit at ∅ is the normalized gain of F t at Si−1 . This property will be crucial. ˆ ˆ ˆ We next deﬁne truncated versions of h∗ : h1 , h2 , . . . hn which correspond to the loss of S ∗ for the ˆ ˆ t . For each j ∈ 1 . . . n, let hi have T columns of height j easier covering problems involving Fi t ∗ t ∗ ˆ ˆ with the tth such column of width Fi (Sj ) − Fi (Sj−1 ) (some of these columns may have 0 width). ˆ Assume again the columns are ordered by height. Figure 1 shows h∗ and hi . ∗ We assume without loss of generality that F t (Sn ) ≥ 1 for every t (clearly some choice of S ∗ ∗ contains no duplicates, so under our assumption that F t (V ) ≥ 1 we also have F t (Sn ) ≥ 1). Note 4 ˆ that the total width of hi is then the number of functions remaining to be covered after Si−1 is given ˆ for free (i.e., the number of F t with F t (Si−1 ) < 1). It is not hard to see that the total area of hi is ˆ(F t , S ∗ ) where ˆ is the loss function for min-sum submodular set cover (2). From this we know ˆ l i t ˆ hi has area less than h∗ . In fact, Azar and Gamzu [8] show the following. ˆ ˆ Lemma 1 ([8]). hi is completely contained within h∗ when hi and h∗ are aligned along their lower right boundaries. We need one ﬁnal lemma before proving the main result of this section. For a sequence S deﬁne Qi = t δ(F t , Si−1 , vi ) to be the total normalized gain of the ith selected element and let ∆i = n t j=i Qj be the sum of the normalized gains from i to n. Deﬁne Πi = |{t : F (Si−1 ) < 1}| to be the number of functions which are still uncovered before vi is selected (i.e., the loss incurred at step i). [8] show the following result relating ∆i to Πi . Lemma 2 ([8]). For any i, ∆i ≤ (ln 1/ + 2)Πi We now state and prove the main result of this section, that Algorithm 1 is approximately optimal even when the ith greedy selection is preformed with some additive error Ri . This theorem shows that in order to achieve low average cover time it sufﬁces to approximately implement Algorithm 1. Aside from being useful for converting Algorithm 1 into an online algorithm, this theorem may be useful for applications in which the ground set V is very large. In these situations it may be possible to approximate Algorithm 1 (e.g., through sampling). Streeter and Golovin [13] prove similar results for submodular function maximization and min-sum submodular set cover. Our result is similar, but t the proof is non trivial. The loss function is highly non linear with respect to changes in F t (Si ), so it is conceivable that small additive errors in the greedy selection could have a large effect. The analysis of Im and Nagarajan [9] involves a version of Algorithm 1 which is robust to a sort of multplicative error in each stage of the greedy selection. Theorem 2. Let S = (v1 , v2 , . . . vn ) be any sequence for which δ(F t , Si−1 , vi ) + Ri ≥ max Then t δ(F t , Si−1 , v) v∈V t (F t , S t ) ≤ 4(ln 1/ + 2) t (F t , S ∗ ) + n t i Ri Proof. Let h be a histogram with a column for each Πi with Πi = 0. Let γ = (ln 1/ + 2). Let the ith column have width (Qi + Ri )/(2γ) and height max(Πi − j Rj , 0)/(2(Qi + Ri )). Note that Πi = 0 iff Qi + Ri = 0 as if there are functions not yet covered then there is some set element with non zero gain (and vice versa). The area of h is i:Πi =0 max(Πi − j Rj , 0) 1 1 (Qi + Ri ) ≥ 2γ 2(Qi + Ri ) 4γ (F t , S) − t n 4γ Rj j ˆ Assume h and every hi are aligned along their lower right boundaries. We show that if the ith ˆ column of h has non-zero area then it is contained within hi . Then, it follows from Lemma 1 that h ∗ is contained within h , completing the proof. Consider the ith column in h. Assume this column has non zero area so Πi ≥ j Rj . This column is at most (∆i + j≥i Rj )/(2γ) away from the right hand boundary. To show that this column is in ˆ hi it sufﬁces to show that after selecting the ﬁrst k = (Πi − j Rj )/(2(Qi + Ri )) items in S ∗ we ˆ ∗ ˆ still have t (1 − Fit (Sk )) ≥ (∆i + j≥i Rj )/(2γ) . The most that t Fit can increase through ˆ the addition of one item is Qi + Ri . Therefore, using the submodularity of Fit , ˆ ∗ Fit (Sk ) − t Therefore t (1 ˆ Fit (∅) ≤ k(Qi + Ri ) ≤ Πi /2 − t ˆ ∗ − Fit (Sk )) ≥ Πi /2 + j Rj /2 since Rj /2 ≥ ∆i /(2γ) + Πi /2 + Rj /2 j t (1 ˆ − Fit (∅)) = Πi . Using Lemma 2 Rj /2 ≥ (∆i + j j 5 Rj )/(2γ) j≥i Algorithm 2 Online Adaptive Residual Input: Integer T Initialize n online learning algorithms E1 , E2 , . . . En with A = V for t = 1 → T do t ∀i ∈ 1 . . . n predict vi with Ei t t S t ← (v1 , . . . vn ) Receive F t , pay loss l(F t , S t ) t For Ei , t (v) ← (1 − δ(F t , Si−1 , v)) end for 4 Figure 2: Ei selects the ith element in S t . Online Analysis We now show how to convert Algorithm 1 into an online algorithm. We use the same idea used by Streeter and Golovin [13] and Radlinski et al. [14] for online submodular function maximization: we run n copies of some low regret online learning algorithm, E1 , E2 , . . . En , each with action space A = V . We use the ith copy Ei to select the ith item in each predicted sequence S t . In other 1 2 T words, the predictions of Ei will be vi , vi , . . . vi . Figure 2 illustrates this. Our algorithm assigns loss values to each Ei so that, assuming Ei has low regret, Ei approximately implements the ith greedy selection in Algorithm 1. Algorithm 2 shows this approach. Note that under our assumption that F 1 , F 2 , . . . F T is chosen by an oblivious adversary, the loss values for the ith copy of the online algorithm are oblivious to the predictions of that run of the algorithm. Therefore we can use any algorithm for learning against an oblivious adversary. Theorem 3. Assume we use as a subroutine an online prediction algorithm with expected regret √ √ E[R] ≤ T ln n. Algorithm 2 has expected α-regret E[Rα ] ≤ n2 T ln n for α = 4(ln(1/ ) + 2) 1 2 T Proof. Deﬁne a meta-action vi for the sequence of actions chosen by Ei , vi = (vi , vi , . . . vi ). We ˜ ˜ t t t t ˜ can extend the domain of F to allow for meta-actions F (S ∪ {ˆi }) = F (S ∪ {vi }). Let S be v ˜ the sequence of meta actions S = (v1 , v2 , . . . vn ). Let Ri be the regret of Ei . Note that from the ˜ ˜ ˜ deﬁnition of regret and our choice of loss values we have that ˜ δ(F t , Si−1 , v) − max v∈V t ˜ δ(F t , Si−1 , vi ) = Ri ˜ t ˜ Therefore, S approximates the greedy solution in the sense required by Theorem 2. Theorem 2 did not require that S be constructed V . From Theorem 2 we then have ˜ (F t , S) ≤ α (F t , S t ) = t t The expected α-regret is then E[n i (F t , S ∗ ) + n t Ri i √ Ri ] ≤ n2 T ln n We describe several variations and extensions of this analysis, some of which mirror those for related work [13, 14, 15]. Avoiding Duplicate Items Since each run of the online prediction algorithm is independent, Algorithm 2 may select the same ground set element multiple times. This drawback is easy to ﬁx. We can simply select any arbitrary vi ∈ Si−1 if Ei selects a vi ∈ Si−i . This modiﬁcation does not affect / the regret guarantee as selecting a vi ∈ Si−1 will always result in a gain of zero (loss of 1). Truncated Loss In some applications we only care about the ﬁrst k items in the sequence S t . For these applications it makes sense to consider a truncated version of l(F t , S t ) with parameter k k (F t , S t ) t t min {k} ∪ {|Si | : F t (Si ) ≥ 1} This is cover time computed up to the kth element in S t . The analysis for Theorem 2 also shows k k (F t , S t ) ≤ 4(ln 1/ + 2) t (F t , S ∗ ) + k i=1 Ri . The corresponding regret bound is then t 6 √ k 2 T ln n. Note here we are bounding truncated loss t k (F t , S t ) in terms of untruncated loss t ∗ 2 2 t (F , S ). In this sense this bound is weaker. However, we replace n with k which may be much smaller. Algorithm 2 achieves this bound simultaneously for all k. Multiple Objectives per Round Consider a variation of online submodular set cover in which int t t stead of receiving a single objective F t each round we receive a batch of objectives F1 , F2 , . . . Fm m t t and incur loss i=1 (Fi , S ). In other words, each rounds corresponds to a ranking with submodular valuations problem. It is easy to extend Algorithm 2 to this√setting by using 1 − m t (1/m) i=1 δ(Fit , Si−1 , v) for the loss of action v in Ei . We then get O(k 2 mL∗ ln n+k 2 m ln n) T m ∗ total regret where L = t=1 i=1 (Fit , S ∗ ) (Section 2.6 of [10]). Bandit Setting Consider a setting where instead of receiving full access to F t we only observe t t t the sequence of objective function values F t (S1 ), F t (S2 ), . . . F t (Sn ) (or in the case of multiple t t objectives per round, Fi (Sj ) for every i and j). We can extend Algorithm 2 to this setting using a nonstochastic multiarmed bandits algorithm [12]. We note duplicate removal becomes more subtle in the bandit setting: should we feedback a gain of zero when a duplicate is selected or the gain of the non-duplicate replacement? We propose either is valid if replacements are chosen obliviously. Bandit Setting with Expert Advice We can further generalize the bandit setting to the contextual bandit setting [18] (e.g., the bandit setting with expert advice [12]). Say that we have access at time step t to predictions from a set of m experts. Let vj be the meta action corresponding to the sequence ˜ ˜ of predictions from the jth expert and V be the set of all vj . Assume that Ei guarantees low regret ˜ ˜ with respect to V t t δ(F t , Si−1 , vi ) + Ri ≥ max v ∈V ˜ ˜ t t δ(F t , Si−1 , v ) ˜ (3) t where we have extended the domain of each F t to include meta actions as in the proof of Theorem ˜ 3. Additionally assume that F t (V ) ≥ 1 for every t. In this case we can show t k (F t , S t ) ≤ √ k m t ∗ minS ∗ ∈V m t (F , S ) + k i=1 Ri . The Exp4 algorithm [12] has Ri = O( nT ln m) giving ˜ √ total regret O(k 2 nT ln m). Experts may use context in forming recommendations. For example, in a search ranking problem the context could be the query. 5 5.1 Experimental Results Synthetic Example We present a synthetic example for which the online cumulative greedy algorithm [13] fails, based on the example in Azar and Gamzu [8] for the ofﬂine setting. Consider an online ad placement problem where the ground set V is a set of available ad placement actions (e.g., a v ∈ V could correspond to placing an ad on a particular web page for a particular length of time). On round t, we receive an ad from an advertiser, and our goal is to acquire λ clicks for the ad using as few t advertising actions as possible. Deﬁne F t (Si ) to be min(ct , λ)/λ where ct is number of clicks i i t acquired from the ad placement actions Si . Say that we have n advertising actions of two types: 2 broad actions and n − 2 narrow actions. Say that the ads we receive are also of two types. Common type ads occur with probability (n − 1)/n and receive 1 and λ − 1 clicks respectively from the two broad actions and 0 clicks from narrow actions. Uncommon type ads occur with probability 1/n and receive λ clicks from one randomly chosen narrow action and 0 clicks from all other actions. Assume λ ≥ n2 . Intuitively broad actions could correspond to ad placements on sites for which many ads are relevant. The optimal strategy giving an average cover time O(1) is to ﬁrst select the two broad actions covering all common ads then select the narrow actions in any order. However, the ofﬂine cumulative greedy algorithm will pick all narrow actions before picking the broad action with gain 1 giving average cover time O(n). The left of Figure 3 shows average cover time for our proposed algorithm and the cumulative greedy algorithm of [13] on the same sequences of random objectives. For this example we use n = 25 and the bandit version of the problem with the Exp3 algorithm [12]. We also plot the average cover times for ofﬂine solutions as baselines. As seen in the ﬁgure, the cumulative algorithms converge to higher average cover times than the adaptive residual algorithms. Interestingly, the online cumulative algorithm does better than the ofﬂine cumulative algorithm: it seems added randomization helps. 7 Figure 3: Average cover time 5.2 Repeated Active Learning for Movie Recommendation Consider a movie recommendation website which asks users a sequence of questions before they are given recommendations. We deﬁne an online submodular set cover problem for choosing sequences of questions in order to quickly eliminate a large number of movies from consideration. This is similar conceptually to the diagnosis problem discussed in the introduction. Deﬁne the ground set V to be a set of questions (for example “Do you want to watch something released in the past 10 years?” or “Do you want to watch something from the Drama genre?”). Deﬁne F t (S) to be proportional to the number of movies eliminated from consideration after asking the tth user S. Speciﬁcally, let H be the set of all movies in our database and V t (S) be the subset of movies consistent with the tth user’s responses to S. Deﬁne F t (S) min(|H \ V t (S)|/c, 1) where c is a constant. F t (S) ≥ iff after asking the set of questions S we have eliminated at least c movies. We set H to be a set of 11634 movies available on Netﬂix’s Watch Instantly service and use 803 questions based on those we used for an ofﬂine problem [7]. To simulate user responses to questions, on round t we randomly select a movie from H and assume the tth user answers questions consistently with this movie. We set c = |H| − 500 so the goal is to eliminate about 95% of all movies. We evaluate in the full information setting: this makes sense if we assume we receive as feedback the movie the user actually selected. As our online prediction subroutine we tried Normal-Hedge [19], a second order multiplicative weights method [20], and a version of multiplicative weights for small gains using the doubling trick (Section 2.6 of [10]). We also tried a heuristic modiﬁcation of Normal-Hedge which ﬁxes ct = 1 for a ﬁxed, more aggressive learning rate than theoretically justiﬁed. The right of Figure 3 shows average cover time for 100 runs of T = 10000 iterations. Note the different scale in the bottom row–these methods performed signiﬁcantly worse than Normal-Hedge. The online cumulative greedy algorithm converges to a average cover time close to but slightly worse than that of the adaptive greedy method. The differences are more dramatic for prediction subroutines that converge slowly. The modiﬁed Normal-Hedge has no theoretical justiﬁcation, so it may not generalize to other problems. For the modiﬁed Normal-Hedge the ﬁnal average cover times are 7.72 adaptive and 8.22 cumulative. The ofﬂine values are 6.78 and 7.15. 6 Open Problems It is not yet clear what practical value our proposed approach will have for web search result ranking. A drawback to our approach is that we pick a ﬁxed order in which to ask questions. For some problems it makes more sense to consider adaptive strategies [5, 6]. Acknowledgments This material is based upon work supported in part by the National Science Foundation under grant IIS-0535100, by an Intel research award, a Microsoft research award, and a Google research award. 8 References [1] H. Lin and J. Bilmes. A class of submodular functions for document summarization. In HLT, 2011. ´ [2] D. Kempe, J. Kleinberg, and E. Tardos. Maximizing the spread of inﬂuence through a social network. In KDD, 2003. [3] A. Krause, A. Singh, and C. Guestrin. Near-optimal sensor placements in Gaussian processes: Theory, efﬁcient algorithms and empirical studies. JMLR, 2008. [4] L.A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2(4), 1982. [5] D. Golovin and A. Krause. Adaptive submodularity: A new approach to active learning and stochastic optimization. In COLT, 2010. [6] Andrew Guillory and Jeff Bilmes. Interactive submodular set cover. In ICML, 2010. [7] Andrew Guillory and Jeff Bilmes. Simultaneous learning and covering with adversarial noise. In ICML, 2011. [8] Yossi Azar and Iftah Gamzu. Ranking with Submodular Valuations. In SODA, 2011. [9] S. Im and V. Nagarajan. Minimum Latency Submodular Cover in Metrics. ArXiv e-prints, October 2011. [10] N. Cesa-Bianchi and G. Lugosi. Prediction, learning, and games. Cambridge University Press, 2006. [11] Y. Freund and R. Schapire. A desicion-theoretic generalization of on-line learning and an application to boosting. In Computational learning theory, pages 23–37, 1995. [12] P. Auer, N. Cesa-Bianchi, Y. Freund, and R.E. Schapire. The nonstochastic multiarmed bandit problem. SIAM Journal on Computing, 32(1):48–77, 2003. [13] M. Streeter and D. Golovin. An online algorithm for maximizing submodular functions. In NIPS, 2008. [14] F. Radlinski, R. Kleinberg, and T. Joachims. Learning diverse rankings with multi-armed bandits. In ICML, 2008. [15] A. Slivkins, F. Radlinski, and S. Gollapudi. Learning optimally diverse rankings over large document collections. In ICML, 2010. [16] N. Alon, B. Awerbuch, and Y. Azar. The online set cover problem. In STOC, 2003. [17] Sham M. Kakade, Adam Tauman Kalai, and Katrina Ligett. Playing games with approximation algorithms. In STOC, 2007. [18] J. Langford and T. Zhang. The epoch-greedy algorithm for contextual multi-armed bandits. In NIPS, 2007. [19] K. Chaudhuri, Y. Freund, and D. Hsu. A parameter-free hedging algorithm. In NIPS, 2009. [20] N. Cesa-Bianchi, Y. Mansour, and G. Stoltz. Improved second-order bounds for prediction with expert advice. Machine Learning, 2007. 9

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