nips nips2011 nips2011-204 knowledge-graph by maker-knowledge-mining

204 nips-2011-Online Learning: Stochastic, Constrained, and Smoothed Adversaries

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Author: Alexander Rakhlin, Karthik Sridharan, Ambuj Tewari

Abstract: Learning theory has largely focused on two main learning scenarios: the classical statistical setting where instances are drawn i.i.d. from a ﬁxed distribution, and the adversarial scenario wherein, at every time step, an adversarially chosen instance is revealed to the player. It can be argued that in the real world neither of these assumptions is reasonable. We deﬁne the minimax value of a game where the adversary is restricted in his moves, capturing stochastic and non-stochastic assumptions on data. Building on the sequential symmetrization approach, we deﬁne a notion of distribution-dependent Rademacher complexity for the spectrum of problems ranging from i.i.d. to worst-case. The bounds let us immediately deduce variation-type bounds. We study a smoothed online learning scenario and show that exponentially small amount of noise can make function classes with inﬁnite Littlestone dimension learnable. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We deﬁne the minimax value of a game where the adversary is restricted in his moves, capturing stochastic and non-stochastic assumptions on data. [sent-11, score-0.755]

2 Building on the sequential symmetrization approach, we deﬁne a notion of distribution-dependent Rademacher complexity for the spectrum of problems ranging from i. [sent-12, score-0.348]

3 We study a smoothed online learning scenario and show that exponentially small amount of noise can make function classes with inﬁnite Littlestone dimension learnable. [sent-17, score-0.323]

4 1 Introduction In the papers [1, 10, 11], an array of tools has been developed to study the minimax value of diverse sequential problems under the worst-case assumption on Nature. [sent-18, score-0.29]

5 The process of sequential symmetrization emerged as a key technique for dealing with complicated nested minimax expressions. [sent-20, score-0.411]

6 By not placing any restrictions on Nature, we recover the worst-case results of [10]. [sent-32, score-0.253]

7 Between these two endpoints of the spectrum, particular assumptions on the adversary yield interesting bounds on the minimax value of the associated problem. [sent-33, score-0.62]

8 Once again, the sequential symmetrization technique arises as the main tool for dealing with the minimax value, but the proofs require more care than in the i. [sent-34, score-0.411]

9 1 Adapting the game-theoretic language, we will think of the learner and the adversary as the two players of a zero-sum repeated game. [sent-38, score-0.563]

10 Adversary’s moves will be associated with “data”, while the moves of the learner – with a function or a parameter. [sent-39, score-0.244]

11 In fact, there is a well-developed theory of minimax estimation when restrictions are put on either the choice of the adversary or the allowed estimators by the player. [sent-41, score-0.902]

12 The main contribution of this paper is the development of tools for the analysis of online scenarios where the adversary’s moves are restricted in various ways. [sent-46, score-0.239]

13 , T , the learner chooses ft ∈ F, the adversary simultaneously picks xt ∈ X , and the learner suffers loss ft (xt ). [sent-54, score-1.461]

14 The goal of the learner is to minimize regret, deﬁned as T T t=1 ft (xt ) − inf f ∈F t=1 f (xt ). [sent-55, score-0.36]

15 It is a standard fact that simultaneity of the choices can be formalized by the ﬁrst player choosing a mixed strategy; the second player then picks an action based on this mixed strategy, but not on its realization. [sent-56, score-0.322]

16 This is achieved by deﬁning restrictions on the adversary, that is, subsets of “allowed” distributions for each round. [sent-63, score-0.298]

17 These restrictions limit the scope of available mixed strategies for the adversary. [sent-64, score-0.325]

18 A restriction P1:T on the adversary is a sequence P1 , . [sent-66, score-0.541]

19 Note that the restrictions depend on the past moves of the adversary, but not on those of the player. [sent-70, score-0.368]

20 (1) A worst-case adversary is deﬁned by vacuous restrictions Pt (x1:t−1 ) = P. [sent-73, score-0.723]

21 (2) A constrained adversary is deﬁned by Pt (x1:xt−1 ) being the set of all distributions supported on the set {x ∈ X : Ct (x1 , . [sent-75, score-0.559]

22 (3) A smoothed adversary picks the worst-case sequence which gets corrupted by i. [sent-83, score-0.704]

23 Equivalently, we can view this as restrictions on the adversary who chooses the “center” (or a parameter) of the noise distribution. [sent-87, score-0.805]

24 Using techniques developed in this paper, we can also study the following adversaries (omitted due to lack of space): (4) A hybrid adversary in the supervised learning game picks the worst-case label yt , but is forced to draw the xt -variable from a ﬁxed distribution [6]. [sent-88, score-1.235]

25 adversary is deﬁned by a time-invariant restriction Pt (x1:t−1 ) = {p} for every t and some p ∈ P. [sent-92, score-0.499]

26 As in [10], the adversary is adaptive, that is, chooses pt based on the history of moves f1:t−1 and x1:t−1 . [sent-94, score-0.876]

27 At this point, the only difference from 2 the setup of [10] is in the restrictions Pt on the adversary. [sent-95, score-0.253]

28 Because these restrictions might not allow point distributions, suprema over pt ’s in (1) cannot be equivalently written as the suprema over xt ’s. [sent-96, score-1.095]

29 In the present paper, the main focus is indeed on the restrictions on the adversary, justifying our choice VT (P1:T ) for the notation. [sent-100, score-0.253]

30 Let F and X be the sets of moves for the two players, satisfying the necessary conditions for the minimax theorem to hold. [sent-107, score-0.293]

31 sup ExT ∼pT p1 ∈P1 pT ∈PT t=1 T inf Ext ∼pt [ft (xt )] − inf ft ∈F f ∈F f (xt ) . [sent-112, score-0.582]

32 (2) t=1 The nested sequence of suprema and expected values in Theorem 1 can be re-written succinctly as T VT (P1:T ) = sup Ex1 ∼p1 Ex2 ∼p2 (·|x1 ) . [sent-113, score-0.266]

33 , xT ) ∈ X T , we denote the associated conditional distributions by pt (·|x1:t−1 ). [sent-120, score-0.342]

34 We can think of the choice p as a sequence of oblivious strategies {pt : X t−1 → P}T , mapping the preﬁx x1:t−1 to a conditional t=1 distribution pt (·|x1:t−1 ) ∈ Pt (x1:t−1 ). [sent-121, score-0.413]

35 We say that a joint distribution p satisﬁes restrictions if for any t and any x1:t−1 ∈ X t−1 , pt (·|x1:t−1 ) ∈ Pt (x1:t−1 ). [sent-123, score-0.544]

36 The set of all joint distributions satisfying the restrictions is denoted by P. [sent-124, score-0.342]

37 We note that Theorem 1 cannot be deduced immediately from the analogous result in [10], as it is not clear how the restrictions on the adversary per each round come into play after applying the minimax theorem. [sent-125, score-0.975]

38 Nevertheless, it is comforting that the restrictions directly translate into the set P of oblivious strategies satisfying the restrictions. [sent-126, score-0.384]

39 There is an oblivious minimax optimal strategy for the adversary, and there is a corresponding minimax optimal strategy for the player that does not depend on its own moves. [sent-129, score-0.511]

40 The proposition holds for all online learning settings with legal restrictions P1:T , encompassing also the no-restrictions setting of worst-case online learning [10]. [sent-131, score-0.478]

41 This process can be termed sequential symmetrization and has been exploited in [1, 10, 11]. [sent-136, score-0.261]

42 The restrictions Pt , however, make sequential symmetrization considerably more involved than in the papers cited above. [sent-137, score-0.514]

43 is introduced such that xt and x′ are independent and t 1 2 3 identically distributed given “the past”. [sent-142, score-0.425]

44 When xt and x′ are understood from the context, we will use the shorthand χt (ǫ) := t χ(xt , x′ , ǫ). [sent-145, score-0.425]

45 In other words, χt selects between xt and x′ depending on the sign of ǫ. [sent-146, score-0.425]

46 However, in-between these extremes the notational complexity seems to be unavoidable if we are to employ symmetrization and obtain a version of Rademacher complexity. [sent-175, score-0.21]

47 The probability tree ρ on the “left-most” path is, therefore, deﬁned by the conditional distributions pt (·|x1:t−1 ); on the path ǫ = 1, the conditional distributions are pt (·|x′ 1:t−1 ). [sent-184, score-0.765]

48 By saying that ρ satisﬁes restrictions we then mean that p ∈ P. [sent-191, score-0.253]

49 The distribution-dependent sequential Rademacher complexity of a function class F ⊆ RX is deﬁned as T △ RT (F, p) = E(x,x′ )∼ρ Eǫ sup ǫt f (xt (ǫ)) f ∈F t=1 where ǫ = (ǫ1 , . [sent-208, score-0.334]

50 We now prove an upper bound on the value VT (P1:T ) of the game in terms of this distributiondependent sequential Rademacher complexity. [sent-215, score-0.276]

51 The minimax value is bounded as VT (P1:T ) ≤ 2 sup RT (F, p). [sent-218, score-0.333]

52 For the game with restrictions P1:T , T VT (P1:T ) ≤ 2 sup E(x,x′ )∼ρ Eǫ sup f ∈F t=1 p∈P ǫt f (xt (ǫ)) − Et−1 f (xt (ǫ)) where Et−1 denotes the conditional expectation of xt (ǫ). [sent-222, score-1.164]

53 Then T VT (P1:T ) ≤ 2 sup E(x,x′ )∼ρ Eǫ p∈P t=1 ǫt xt (ǫ) − Et−1 xt (ǫ) Suppose the adversary plays a simple random walk (e. [sent-225, score-1.481]

54 , xt−1 ) = pt (x|xt−1 ) is uniform on a unit sphere). [sent-230, score-0.268]

55 Then xt (ǫ) − Et−1 xt (ǫ) are independent increments when conditioned on the history. [sent-232, score-0.85]

56 data is precisely the classical Rademacher complexity, and further show that the distribution-dependent sequential Rademacher complexity is always upper bounded by the worst-case sequential Rademacher complexity deﬁned in [10]. [sent-238, score-0.43]

57 restrictions Pt = {p} for all t, where p is some ﬁxed distribution on X , and let ρ be the process associated with the joint distribution p = pT . [sent-243, score-0.276]

58 ,xT ∼p Eǫ sup ǫt f (xt ) (6) f ∈F t=1 is the classical Rademacher complexity. [sent-247, score-0.194]

59 Second, for any joint distribution p, T RT (F, p) ≤ RT (F), where △ RT (F) = sup Eǫ sup x ǫt f (xt (ǫ)) (7) f ∈F t=1 is the sequential Rademacher complexity deﬁned in [10]. [sent-248, score-0.518]

60 In the case of hybrid learning, adversary chooses a sequence of pairs (xt , yt ) where the instance xt ’s are i. [sent-249, score-1.063]

61 The distribution-dependent Rademacher complexity in such a hybrid case can be upper bounded by a vary natural quantity: a random average where expectation is taken over xt ’s and a supremum over Y-valued trees. [sent-253, score-0.611]

62 So, the distribution dependent Rademacher complexity itself becomes a hybrid between the classical Rademacher complexity and the worst case sequential Rademacher complexity. [sent-254, score-0.309]

63 Distribution-dependent sequential Rademacher complexity enjoys many of the nice properties satisﬁed by both classical and worst-case Rademacher complexities. [sent-256, score-0.206]

64 This notion of a cover captures the sequential complexity of a function class on a given X -valued tree x. [sent-260, score-0.208]

65 It is often useful to consider scenarios where the adversary is worst case, yet has some budget or constraint to satisfy while picking the actions. [sent-263, score-0.505]

66 Examples of such scenarios include, for instance, games where the adversary is constrained to make moves that are close in some fashion to the previous move, linear games with bounded variance, and so on. [sent-264, score-0.725]

67 Below we formulate such games quite generally through arbitrary constraints that the adversary has to satisfy on each round. [sent-265, score-0.5]

68 For a T round game consider an adversary who is only allowed to play sequences x1 , . [sent-267, score-0.754]

69 , xt ) = 1 is satisﬁed, where Ct : X t → {0, 1} represents the constraint on the sequence played so far. [sent-273, score-0.512]

70 The constrained adversary can be viewed as a stochastic adversary with restrictions on the conditional distribution at time t given by the set of all Borel △ distributions on the set Xt (x1:t−1 ) = {x ∈ X : Ct (x1 , . [sent-274, score-1.311]

71 Since this set includes all point distributions on each x ∈ Xt , the sequential complexity simpliﬁes in a way similar to worst-case adversaries. [sent-278, score-0.218]

72 Let F and X be the sets of moves for the two players, satisfying the necessary conditions for the minimax theorem to hold. [sent-284, score-0.293]

73 Then t=1 T T VT (C1:T ) = sup E p∈P t=1 inf Ext ∼pt [ft (xt )] − inf ft ∈F f ∈F f (xt ) (8) t=1 where p ranges over all distributions over sequences (x1 , . [sent-286, score-0.672]

74 The minimax value is bounded as t VT (C1:T ) ≤ 2 sup RT (F, p). [sent-298, score-0.333]

75 (x,x′ )∈T More generally, for any measurable function Mt such that Mt (f, x, x′ , ǫ) = Mt (f, x′ , x, −ǫ), T VT (C1:T ) ≤ 2 sup (x,x′ )∈T Eǫ sup f ∈F t=1 ǫt (f (xt (ǫ)) − Mt (f, x, x′ , ǫ)) . [sent-299, score-0.347]

76 The ﬁrst result says that if the adversary is not allowed to move by more than σt away from its previous average of decisions, the player has a strategy to exploit this fact and obtain lower regret. [sent-301, score-0.684]

77 In game where the move xt played by adversary at time t satisﬁes xt − t−1 τ =1 xτ 2 √ T 2 this case we can conclude that VT (C1:T ) ≤ 2 2 t=1 σt . [sent-315, score-1.583]

78 Consider the constrained game where the move xt played by adversary at time t satisﬁes maxj∈[d] xt [j] − 1 t−1 √ conclude that VT (C1:T ) ≤ 2 2 t−1 τ =1 xτ [j] ≤ σt . [sent-318, score-1.627]

79 The next Proposition gives a bound whenever the adversary is constrained to choose his decision from a small ball around the previous decision. [sent-320, score-0.514]

80 Consider the online linear optimization setting where adversary’s move at any time is close to the move during the previous time step. [sent-322, score-0.198]

81 Consider the constrained 2 game where the move xt played by adversary at time t satisﬁes xt − xt−1 2 ≤ δ . [sent-332, score-1.602]

82 Consider the constrained game where the move xt played by adversary at time t satisﬁes xt − xt−1 ∞ ≤ δ. [sent-336, score-1.602]

83 5 Application: Smoothed Adversaries The development of smoothed analysis over the past decade is arguably one of the landmarks in the study of complexity of algorithms. [sent-338, score-0.238]

84 In contrast to the overly optimistic average complexity and the overly pessimistic worst-case complexity, smoothed complexity can be seen as a more realistic measure of algorithm’s performance. [sent-339, score-0.324]

85 In their groundbreaking work, Spielman and Teng [13] showed that the smoothed running time complexity of the simplex method is polynomial. [sent-340, score-0.242]

86 The idea of limiting the power of the malicious adversary through perturbing the sequence can be traced back to Posner and Kulkarni [9]. [sent-354, score-0.512]

87 We deﬁne the smoothed online learning model as the following T -round interaction between the learner and the adversary. [sent-356, score-0.294]

88 On round t, the learner chooses ft ∈ F; the adversary simultaneously ˜ chooses xt ∈ X , which is then perturbed by some noise st ∼ σ, yielding a value xt = ω(xt , st ); and the player suffers ft (˜t ). [sent-357, score-2.197]

89 If ω keeps xt unchanged, that is ω(xt , st ) = xt , the setting is precisely ˜ the standard online learning model. [sent-360, score-1.006]

90 In the full information version, we assume that the choice xt ˜ is revealed to the player at the end of round t. [sent-361, score-0.59]

91 That is, the choice xt ∈ X deﬁnes a parameter of a mixed strategy from which a actual move ω(xt , st ) is drawn; for instance, for additive zero-mean Gaussian noise, xt deﬁnes the center of the distribution from which xt + st is drawn. [sent-363, score-1.594]

92 In other words, noise does not allow the adversary to play any desired mixed strategy. [sent-364, score-0.583]

93 Using sequential symmetrization, we deduce the following upper bound on the value of the smoothed online learning game. [sent-366, score-0.379]

94 The value of the smoothed online learning game is bounded above as T VT ≤ 2 sup E Eǫ1 . [sent-368, score-0.556]

95 sup x1 ∈X s1 ∼σ E EǫT xT ∈X sT ∼σ sup ǫt f (ω(xt , st )) f ∈F t=1 We now demonstrate how Theorem 10 can be used to show learnability for smoothed learning of threshold functions. [sent-371, score-0.607]

96 The moves of the adversary are pairs x = (z, y) with z ∈ [0, 1] and y ∈ {0, 1}, and the binary-valued function class F is deﬁned by (9) F = {fθ (z, y) = |y − 1 {z < θ}| : θ ∈ [0, 1]} , that is, every function is associated with a threshold θ ∈ [0, 1]. [sent-373, score-0.564]

97 Consider a smoothed scenario, with the z-variable of the adversarial move (z, y) perturbed by an additive uniform noise σ = Unif[−γ/2, γ/2] for some γ ≥ 0. [sent-375, score-0.356]

98 That is, the actual move revealed to the player at time t is (zt + st , yt ), with st ∼ σ. [sent-376, score-0.362]

99 For the problem of smoothed online learning of half-spaces, VT = O dT log 1 γ 3 log T + d−1 + vd−2 · 1 γ 3 d−1 where vd−2 is constant depending only on the dimension d. [sent-408, score-0.26]

100 We conclude that half spaces are online learnable in the smoothed model, since the upper bound of Proposition 13 guarantees existence of an algorithm which achieves this regret. [sent-409, score-0.366]

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1 0.93514723 232 nips-2011-Ranking annotators for crowdsourced labeling tasks

Author: Vikas C. Raykar, Shipeng Yu

Abstract: With the advent of crowdsourcing services it has become quite cheap and reasonably effective to get a dataset labeled by multiple annotators in a short amount of time. Various methods have been proposed to estimate the consensus labels by correcting for the bias of annotators with different kinds of expertise. Often we have low quality annotators or spammers–annotators who assign labels randomly (e.g., without actually looking at the instance). Spammers can make the cost of acquiring labels very expensive and can potentially degrade the quality of the consensus labels. In this paper we formalize the notion of a spammer and deﬁne a score which can be used to rank the annotators—with the spammers having a score close to zero and the good annotators having a high score close to one. 1 Spammers in crowdsourced labeling tasks Annotating an unlabeled dataset is one of the bottlenecks in using supervised learning to build good predictive models. Getting a dataset labeled by experts can be expensive and time consuming. With the advent of crowdsourcing services (Amazon’s Mechanical Turk being a prime example) it has become quite easy and inexpensive to acquire labels from a large number of annotators in a short amount of time (see [8], [10], and [11] for some computer vision and natural language processing case studies). One drawback of most crowdsourcing services is that we do not have tight control over the quality of the annotators. The annotators can come from a diverse pool including genuine experts, novices, biased annotators, malicious annotators, and spammers. Hence in order to get good quality labels requestors typically get each instance labeled by multiple annotators and these multiple annotations are then consolidated either using a simple majority voting or more sophisticated methods that model and correct for the annotator biases [3, 9, 6, 7, 14] and/or task complexity [2, 13, 12]. In this paper we are interested in ranking annotators based on how spammer like each annotator is. In our context a spammer is a low quality annotator who assigns random labels (maybe because the annotator does not understand the labeling criteria, does not look at the instances when labeling, or maybe a bot pretending to be a human annotator). Spammers can signiﬁcantly increase the cost of acquiring annotations (since they need to be paid) and at the same time decrease the accuracy of the ﬁnal consensus labels. A mechanism to detect and eliminate spammers is a desirable feature for any crowdsourcing market place. For example one can give monetary bonuses to good annotators and deny payments to spammers. The main contribution of this paper is to formalize the notion of a spammer for binary, categorical, and ordinal labeling tasks. More speciﬁcally we deﬁne a scalar metric which can be used to rank the annotators—with the spammers having a score close to zero and the good annotators having a score close to one (see Figure 4). We summarize the multiple parameters corresponding to each annotator into a single score indicative of how spammer like the annotator is. While this spammer score was implicit for binary labels in earlier works [3, 9, 2, 6] the extension to categorical and ordinal labels is novel and is quite different from the accuracy computed from the confusion rate matrix. An attempt to quantify the quality of the workers based on the confusion matrix was recently made by [4] where they transformed the observed labels into posterior soft labels based on the estimated confusion 1 matrix. While we obtain somewhat similar annotator rankings, we differ from this work in that our score is directly deﬁned in terms of the annotator parameters (see § 5 for more details). The rest of the paper is organized as follows. For ease of exposition we start with binary labels (§ 2) and later extend it to categorical (§ 3) and ordinal labels (§ 4). We ﬁrst specify the annotator model used, formalize the notion of a spammer, and propose an appropriate score in terms of the annotator model parameters. We do not dwell too much on the estimation of the annotator model parameters. These parameters can either be estimated directly using known gold standard 1 or the iterative algorithms that estimate the annotator model parameters without actually knowing the gold standard [3, 9, 2, 6, 7]. In the experimental section (§ 6) we obtain rankings for the annotators using the proposed spammer scores on some publicly available data from different domains. 2 Spammer score for crowdsourced binary labels j Annotator model Let yi ∈ {0, 1} be the label assigned to the ith instance by the j th annotator, and let yi ∈ {0, 1} be the actual (unobserved) binary label. We model the accuracy of the annotator separately on the positive and the negative examples. If the true label is one, the sensitivity (true positive rate) αj for the j th annotator is deﬁned as the probability that the annotator labels it as one. j αj := Pr[yi = 1|yi = 1]. On the other hand, if the true label is zero, the speciﬁcity (1−false positive rate) β j is deﬁned as the probability that annotator labels it as zero. j β j := Pr[yi = 0|yi = 0]. Extensions of this basic model have been proposed to include item level difﬁculty [2, 13] and also to model the annotator performance based on the feature vector [14]. For simplicity we use the basic model proposed in [7] in our formulation. Based on many instances labeled by multiple annotators the maximum likelihood estimator for the annotator parameters (αj , β j ) and also the consensus ground truth (yi ) can be estimated iteratively [3, 7] via the Expectation Maximization (EM) algorithm. The EM algorithm iteratively establishes a particular gold standard (initialized via majority voting), measures the performance of the annotators given that gold standard (M-step), and reﬁnes the gold standard based on the performance measures (E-step). Who is a spammer? Intuitively, a spammer assigns labels randomly—maybe because the annotator does not understand the labeling criteria, does not look at the instances when labeling, or maybe a bot pretending to be a human annotator. More precisely an annotator is a spammer if the probability j of observed label yi being one given the true label yi is independent of the true label, i.e., j j Pr[yi = 1|yi ] = Pr[yi = 1]. This means that the annotator is assigning labels randomly by ﬂipping a coin with bias without actually looking at the data. Equivalently (1) can be written as j j Pr[yi = 1|yi = 1] = Pr[yi = 1|yi = 0] which implies αj = 1 − β j . (1) j Pr[yi = 1] (2) Hence in the context of the annotator model deﬁned earlier a perfect spammer is an annotator for whom αj + β j − 1 = 0. This corresponds to the diagonal line on the Receiver Operating Characteristic (ROC) plot (see Figure 1(a)) 2 . If αj + β j − 1 < 0 then the annotators lies below the diagonal line and is a malicious annotator who ﬂips the labels. Note that a malicious annotator has discriminatory power if we can detect them and ﬂip their labels. In fact the methods proposed in [3, 7] can automatically ﬂip the labels for the malicious annotators. Hence we deﬁne the spammer score for an annotator as S j = (αj + β j − 1)2 (3) An annotator is a spammer if S j is close to zero. Good annotators have S j > 0 while a perfect annotator has S j = 1. 1 One of the commonly used strategy to ﬁlter out spammers is to inject some items into the annotations with known labels. This is the strategy used by CrowdFlower (http://crowdflower.com/docs/gold). 2 Also note that (αj + β j )/2 is equal to the area shown in the plot and can be considered as a non-parametric approximation to the area under the ROC curve (AUC) based on one observed point. It is also equal to the Balanced Classiﬁcation Rate (BCR). So a spammer can also be deﬁned as having BCR or AUC equal to 0.5. 2 Equal accuracy contours (prevalence=0.5) 0.9 1 Good Annotators Biased Annotators 0.8 0.9 Sensitivity Sensitivity ( αj ) Spammers 0.5 0.4 j 0.3 j 0.6 0.5 0.4 4 0. 5 0. 3 0. 7 0. 6 0. 4 0. 5 0. 2 0. 3 0. Malicious Annotators 0.2 0.4 0.6 1−Specificity ( βj ) 0.8 0.1 1 0. 0. 2 0. 3 0. 1 0. 0.6 0.5 1 0. 2 0. 2 0. 1 0. 0.4 3 1 0. 0. 2 0. 4 0. 0.2 4 0. 5 0. 0 0 1 3 0. 0.3 0.2 Biased Annotators 4 0.7 6 0. 4 0. 0.8 7 0.3 Area = (α +β )/2 0.2 0 0 0.9 0. 8 0. 8 0. 0.7 6 0. .5 0 5 0. 0.7 [ 1−βj, αj ] 0.6 0.1 6 0. 0.8 0.7 Equal spammer score contours 1 7 0. 8 0. 9 0. Sensitivity 1 (a) Binary annotator model 0.1 1 0. 2 0. 3 0. 0.2 0.4 0.6 1−Specificity 0.8 1 1 0. 0 0 (b) Accuracy 0.2 3 0. 4 0. 0.4 0.6 1−Specificity 5 0. .6 7 0 0. 8 0. 0.8 1 (c) Spammer score Figure 1: (a) For binary labels an annotator is modeled by his/her sensitivity and speciﬁcity. A perfect spammer lies on the diagonal line on the ROC plot. (b) Contours of equal accuracy (4) and (c) equal spammer score (3). Accuracy This notion of a spammer is quite different for that of the accuracy of an annotator. An annotator with high accuracy is a good annotator but one with low accuracy is not necessarily a spammer. The accuracy is computed as 1 j Accuracyj = Pr[yi = yi ] = j Pr[yi = 1|yi = k]Pr[yi = k] = αj p + β j (1 − p), (4) k=0 where p := Pr[yi = 1] is the prevalence of the positive class. Note that accuracy depends on prevalence. Our proposed spammer score does not depend on prevalence and essentially quantiﬁes the annotator’s inherent discriminatory power. Figure 1(b) shows the contours of equal accuracy on the ROC plot. Note that annotators below the diagonal line (malicious annotators) have low accuracy. The malicious annotators are good annotators but they ﬂip their labels and as such are not spammers if we can detect them and then correct for the ﬂipping. In fact the EM algorithms [3, 7] can correctly ﬂip the labels for the malicious annotators and hence they should not be treated as spammers. Figure 1(c) also shows the contours of equal score for our proposed score and it can be seen that the malicious annotators have a high score and only annotators along the diagonal have a low score (spammers). Log-odds Another interpretation of a spammer can be seen from the log odds. Using Bayes’ rule the posterior log-odds can be written as log j Pr[yi = 1|yi ] Pr[yi = j 0|yi ] = log j Pr[yi |yi = 1] j Pr[yi |yi = 0] + log p . 1−p Pr[y =1|y j ] p If an annotator is a spammer (i.e., (2) holds) then log Pr[yi =0|yi ] = log 1−p . Essentially the annotator j i i provides no information in updating the posterior log-odds and hence does not contribute to the estimation of the actual true label. 3 Spammer score for categorical labels Annotator model Suppose there are K ≥ 2 categories. We introduce a multinomial parameter αj = (αj , . . . , αj ) for each annotator, where c c1 cK K j αj := Pr[yi = k|yi = c] ck αj = 1. ck and k=1 αj ck The term denotes the probability that annotator j assigns class k to an instance given that the true class is c. When K = 2, αj and αj are sensitivity and speciﬁcity, respectively. 11 00 Who is a spammer? As earlier a spammer assigns labels randomly, i.e., j j Pr[yi = k|yi ] = Pr[yi = k], ∀k. 3 j j This is equivalent to Pr[yi = k|yi = c] = Pr[yi = k|yi = c ], ∀c, c , k = 1, . . . , K— which means knowing the true class label being c or c does not change the probability of the annotator’s assigned label. This indicates that the annotator j is a spammer if αj = αj k , ∀c, c , k = 1, . . . , K. ck c (5) Let Aj be the K × K confusion rate matrix with entries [Aj ]ck = αck —a spammer would have 0.50 0.50 0.50 all the rows of Aj equal, for example, Aj = 0.25 0.25 0.25 0.25 0.25 0.25 , for a three class categorical annotation problem. Essentially Aj is a rank one matrix of the form Aj = evj , for some column vector vj ∈ RK that satisﬁes vj e = 1, where e is column vector of ones. In the binary case we had this natural notion of spammer as an annotator for whom αj + β j − 1 was close to zero. One natural way to summarize (5) would be in terms of the distance (Frobenius norm) of the confusion matrix to the closest rank one approximation, i.e, S j := Aj − eˆj v 2 F, (6) where ˆj solves v ˆj = arg min Aj − evj v vj 2 F s.t. vj e = 1. (7) Solving (7) yields ˆj = (1/K)Aj e, which is the mean of the rows of Aj . Then from (6) we have v Sj = I− 1 ee K 2 Aj = F 1 K (αj − αj k )2 . ck c c < . . . < K. Annotator model It is conceptually easier to think of the true label to be binary, that is, yi ∈ {0, 1}. For example in mammography a lesion is either malignant (1) or benign (0) (which can be conﬁrmed by biopsy) and the BIRADS ordinal scale is a means for the radiologist to quantify the uncertainty based on the digital mammogram. The radiologist assigns a higher value of the label if he/she thinks the true label is closer to one. As earlier we characterize each annotator by the sensitivity and the speciﬁcity, but the main difference is that we now deﬁne the sensitivity and speciﬁcity for j each ordinal label (or threshold) k ∈ {1, . . . , K}. Let αj and βk be the sensitivity and speciﬁcity k th respectively of the j annotator corresponding to the threshold k, that is, j j j αj = Pr[yi ≥ k | yi = 1] and βk = Pr[yi < k | yi = 0]. k j j Note that αj = 1, β1 = 0 and αj 1 K+1 = 0, βK+1 = 1 from this deﬁnition. Hence each annotator j j is parameterized by a set of 2(K − 1) parameters [αj , β2 , . . . , αj , βK ]. This corresponds to an 2 K empirical ROC curve for the annotator (Figure 2). 4 Who is a spammer? As earlier we deﬁne an an1 j k=1 notator j to be a spammer if Pr[yi = k|yi = 1] = 0.9 j k=2 0.8 Pr[yi = k|yi = 0] ∀k = 1, . . . , K. Note that from j 0.7 k=3 [ 1−β , α ] the annotation model we have 3 Pr[yi = k | yi = 0.6 j j j 1] = αk − αk+1 and Pr[yi = k | yi = 0] = 0.5 k=4 j j 0.4 βk+1 − βk . This implies that annotator j is a spam0.3 j j mer if αj − αj k k+1 = βk+1 − βk , ∀k = 1, . . . , K, 0.2 j j 0.1 which leads to αj + βk = αj + β1 = 1, ∀k. This 1 k j 0 0 0.2 0.4 0.6 0.8 1 means that for every k, the point (1 − βk , αj ) lies on k 1−Specificity ( β ) the diagonal line in the ROC plot shown in Figure 2. The area under the empirical ROC curve can be comFigure 2: Ordinal labels: An annotator is modK 1 puted as (see Figure 2) AUCj = 2 k=1 (αj + eled by sensitivity/speciﬁcity for each threshold. k+1 j j αj )(βk+1 − βk ), and can be used to deﬁne the folk lowing spammer score as (2AUCj − 1)2 to rank the different annotators. 3 Sensitivity ( αj ) 3 j 2 K (αj k+1 k=1 j S = + j αj )(βk+1 k − j βk ) −1 (9) With two levels this expression defaults to the binary case. An annotator is a spammer if S j is close to zero. Good annotators have S j > 0 while a perfect annotator has S j = 1. 5 Previous work Recently Ipeirotis et.al. [4] proposed a score for categorical labels based on the expected cost of the posterior label. In this section we brieﬂy describe their approach and compare it with our proposed score. For each instance labeled by the annotator they ﬁrst compute the posterior (soft) label j j Pr[yi = c|yi ] for c = 1, . . . , K, where yi is the label assigned to the ith instance by the j th annotator and yi is the true unknown label. The posterior label is computed via Bayes’ rule as j j j Pr[yi = c|yi ] ∝ Pr[yi |yi = c]Pr[yi = c] = (αj )δ(yi ,k) pc , where pc = Pr[yi = c] is the prevack lence of class c. The score for a spammer is based on the intuition that the posterior label vector j j (Pr[yi = 1|yi ], . . . , Pr[yi = K|yi ]) for a good annotator will have all the probability mass concentrated on single class. For example for a three class problem (with equal prevalence), a posterior label vector of (1, 0, 0) (certain that the class is one) comes from a good annotator while a (1/3, 1/3, 1/3) (complete uncertainty about the class label) comes from spammer. Based on this they deﬁne the following score for each annotator 1 Score = N N K K j j costck Pr[yi = k|yi ]Pr[yi = c|yi ] j i=1 . (10) c=1 k=1 where costck is the misclassiﬁcation cost when an instance of class c is classiﬁed as k. Essentially this is capturing some sort of uncertainty of the posterior label averaged over all the instances. Perfect workers have a score Scorej = 0 while spammers will have high score. An entropic version of this score based on similar ideas has also been recently proposed in [5]. Our proposed spammer score differs from this approach in the following aspects: (1) Implicit in the score deﬁned above (10) j is the assumption that an annotator is a spammer when Pr[yi = c|yi ] = Pr[yi = c], i.e., the estimated posterior labels are simply based on the prevalence and do not depend on the observed labels. By j j Bayes’ rule this is equivalent to Pr[yi |yi = c] = Pr[yi ] which is what we have used to deﬁne our spammer score. (2) While both notions of a spammer are equivalent, the approach of [4] ﬁrst computes the posterior labels based on the observed data, the class prevalence and the annotator j j j j This can be seen as follows: Pr[yi = k | yi = 1] = Pr[(yi ≥ k) AND (yi < k + 1) | yi = 1] = Pr[yi ≥ j j j j k | yi = 1] + Pr[yi < k + 1 | yi = 1] − Pr[(yi ≥ k) OR (yi < k + 1) | yi = 1] = Pr[yi ≥ k | yi = j j j 1] − Pr[yi ≥ k + 1 | yi = 1] = αj − αj . Here we used the fact that Pr[(yi ≥ k) OR (yi < k + 1)] = 1. k k+1 3 5 simulated | 500 instances | 30 annotators simulated | 500 instances | 30 annotators 1 12 0.8 Spammer Score 18 0.6 0.5 22 24 23 25 0.3 29 20 0.2 0.4 0.2 30 16 14 0.1 26 21 27 28 19 0 0 13 0 0.2 0.4 0.6 1−Specificity 0.8 1 500 500 500 500 500 500 500 500 500 500 0.4 0.6 500 500 500 1 0.7 500 500 500 500 500 500 500 500 500 500 500 500 500 500 3 1 500 500 500 2 8 510 7 17 4 9 27 8 30 6 3 28 7 10 2 23 22 26 24 5 1 21 29 25 14 12 17 11 18 20 19 15 16 13 4 0.8 Sensitivity 6 9 0.9 15 11 Annotator (a) Simulation setup (b) Annotator ranking Annotator rank (median) via accuracy simulated | 500 instances | 30 annotators Annotator rank (median) via Ipeirotis et.al.[4] simulated | 500 instances | 30 annotators 27 30 30 28 23 22 26 25 24 21 29 25 20 14 17 18 15 20 16 15 11 13 12 19 1 10 5 10 2 7 6 3 5 8 9 4 0 0 5 10 15 20 25 Annotator rank (median) via spammer score 30 (c) Comparison with accuracy 30 18 20 16 15 19 13 12 25 11 14 17 25 20 29 21 51 15 26 22 23 24 10 2 7 10 28 6 3 8 30 5 27 9 4 0 0 5 10 15 20 25 Annotator rank (median) via spammer score 30 (d) Comparison with Ipeirotis et. al. [4] Figure 3: (a) The simulation setup consisting of 10 good annotators (annotators 1 to 10), 10 spammers (11 to 20), and 10 malicious annotators (21 to 30). (b) The ranking of annotators obtained using the proposed spammer score. The spammer score ranges from 0 to 1, the lower the score, the more spammy the annotator. The mean spammer score and the 95% conﬁdence intervals (CI) are shown—obtained from 100 bootstrap replications. The annotators are ranked based on the lower limit of the 95% CI. The number at the top of the CI bar shows the number of instances annotated by that annotator. (c) and (d) Comparison of the median rank obtained via the spammer score with the rank obtained using (c) accuracy and (d) the method proposed by Ipeirotis et. al. [4]. parameters and then computes the expected cost. Our proposed spammer score does not depend on the prevalence of the class. Our score is also directly deﬁned only in terms of the annotator confusion matrix and does not need the observed labels. (3) For the score deﬁned in (10) while perfect annotators have a score of 0 it is not clear what should be a good baseline for a spammer. The authors suggest to compute the baseline by assuming that a worker assigns as label the class with maximum prevalence. Our proposed score has a natural scale with a perfect annotator having a score of 1 and a spammer having a score of 0. (4) However one advantage of the approach in [4] is that they can directly incorporate varied misclassiﬁcation costs. 6 Experiments Ranking annotators based on the conﬁdence interval As mentioned earlier the annotator model parameters can be estimated using the iterative EM algorithms [3, 7] and these estimated annotator parameters can then be used to compute the spammer score. The spammer score can then be used to rank the annotators. However one commonly observed phenomenon when working with crowdsourced data is that we have a lot of annotators who label only a very few instances. As a result the annotator parameters cannot be reliably estimated for these annotators. In order to factor this uncertainty in the estimation of the model parameters we compute the spammer score for 100 bootstrap replications. Based on this we compute the 95% conﬁdence intervals (CI) for the spammer score for each annotator. We rank the annotators based on the lower limit of the 95% CI. The CIs are wider 6 Table 1: Datasets N is the number of instances. M is the number of annotators. M ∗ is the mean/median number of annotators per instance. N ∗ is the mean/median number of instances labeled by each annotator. Dataset Type N M M∗ N∗ bluebird binary 108 39 39/39 108/108 temp binary 462 76 10/10 61/16 Brief Description wsd categorical/3 177 34 10/10 52/20 sentiment categorical/3 1660 33 6/6 291/175 30 100 10 38 10/10 10/10 bird identiﬁcation [12] The annotator had to identify whether there was an Indigo Bunting or Blue Grosbeak in the image. event annotation [10] Given a dialogue and a pair of verbs annotators need to label whether the event described by the ﬁrst verb occurs before or after the second. 30/30 26/20 30 30 30 30 30 30 3 30 30 1 30 20 20 20 20 20 77 117 20 60 Spammer Score 0.4 10 7 9 8 6 5 0 2 0.2 13 31 10 23 29 1 2 4 6 8 9 14 15 17 22 32 5 18 16 19 11 12 20 21 24 25 26 27 28 30 33 34 7 3 0 0.6 20 20 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 20 20 20 20 17 17 40 20 20 100 Spammer Score 0.4 108 108 108 108 108 108 108 108 108 108 108 108 108 0.8 0.6 0.2 17 8 27 30 25 35 1 12 32 37 38 16 22 9 29 15 20 19 5 39 3 21 23 14 2 10 24 7 33 13 36 31 4 34 28 18 11 6 26 0.2 30 77 77 4 108 108 108 108 0.4 0 wosi | 30 instances | 10 annotators 1 0.8 108 108 0.6 108 108 108 Spammer Score 0.8 wsd | 177 instances | 34 annotators 1 80 177 157 177 157 bluebird | 108 instances | 39 annotators 1 word similarity [10] Numeric judgements of word similarity. affect recognition [10] Each annotator is presented with a short headline and asked to rate it on a scale [-100,100] to denote the overall positive or negative valence. 40 40 20 ordinal/[0 10] ordinal[-100 100] 20 20 20 wosi valence word sense disambiguation [10] The labeler is given a paragraph of text containing the word ”president” and asked to label one of the three appropriate senses. irish economic sentiment analysis [1] Articles from three Irish online news sources were annotated by volunteer users as positive, negative, or irrelevant. 20 20 20 20 20 60 20 20 20 40 40 100 0.4 Annotator Annotator 0 1 26 10 18 28 15 5 36 23 12 8 32 31 38 13 17 27 11 2 35 24 19 9 6 30 33 37 14 29 4 3 20 34 22 25 7 16 21 40 20 0.2 26 2 6 11 5 14 3 20 9 22 31 10 12 18 8 13 30 4 1 29 19 17 27 28 21 15 25 23 7 33 16 24 32 10 132 10 360 10 0 13 18 52 75 33 32 12 74 31 51 41 55 7 14 70 42 58 65 43 1 10 47 61 73 25 37 76 67 24 46 54 48 39 56 15 62 68 44 53 64 40 9 28 6 2 57 3 4 5 8 11 16 17 19 20 21 22 23 26 27 29 30 34 35 36 38 45 49 50 59 60 63 66 69 71 72 442 462 452 10 10 0.6 238 171 75 654 20 0.2 0.2 0 12 77 67 374 249 229 453 346 428 0.4 Spammer Score 43 175 119 541 525 437 0.8 917 104 284 0.4 0.6 1211 1099 10 Spammer Score 0.8 572 30 52 402 60 0.6 30 Spammer Score 0.8 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 60 40 20 15 7 7 11 12 35 29 1 87 10 10 10 10 10 10 10 12 1 30 10 10 10 10 10 10 10 10 10 10 10 10 10 valence | 100 instances | 38 annotators 20 22 10 10 10 10 sentiment | 1660 instances | 33 annotators 10 10 30 20 10 Annotator temp | 462 instances | 76 annotators 1 Annotator 10 50 10 10 40 10 70 350 80 40 100 192 190 40 32 60 70 20 20 40 80 20 50 50 50 30 Annotator Annotator Figure 4: Annotator Rankings The rankings obtained for the datasets in Table 1. The spammer score ranges from 0 to 1, the lower the score, the more spammy the annotator. The mean spammer score and the 95% conﬁdence intervals (CI) are shown—obtained from 100 bootstrap replications. The annotators are ranked based on the lower limit of the 95% CI. The number at the top of the CI bar shows the number of instances annotated by that annotator. Note that the CIs are wider when the annotator labels only a few instances. when the annotator labels only a few instances. For a crowdsourced labeling task the annotator has to be good and also label a reasonable number of instances in order to be reliably identiﬁed. Simulated data We ﬁrst illustrate our proposed spammer score on simulated binary data (with equal prevalence for both classes) consisting of 500 instances labeled by 30 annotators of varying sensitivity and speciﬁcity (see Figure 3(a) for the simulation setup). Of the 30 annotators we have 10 good annotators (annotators 1 to 10 who lie above the diagonal in Figure 3(a)), 10 spammers (annotators 11 to 20 who lie around the diagonal), and 10 malicious annotators (annotators 21 to 30 who lie below the diagonal). Figure 3(b) plots the ranking of annotators obtained using the proposed spammer score with the annotator model parameters estimated via the EM algorithm [3, 7]. The spammer score ranges from 0 to 1, the lower the score, the more spammy the annotator. The mean spammer score and the 95% conﬁdence interval (CI) obtained via bootstrapping are shown. The annotators are ranked based on the lower limit of the 95% CI. As can be seen all the spammers (annotators 11 to 20) have a low spammer score and appear at the bottom of the list. The malicious annotators have higher score than the spammers since we can correct for their ﬂipping. The malicious annotators are good annotators but they ﬂip their labels and as such are not spammers if we detect that they are malicious. Figure 3(c) compares the (median) rank obtained via the spammer score with the (median) rank obtained using accuracy as the score to rank the annotators. While the good annotators are ranked high by both methods the accuracy score gives a low rank to the malicious annotators. Accuracy does not capture the notion of a spammer. Figure 3(d) compares the ranking with the method proposed by Ipeirotis et. al. [4] which gives almost similar rankings as our proposed score. 7 21 23 10 6 35 4 34 1126 18 147 30 3 31 13 2436 33 25 5 2 20 15 39 19 15 20 28 22 299 12 37 16 38 10 32 1 5 27 25 35 30 8 17 0 0 5 10 15 20 25 30 35 Annotator rank (median) via spammer score 40 bluebird | 108 instances | 39 annotators 40 1 6 34 112618 4 31 1013 7 30 2 28 21 5 20 15 39 19 20 15 22 37 16 299 12 38 10 5 8 17 0 0 27 25 35 30 0.6 0.5 35 32 2 0.4 36 11 13 31 24 10 33 28 21 26 18 0 0 40 34 15 19 39 0.1 (a) 22 37 20 38 29 9 0.2 5 10 15 20 25 30 35 Annotator rank (median) via spammer score 6 4 16 0.3 32 1 7 30 25 1 3 14 3 27 5 0.7 24 33 14 36 23 25 12 8 0.9 17 0.8 35 Sensitivity Annotator rank (median) via accuracy bluebird | 108 instances | 39 annotators Annotator rank (median) via Ipeirotis et.al.[4] bluebird | 108 instances | 39 annotators 40 23 0.2 0.4 0.6 1−Specificity (b) 0.8 1 (c) Figure 5: Comparison of the rank obtained via the spammer score with the rank obtained using (a) accuracy and (b) the method proposed by Ipeirotis et. al. [4] for the bluebird binary dataset. (c) The annotator model parameters as estimated by the EM algorithm [3, 7]. 19 25 12 18 7 3 14 20 32 5 8 1 16 20 9 21 15 34 10 31 29 17 28 22 26 2315 5 2 0 0 4 6 13 10 5 10 15 20 25 30 Annotator rank (median) via spammer score 35 30 25 16 19 7 25 8 9 27 14 3 28 17 18 32 5 10 4 2 10 6 1529 31 23 22 21 15 0 0 33 30 11 1 20 5 sentiment | 1660 instances | 33 annotators 24 35 12 20 24 34 26 13 5 10 15 20 25 30 Annotator rank (median) via spammer score 35 33 7 30 15 17 25 28 2719 2223 20 8 1 4 1812 15 13 10 20 32 30 10 3 29 9 31 16 5 6 2 5 14 11 26 0 0 5 10 15 20 25 30 Annotator rank (median) via spammer score 25 21 Annotator rank (median) via Ipeirotis et.al.[4] 25 24 27 Annotator rank (median) via accuracy Annotator rank (median) via accuracy 30 sentiment | 1660 instances | 33 annotators wsd | 177 instances | 34 annotators 33 30 11 Annotator rank (median) via Ipeirotis et.al.[4] wsd | 177 instances | 34 annotators 35 7 30 15 19 17 27 25 21 25 8 12 4 18 20 24 15 20 33 10 3 13 9 28 1 29 23 10 1632 11 14 5 6 2 5 31 30 22 26 0 0 5 10 15 20 25 30 Annotator rank (median) via spammer score Figure 6: Comparison of the median rank obtained via the spammer score with the rank obtained using accuracy and he method proposed by Ipeirotis et. al. [4] for the two categorial datasets in Table 1. Mechanical Turk data We report results on some publicly available linguistic and image annotation data collected using the Amazon’s Mechanical Turk (AMT) and other sources. Table 1 summarizes the datasets. Figure 4 plots the spammer scores and rankings obtained. The mean and the 95% CI obtained via bootstrapping are also shown. The number at the top of the CI bar shows the number of instances annotated by that annotator. The rankings are based on the lower limit of the 95% CI which factors the number of instances labeled by the annotator into the ranking. An annotator who labels only a few instances will have very wide CI. Some annotators who label only a few instances may have a high mean spammer score but the CI will be wide and hence ranked lower. Ideally we would like to have annotators with a high score and at the same time label a lot of instances so that we can reliablly identify them. The authors [1] for the sentiment dataset shared with us some of the qualitative observations regarding the annotators and they somewhat agree with our rankings. For example the authors made the following comments about Annotator 7 ”Quirky annotator - had a lot of debate about what was the meaning of the annotation question. I’d say he changed his labeling strategy at least once during the process”. Our proposed score gave a low rank to this annotator. Comparison with other approaches Figure 5 and 6 compares the proposed ranking with the rank obtained using accuracy and the method proposed by Ipeirotis et. al. [4] for some binary and categorical datasets in Table 1. Our proposed ranking is somewhat similar to that obtained by Ipeirotis et. al. [4] but accuracy does not quite capture the notion of spammer. For example for the bluebird dataset for annotator 21 (see Figure 5(a)) accuracy ranks it at the bottom of the list while the proposed score puts is in the middle of the list. From the estimated model parameters it can be seen that annotator 21 actually ﬂips the labels (below the diagonal in Figure 5(c)) but is a good annotator. 7 Conclusions We proposed a score to rank annotators for crowdsourced binary, categorical, and ordinal labeling tasks. The obtained rankings and the scores can be used to allocate monetary bonuses to be paid to different annotators and also to eliminate spammers from further labeling tasks. A mechanism to rank annotators should be desirable feature of any crowdsourcing service. The proposed score should also be useful to specify the prior for Bayesian approaches to consolidate annotations. 8 References [1] A. Brew, D. Greene, and P. Cunningham. Using crowdsourcing and active learning to track sentiment in online media. In Proceedings of the 6th Conference on Prestigious Applications of Intelligent Systems (PAIS’10), 2010. [2] B. Carpenter. Multilevel bayesian models of categorical data annotation. Technical Report available at http://lingpipe-blog.com/lingpipe-white-papers/, 2008. [3] A. P. Dawid and A. M. Skene. Maximum likeihood estimation of observer error-rates using the EM algorithm. Applied Statistics, 28(1):20–28, 1979. [4] P. G. Ipeirotis, F. Provost, and J. Wang. Quality management on Amazon Mechanical Turk. In Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP’10), pages 64–67, 2010. [5] V. C. Raykar and S. Yu. An entropic score to rank annotators for crowdsourced labelling tasks. In Proceedings of the Third National Conference on Computer Vision, Pattern Recognition, Image Processing and Graphics (NCVPRIPG), 2011. [6] V. C. Raykar, S. Yu, L .H. Zhao, A. Jerebko, C. Florin, G. H. Valadez, L. Bogoni, and L. Moy. Supervised learning from multiple experts: Whom to trust when everyone lies a bit. In Proceedings of the 26th International Conference on Machine Learning (ICML 2009), pages 889– 896, 2009. [7] V. C. Raykar, S. Yu, L. H. Zhao, G. H. Valadez, C. Florin, L. Bogoni, and L. Moy. Learning from crowds. Journal of Machine Learning Research, 11:1297–1322, April 2010. [8] V. S. Sheng, F. Provost, and P. G. Ipeirotis. Get another label? Improving data quality and data mining using multiple, noisy labelers. In Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 614–622, 2008. [9] P. Smyth, U. Fayyad, M. Burl, P. Perona, and P. Baldi. Inferring ground truth from subjective labelling of venus images. In Advances in Neural Information Processing Systems 7, pages 1085–1092. 1995. [10] R. Snow, B. O’Connor, D. Jurafsky, and A. Y. Ng. Cheap and Fast—but is it good? Evaluating Non-Expert Annotations for Natural Language Tasks. In Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP ’08), pages 254–263, 2008. [11] A. Sorokin and D. Forsyth. Utility data annotation with Amazon Mechanical Turk. In Proceedings of the First IEEE Workshop on Internet Vision at CVPR 08, pages 1–8, 2008. [12] P. Welinder, S. Branson, S. Belongie, and P. Perona. The multidimensional wisdom of crowds. In Advances in Neural Information Processing Systems 23, pages 2424–2432. 2010. [13] J. Whitehill, P. Ruvolo, T. Wu, J. Bergsma, and J. Movellan. Whose vote should count more: Optimal integration of labels from labelers of unknown expertise. In Advances in Neural Information Processing Systems 22, pages 2035–2043. 2009. [14] Y. Yan, R. Rosales, G. Fung, M. Schmidt, G. Hermosillo, L. Bogoni, L. Moy, and J. Dy. Modeling annotator expertise: Learning when everybody knows a bit of something. In Proceedings of the Thirteenth International Conference on Artiﬁcial Intelligence and Statistics (AISTATS 2010), pages 932–939, 2010. 9

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