jmlr jmlr2010 jmlr2010-59 knowledge-graph by maker-knowledge-mining

59 jmlr-2010-Large Scale Online Learning of Image Similarity Through Ranking

Source: pdf

Author: Gal Chechik, Varun Sharma, Uri Shalit, Samy Bengio

Abstract: Learning a measure of similarity between pairs of objects is an important generic problem in machine learning. It is particularly useful in large scale applications like searching for an image that is similar to a given image or ﬁnding videos that are relevant to a given video. In these tasks, users look for objects that are not only visually similar but also semantically related to a given object. Unfortunately, the approaches that exist today for learning such semantic similarity do not scale to large data sets. This is both because typically their CPU and storage requirements grow quadratically with the sample size, and because many methods impose complex positivity constraints on the space of learned similarity functions. The current paper presents OASIS, an Online Algorithm for Scalable Image Similarity learning that learns a bilinear similarity measure over sparse representations. OASIS is an online dual approach using the passive-aggressive family of learning algorithms with a large margin criterion and an efﬁcient hinge loss cost. Our experiments show that OASIS is both fast and accurate at a wide range of scales: for a data set with thousands of images, it achieves better results than existing state-of-the-art methods, while being an order of magnitude faster. For large, web scale, data sets, OASIS can be trained on more than two million images from 150K text queries within 3 days on a single CPU. On this large scale data set, human evaluations showed that 35% of the ten nearest neighbors of a given test image, as found by OASIS, were semantically relevant to that image. This suggests that query independent similarity could be accurately learned even for large scale data sets that could not be handled before. Keywords: large scale, metric learning, image similarity, online learning ∗. Varun Sharma and Uri Shalit contributed equally to this work. †. Also at ICNC, The Hebrew University of Jerusalem, 91904, Israel. c 2010 Gal Chechik, Varun Sharma, Uri Shalit

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 It is particularly useful in large scale applications like searching for an image that is similar to a given image or ﬁnding videos that are relevant to a given video. [sent-9, score-0.266]

2 For large, web scale, data sets, OASIS can be trained on more than two million images from 150K text queries within 3 days on a single CPU. [sent-16, score-0.437]

3 On this large scale data set, human evaluations showed that 35% of the ten nearest neighbors of a given test image, as found by OASIS, were semantically relevant to that image. [sent-17, score-0.254]

4 This suggests that query independent similarity could be accurately learned even for large scale data sets that could not be handled before. [sent-18, score-0.286]

5 Learning a pairwise similarity measure is a particularly challenging large scale task: since pairs of samples have to be considered, the large scale regime is reached even for fairly small data sets, and learning similarity for large data sets becomes exceptionally hard to handle. [sent-26, score-0.357]

6 In all these applications, similarity could have different ﬂavors: a user may search for images that are similar visually, or semantically, or anywhere in between. [sent-31, score-0.378]

7 For large scale data, where man-in-the-loop experiments are prohibitively costly, relative similarities can be extracted from analyzing pairs of images that are returned in response to the same text query (Schultz and Joachims, 2004). [sent-36, score-0.449]

8 For instance, the images that are ranked highly by one of the image search engines for the query “cute kitty” are likely to be semantically more similar than a random pair of images. [sent-37, score-0.605]

9 The current paper focuses on this setting: similarity information is extracted from pairs of images that share a common label or are retrieved in response to a common text query. [sent-38, score-0.464]

10 Given two images p1 and p2 we measure similarity through a bilinear form pT Wp2 , where the 1 matrix W is not required to be positive, or even symmetric. [sent-53, score-0.378]

11 For web-scale data sets, OASIS can be trained on more than two million images within three days on a single CPU, and its training time grows linearly with the size of the data. [sent-59, score-0.336]

12 On this large scale data set, human evaluations of OASIS learned similarity show that 35% of the ten nearest neighbors of a given image are semantically relevant to that image. [sent-60, score-0.49]

13 Learning Relative Similarity We consider the problem of learning a pairwise similarity function S, given data on the relative similarity of pairs of images. [sent-67, score-0.311]

14 This relevance measure could encode the fact that two images belong to the same category or were appropriate for the same query. [sent-69, score-0.297]

15 We also assume that when r(pi , p j ) is not available, its value is zero (since the vast majority of images are not related to each other). [sent-72, score-0.248]

16 Our goal is to learn a similarity function S(pi , p j ) that assigns higher similarity scores to pairs of more relevant images, S(pi , p+ ) > S(pi , p− ) , i i ∀pi , p+ , p− ∈ P such that r(pi , p+ ) > r(pi , p− ). [sent-73, score-0.312]

17 i i i i (1) In this paper we overload notation by using pi to denote both the image and its representation as a column vector pi ∈ Rd . [sent-74, score-0.36]

18 Importantly, if the images pi are represented as sparse vectors, namely, only a number ki ≪ d of the d entries in the vector pi are non-zeroes, then the value of Equation (2) can be computed very efﬁciently even when d is large. [sent-76, score-0.502]

19 Here we consider an algorithm that uses triplets of images pi , p+ , p− ∈ P such that r(pi , p+ ) > r(pi , p− ). [sent-81, score-0.428]

20 OASIS Initialization: Initialize W0 = I Iterations repeat Sample three images p, p+ , p− , such that r(pi , p+ ) > r(pi , p− ). [sent-92, score-0.248]

21 Instead, we use the fact that the number of relevant images for a category or a query is typically small, and keep a list of relevant images for each query or category. [sent-145, score-0.824]

22 For the case of single-labeled images, we can efﬁciently retrieve an image that is relevant to a given image, by ﬁrst ﬁnding its class, and then ﬁnding another image from that class. [sent-146, score-0.243]

23 The case of multi-labeled images is described in Section 5. [sent-147, score-0.248]

24 Speciﬁcally, to sample a triplet (pi , p+ , p− ) during training, we ﬁrst uniformly sample an image i i pi from P . [sent-149, score-0.255]

25 Then we uniformly sample an image p+ from the images sharing the same categories i 1114 L ARGE S CALE O NLINE L EARNING OF I MAGE S IMILARITY or queries as pi . [sent-150, score-0.579]

26 Finally, we uniformly sample an image p− from the images that share no category i or query with pi . [sent-151, score-0.614]

27 When the set P is very large and the number of categories or queries is also very large, one does not need to maintain the set of non-relevant images for each image: sampling directly from P instead only adds a small amount of noise to the training procedure and is not really harmful. [sent-152, score-0.376]

28 Image Representation The problem of selecting an informative representation of images is still an unsolved computer vision challenge, and an ongoing research topic. [sent-156, score-0.272]

29 We ﬁrst select a palette or typical colors by training a color codebook from the Red-Green-Blue pixels of a large training set of images using K-means. [sent-183, score-0.351]

30 For this purpose, each local descriptor of an image p is represented as a discrete index, called visual term or visterm, and, like for text data, the image is represented as a bag-of-visterms vector, in which each component pi is related to the presence or absence of visterm i in p. [sent-194, score-0.414]

31 The mapping of the descriptors to discrete indexes is performed according to a codebook C, which is typically learned from the local descriptors of the training images through k-means clustering (Duygulu et al. [sent-195, score-0.352]

32 In the experiments described below, we used a large set of images collected from the web to train the features. [sent-200, score-0.277]

33 Furthermore, in order to be robust to scale, we extracted blocks at various scales by successively down scaling images by a factor of 1. [sent-204, score-0.268]

34 In one such setup, the training set consists of triplets of objects xi1 , xi2 , xi3 and a ranking similarity function, that can tell which of the two pairs (x1 , x2 ) or (x1 , x3 ) is more similar. [sent-212, score-0.255]

35 Finally, multiple similarity learning studies assume that a binary measure of similarity is available yi ∈ {+1, −1}, indicating whether a pair of objects is similar or not. [sent-213, score-0.281]

36 In their case, however, semantic similarity is learned by representing each image by the posterior probability distribution over a predeﬁned set of semantic tags, and then computing the distance between two images as the distance between the two underlying posterior distributions. [sent-239, score-0.664]

37 The representation size of images in this approach is therefore equal to the number of semantic classes, hence it will not scale when the number of semantic classes is very large as in free text search. [sent-240, score-0.432]

38 First, current available benchmarks are limited either in their scale, like 30K images in Caltech256 as described by Grifﬁn et al. [sent-243, score-0.248]

39 (2007), or in their resolution, such as the tiny images data set of Torralba et al. [sent-244, score-0.248]

40 Second, many images on the web cannot be used without explicit permission, hence they cannot be collected and packed into a single database. [sent-247, score-0.277]

41 Large, proprietary collections of images do exist, but are not available freely for academic research. [sent-248, score-0.248]

42 For each query image in the test set, all other test images were ranked according to their similarity to the query image. [sent-256, score-0.78]

43 The number of same-class images among the top k images (the k nearest neighbors) was computed. [sent-257, score-0.554]

44 When averaged across test images (either within or across classes), this yields a measure known as precision-at-top-k, providing a precision curve as a function of the rank k. [sent-258, score-0.384]

45 The average-precision measure is then further averaged across all test image queries, yielding the mean average precision (mAP). [sent-263, score-0.242]

46 + Denote the set of text queries by Q and the set of images by P . [sent-276, score-0.35]

47 For each q ∈ Q , let Pq denote the − set of images that are relevant to the query q, and let Pq denote the set of irrelevant images. [sent-277, score-0.412]

48 We also computed a normalized version of RQI , q q q which can be interpreted as a joint distribution matrix, or the probability to observe a query q and an image p for that query, RQI (q, p) Pr(q, p) = . [sent-279, score-0.239]

49 q∈Q q∈Q To improve scalability, we used a threshold over this joint distribution, and considered two images to be related only if their joint distribution exceeded a cutoff value θ RII (p1 , p2 ) = [Pr(p1 , p2 )]θ (10) where [x]θ = x for x > θ and is zero otherwise. [sent-281, score-0.248]

50 To set the value of θ we have manually inspected a small subset of pairs of related images taken from the training set. [sent-282, score-0.299]

51 In practice, we can use the fact that RQI is very sparse, to quickly create a list with images that are relevant to a given image. [sent-285, score-0.279]

52 To do this given an image pi , we go over all the queries for which it is relevant RQI (q, pi ), and for each of these queries, collect the list of all images that are relevant to that query. [sent-286, score-0.748]

53 The average number of queries relevant for an image in our data is small (about 100), and so is the number of images relevant for a given query. [sent-287, score-0.494]

54 3 million images in the training set using the sampling mechanism based on the relevance of each image, as described in Section 2. [sent-290, score-0.364]

55 Finally, we evaluated the trained model on the 400 thousand images of the test set. [sent-295, score-0.248]

56 Table 2 shows the top ﬁve images as ranked by OASIS on four examples of query-images in the test set. [sent-299, score-0.304]

57 The ﬁrst example (top row), shows a query-image that was originally retrieved 1119 C HECHIK , S HARMA , S HALIT AND B ENGIO Query image Top 5 relevant images retrieved by OASIS Table 2: OASIS: Successful cases from the Web data set in response to the text query “illusion”. [sent-301, score-0.624]

58 All ﬁve images ranked highly by OASIS are semantically related, showing other types of visual illusions. [sent-302, score-0.366]

59 In all these cases, OASIS captures similarity that is both semantic and visual, since the raw visual similarity of these images is not high. [sent-304, score-0.564]

60 It shows three cases where OASIS was biased by visual similarity and provided high rankings to images that were semantically non relevant. [sent-306, score-0.486]

61 The second example presents a query image which in itself has no deﬁnite semantic element. [sent-308, score-0.295]

62 The results retrieved are those that merely match texture of the query image and bear no semantic similarity. [sent-309, score-0.356]

63 To obtain a quantitative evaluation of OASIS we computed the precision at top k, using a threshold θ = 0, which means that an image in the test set is considered relevant to a query image, if there exists at least one text query to which they were both relevant to. [sent-311, score-0.62]

64 This is drastically lower than the precision described below for Caltech256, and could be the result 1120 L ARGE S CALE O NLINE L EARNING OF I MAGE S IMILARITY Query image Top 5 relevant images retrieved by OASIS Table 3: OASIS: Failure cases from the Web data set of multiple reasons. [sent-314, score-0.562]

65 First, the number of unique textual queries in our data is very large (around 150K), hence the images in this data set were signiﬁcantly more heterogeneous than images in the Caltech256 data. [sent-315, score-0.574]

66 This means that many pairs of images that are semantically related are not labeled as such. [sent-317, score-0.357]

67 The query images (like “scottish fold”) have labels that are usually very different from the labels of the retrieved images (as in “humor cat”, “agility”) even if their semantic content is very similar. [sent-319, score-0.726]

68 We also selected randomly a set of 10 negative images p− that were chosen for each of the query images p such that RII (p, p− ) = 0. [sent-327, score-0.629]

69 All 25 query images were presented to twenty human evaluators, asking them to mark which of the 20 candidate images are semantically relevant to the query image. [sent-329, score-0.919]

70 The overall relevance of an image was estimated as the sum of relevances of the image with respect to all queries. [sent-331, score-0.261]

71 We collected the ratings on the positive images and calculated the precision at top k. [sent-335, score-0.41]

72 02 cross validation precision mean human precision Web−scale test set 0. [sent-337, score-0.31]

73 2 0 1 10 number of neighbors 0 0 5 10 15 20 query ID (sorted by precision) 25 Figure 3: (A) Precision at top k as a function of k neighbors computed against RII (θ = 0) for the web-scale test set. [sent-352, score-0.243]

74 To select the queries for this plot, we ﬁrst calculated the meanaverage precision per query, sorted the queries by their mAP, and selected the queries ranked at position 1, 6, 11, 16, and 21. [sent-356, score-0.4]

75 We observed that the variability across different query images was also very high. [sent-360, score-0.381]

76 Some query images were “easy” for OASIS, yielding high scores from most evaluators, while other queries retrieved images that were consistently found to be irrelevant by most evaluators. [sent-364, score-0.748]

77 We ﬁrst calculated the mAP from the precision curves of every query and evaluator, and then calculated the standard deviation in the mAP of every evaluator and of every query. [sent-366, score-0.3]

78 25, suggesting that different evaluators had very different notions of what images should be regarded as “semantically similar” to a query image. [sent-370, score-0.488]

79 Finally, to estimate an “upper bound” on the difﬁculty of the task, we also computed the precision of the human evaluators themselves. [sent-371, score-0.281]

80 As with the ranks of OASIS, we computed the fraction of evaluators that marked an image as relevant, and repeated this separately for every query and human evaluator, providing a measure of “coherence” per query. [sent-373, score-0.384]

81 Figure 3(D) shows the mean precision obtained by OASIS and human evaluators for every query in our data. [sent-374, score-0.414]

82 For some queries OASIS achieves precision that is very close to that of the mean human evaluator. [sent-375, score-0.252]

83 This data set consists of 30607 images that were obtained from Google image search and from PicSearch. [sent-389, score-0.354]

84 After we have pre-processed the images as described in Section 3 and ﬁltered images that were too small, we were left with 29461 images in 256 categories. [sent-392, score-0.744]

85 For OASIS, we treated images that have the same labels as similar. [sent-398, score-0.248]

86 5 hrs 100K 5min 5 min 37sec 9sec 60 600 10K 100K 2M number of images (log scale) Figure 4: Comparison of the runtime of OASIS and fast-LMNN by Weinberger and Saul (2008), over a wide range of scales. [sent-402, score-0.271]

87 As a preprocessing phase, images were projected to a basis of the principal components (PCA) of the data, with no dimensionality reduction, since 1124 L ARGE S CALE O NLINE L EARNING OF I MAGE S IMILARITY this improved the precision results. [sent-424, score-0.384]

88 For each set, images from each class were split into a training set of 40 images and a test set of 25 images, as proposed by Grifﬁn et al. [sent-454, score-0.526]

89 3 Table 4: Mean average precision and precision at top 1, 10, and 50 of all compared methods. [sent-579, score-0.298]

90 1 Random 20 30 number of neighbors 40 50 0 0 OASIS MCML LEGO LMNN Euclidean 10 Random 20 30 number of neighbors 40 50 (C) 50 classes precision 0. [sent-602, score-0.245]

91 Some applications, like ranking images by semantic relevance to a given image query are known to be non-symmetric when based on human judgement (Tversky, 1977). [sent-636, score-0.63]

92 Discussion We have presented OASIS, a scalable algorithm for learning image similarity that captures both semantic and visual aspects of image similarity. [sent-686, score-0.398]

93 Large scale similarity learning, applied to images from a large variety of classes, could therefore be a useful tool to address real-world problems with a large number of classes. [sent-699, score-0.401]

94 Human Evaluation The following text was given as instructions to human evaluators when judging the relevance of images to a query image. [sent-704, score-0.599]

95 Scenario: A user is searching images to use in a presentation he/she plans to give. [sent-705, score-0.248]

96 The user then wishes to refine the search and look for images that are SEMANTICALLY similar to the query image. [sent-707, score-0.381]

97 Your task: You will see a set of query images on the left side of the screen, and a set of potential candidate matches, 5 per row, on the right. [sent-711, score-0.381]

98 Your job is to decide for each of the candidate images if it is a good semantic match to the query image or not. [sent-712, score-0.543]

99 A discriminative kernel-based model to rank images from text queries. [sent-790, score-0.272]

100 Learning to retrieve images from text queries with a discriminative model. [sent-796, score-0.35]

similar papers computed by tfidf model

tfidf for this paper:

wordName wordTfidf (topN-words)

[('oasis', 0.765), ('images', 0.248), ('lmnn', 0.176), ('lego', 0.138), ('precision', 0.136), ('query', 0.133), ('similarity', 0.13), ('pi', 0.127), ('evaluators', 0.107), ('image', 0.106), ('halit', 0.099), ('harma', 0.099), ('hechik', 0.099), ('semantically', 0.088), ('mage', 0.085), ('imilarity', 0.085), ('mcml', 0.084), ('queries', 0.078), ('arge', 0.077), ('cale', 0.077), ('engio', 0.077), ('rqi', 0.069), ('nline', 0.066), ('semantic', 0.056), ('weinberger', 0.056), ('rii', 0.053), ('triplets', 0.053), ('sw', 0.053), ('psd', 0.052), ('lw', 0.051), ('crammer', 0.05), ('relevance', 0.049), ('mahalanobis', 0.046), ('neighbors', 0.042), ('google', 0.041), ('retrieved', 0.041), ('metric', 0.039), ('fro', 0.038), ('kulis', 0.038), ('rca', 0.038), ('truck', 0.038), ('jain', 0.038), ('wi', 0.038), ('human', 0.038), ('online', 0.037), ('million', 0.037), ('earning', 0.034), ('distance', 0.034), ('pr', 0.033), ('ojala', 0.033), ('nearest', 0.032), ('relevant', 0.031), ('evaluator', 0.031), ('frome', 0.031), ('grangier', 0.031), ('proj', 0.031), ('shalit', 0.031), ('visterm', 0.031), ('ranked', 0.03), ('training', 0.03), ('learns', 0.03), ('pairwise', 0.03), ('web', 0.029), ('matlab', 0.029), ('vi', 0.029), ('descriptors', 0.027), ('euclidean', 0.027), ('top', 0.026), ('classes', 0.025), ('vision', 0.024), ('globerson', 0.024), ('text', 0.024), ('scale', 0.023), ('runtime', 0.023), ('block', 0.023), ('logdet', 0.023), ('lwi', 0.023), ('quelhas', 0.023), ('sharma', 0.023), ('takala', 0.023), ('tieu', 0.023), ('color', 0.023), ('pt', 0.022), ('triplet', 0.022), ('hyper', 0.022), ('grif', 0.022), ('pairs', 0.021), ('objects', 0.021), ('positivity', 0.021), ('asymmetric', 0.021), ('days', 0.021), ('categories', 0.02), ('uri', 0.02), ('texture', 0.02), ('scales', 0.02), ('rankings', 0.02), ('rankers', 0.02), ('descriptor', 0.02), ('codebook', 0.02), ('gal', 0.02)]

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Abstract: For many supervised learning tasks it may be infeasible (or very expensive) to obtain objective and reliable labels. Instead, we can collect subjective (possibly noisy) labels from multiple experts or annotators. In practice, there is a substantial amount of disagreement among the annotators, and hence it is of great practical interest to address conventional supervised learning problems in this scenario. In this paper we describe a probabilistic approach for supervised learning when we have multiple annotators providing (possibly noisy) labels but no absolute gold standard. The proposed algorithm evaluates the different experts and also gives an estimate of the actual hidden labels. Experimental results indicate that the proposed method is superior to the commonly used majority voting baseline. Keywords: multiple annotators, multiple experts, multiple teachers, crowdsourcing 1. Supervised Learning From Multiple Annotators/Experts A typical supervised learning scenario consists of a training set D = {(xi , yi )}N containing N i=1 instances, where xi ∈ X is an instance (typically a d-dimensional feature vector) and yi ∈ Y is the corresponding known label. The task is to learn a function f : X → Y which generalizes well on unseen data. Speciﬁcally for binary classiﬁcation the supervision is from the set Y = {0, 1}, for multi-class classiﬁcation Y = {1, . . . , K}, for ordinal regression Y = {1, . . . , K} (with an ordering 1 < . . . < K), and Y = R for regression. c 2010 Vikas C. Raykar, Shipeng Yu, Linda H. Zhao, Gerardo H. Valadez, Charles Florin, Luca Bogoni and Linda Moy. R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY However, for many real life tasks, it may not be possible, or may be too expensive (or tedious) to acquire the actual label yi for training—which we refer to as the gold standard or the objective ground truth. Instead, we may have multiple (possibly noisy) labels y1 , . . . , yR provided by R i i different experts or annotators. In practice, there is a substantial amount of disagreement among the experts, and hence it is of great practical interest to address conventional supervised learning algorithms in this scenario. Our motivation for this work comes from the area of computer-aided diagnosis1 (CAD), where the task is to build a classiﬁer to predict whether a suspicious region on a medical image (like a X-ray, CT scan, or MRI) is malignant (cancerous) or benign. In order to train such a classiﬁer, a set of images is collected from hospitals. The actual gold standard (whether it is cancer or not) can only be obtained from a biopsy of the tissue. Since it is an expensive, invasive, and potentially dangerous process, often CAD systems are built from labels assigned by multiple radiologists who identify the locations of malignant lesions. Each radiologist visually examines the medical images and provides a subjective (possibly noisy) version of the gold standard.2 The radiologist also annotates various descriptors of the potentially malignant lesion, like the size (a regression problem), shape (a multiclass classiﬁcation problem), and also degree of malignancy (an ordinal regression problem). The radiologists come from a diverse pool including luminaries, experts, residents, and novices. Very often there is lot of disagreement among the annotations. For a lot of tasks the labels provided by the annotators are inherently subjective and there will be substantial variation among different annotators. The domain of text classiﬁcation offers such a scenario. In this context the task is to predict the category for a token of text. The labels for training are assigned by human annotators who read the text and attribute their subjective category. With the advent of crowdsourcing (Howe, 2008) services like Amazon’s Mechanical Turk,3 Games with a Purpose,4 and reCAPTCHA5 it is quite inexpensive to acquire labels from a large number of annotators (possibly thousands) in a short time (Sheng et al., 2008; Snow et al., 2008; Sorokin and Forsyth, 2008). Websites such as Galaxy Zoo6 allow the public to label astronomical images over the internet. In situations like these, the performance of different annotators can vary widely (some may even be malicious), and without the actual gold standard, it may not be possible to evaluate the annotators. In this work, we provide principled probabilistic solutions to the following questions: 1. How to adapt conventional supervised learning algorithms when we have multiple annotators providing subjective labels but no objective gold standard? 2. How to evaluate systems when we do not have absolute gold-standard? 3. A closely related problem—particularly relevant when there are a large number of annotators— is to estimate how reliable/trustworthy is each annotator. 1. See Fung et al. (2009) for an overview of the data mining issues in this area. 2. Sometimes even a biopsy cannot conﬁrm whether it is cancer or not and hence all we can hope to get is subjective ground truth. 3. Mechanical Turk found at https://www.mturk.com. 4. Games with a Purpose found at http://www.gwap.com. 5. reCAPTCHA found at http://recaptcha.net/. 6. Galaxy Zoo found at http://galaxyzoo.org. 1298 L EARNING F ROM C ROWDS 1.1 The Problem With Majority Voting When we have multiple labels a commonly used strategy is to use the labels on which the majority of them agree (or average for regression problem) as an estimate of the actual gold standard. For binary classiﬁcation problems this amounts to using the majority label,7 that is, j yi = ˆ 1 if (1/R) ∑R yi > 0.5 j=1 , j 0 if (1/R) ∑R yi < 0.5 j=1 as an estimate of the hidden true label and use this estimate to learn and evaluate classiﬁers/annotators. Another strategy is that of considering every pair (instance, label) provided by each expert as a separate example. Note that this amounts to using a soft probabilistic estimate of the actual ground truth to learn the classiﬁer, that is, R Pr[yi = 1|y1 , . . . , yR ] = (1/R) ∑ yi . i i j j=1 Majority voting assumes all experts are equally good. However, for example, if there is only one true expert and the majority are novices, and if novices give the same incorrect label to a speciﬁc instance, then the majority voting method would favor the novices since they are in a majority. One could address this problem by introducing a weight capturing how good each expert is. But how would one measure the performance of an expert when there is no gold standard available? 1.2 Proposed Approach and Organization To address the apparent chicken-and-egg problem, we present a maximum-likelihood estimator that jointly learns the classiﬁer/regressor, the annotator accuracy, and the actual true label. For ease of exposition we start with binary classiﬁcation problem in § 2. The performance of each annotator is measured in terms of the sensitivity and speciﬁcity with respect to the unknown gold standard (§ 2.1). The proposed algorithm automatically discovers the best experts and assigns a higher weight to them. In order to incorporate prior knowledge about each annotator, we impose a beta prior on the sensitivity and speciﬁcity and derive the maximum-a-posteriori estimate (§ 2.6). The ﬁnal estimation is performed by an Expectation Maximization (EM) algorithm that iteratively establishes a particular gold standard, measures the performance of the experts given that gold standard, and reﬁnes the gold standard based on the performance measures. While the proposed approach is described using logistic regression as the base classiﬁer (§ 2.2), it is quite general, and can be used with any black-box classiﬁer (§ 2.7), and can also handle missing labels (that is, each expert is not required to label all the instances). Furthermore, we extend the proposed algorithm to handle categorical (§ 3), ordinal (§ 4), and regression problems (§ 5). In § 6 section we extensively validate our approach using both simulated data and real data from different domains. 1.3 Related Work and Novel Contributions We ﬁrst summarize the novel contributions of this work in context of other related work in this emerging new area. There has been a long line of work in the biostatistics and epidemiology literature on latent variable models where the task is to get an estimate of the observer error rates based 7. When there is no clear majority among the multiple experts (that is, yi = 0.5) in CAD domain the ﬁnal decision is ˆ often made by an adjudicator or a super-expert. When there is no adjudicator a fair coin toss is used. 1299 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY on the results from multiple diagnostic tests without a gold standard (see Dawid and Skene, 1979, Hui and Walter, 1980, Hui and Zhou, 1998, Albert and Dodd, 2004 and references therein). In the machine learning community Smyth et al. (1995) ﬁrst addressed the same problem in the context of labeling volcanoes in satellite images of Venus. We differ from this previous body of work in the following aspects: 1. Unlike Dawid and Skene (1979) and Smyth et al. (1995) which just focused on estimating the ground truth from multiple noisy labels, we speciﬁcally address the issue of learning a classiﬁer. Estimating the ground truth and the annotator/classifer performance is a byproduct of our proposed algorithm. 2. In order to learn a classiﬁer Smyth (1995) proposed to ﬁrst estimate the ground truth (without using the features) and then use the probabilistic ground truth to learn a classiﬁer. In contrast, our proposed algorithm learns the classiﬁer and the ground truth jointly. Our experiments (§ 6.1.1) show that the classiﬁer learnt and ground truth obtained by the proposed algorithm is superior to that obtained by other procedures which ﬁrst estimates the ground truth and then learns the classiﬁer. 3. Our solution is more general and can be easily extended to categorical(§ 3), ordinal(§ 4), and continuous data(§ 5). It can also be used in conjunction with any supervised learning algorithm. A preliminary version of this paper (Raykar et al., 2009) mainly discussed the binary classiﬁcation problem. 4. Our proposed algorithm is also Bayesian—we impose a prior on the experts. The priors can potential capture the skill of different annotators. In this paper we refrain from doing a full Bayesian inference and use the mode of the posterior as a point estimate. A recent complete Bayesian generalization of these kind of models has been developed by Carpenter (2008). 5. The EM approach used in this paper is similar to that proposed by Jin and Ghahramani (2003). However their motivation is somewhat different. In their setting, each training example is annotated with a set of possible labels, only one of which is correct. There has been recent interest in the natural language processing (Sheng et al., 2008; Snow et al., 2008) and computer vision (Sorokin and Forsyth, 2008) communities where they use Amazon’s Mechanical Turk to collect annotations from many people. They show that it can be potentially as good as that provided by an expert. Sheng et al. (2008) analyzed when it is worthwhile to acquire new labels for some of the training examples. There is also some theoretical work (see Lugosi, 1992 and Dekel and Shamir, 2009a) dealing with multiple experts. Recently Dekel and Shamir (2009b) presented an algorithm which does not resort to repeated labeling, that is, each example does not have to be labeled by multiple teachers. Donmez et al. (2009) address the issue of active learning in this scenario—How to jointly learn the accuracy of labeling sources and obtain the most informative labels for the active learning task? There has also been some work in the medical imaging community (Warﬁeld et al., 2004; Cholleti et al., 2008). 2. Binary Classiﬁcation We ﬁrst describe our proposed noise model for the annotators. The performance of each annotator is measured in terms of the sensitivity and speciﬁcity with respect to the unknown gold standard. 1300 L EARNING F ROM C ROWDS 2.1 A Two-coin Model for Annotators Let y j ∈ {0, 1} be the label assigned to the instance x by the jth annotator/expert. Let y be the actual (unobserved) label for this instance. Each annotator provides a version of this hidden true label based on two biased coins. If the true label is one, she ﬂips a coin with bias α j (sensitivity). If the true label is zero, she ﬂips a coin with bias β j (speciﬁcity). In each case, if she gets heads she keeps the original label, otherwise she ﬂips the label. If the true label is one, the sensitivity (true positive rate) for the jth annotator is deﬁned as the probability that she labels it as one. α j := Pr[y j = 1|y = 1]. (1) On the other hand, if the true label is zero, the speciﬁcity (1−false positive rate) is deﬁned as the probability that she labels it as zero. β j := Pr[y j = 0|y = 0]. (2) The assumption introduced is that α j and β j do not depend on the instance x. For example, in the CAD domain, this means that the radiologist’s performance is consistent across different sub-groups of data.8 2.2 Classiﬁcation Model While the proposed method can be used for any classiﬁer, for ease of exposition, we consider the family of linear discriminating functions: F = { fw}, where for any x, w ∈ Rd , fw(x) = w⊤ x. The ﬁnal classiﬁer can be written in the following form: y = 1 if w⊤ x ≥ γ and 0 otherwise. The ˆ threshold γ determines the operating point of the classiﬁer. The Receiver Operating Characteristic (ROC) curve is obtained as γ is swept from −∞ to ∞. The probability for the positive class is modeled as a logistic sigmoid acting on fw, that is, Pr[y = 1|x, w] = σ(w⊤ x), where the logistic sigmoid function is deﬁned as σ(z) = 1/(1 + e−z ). This classiﬁcation model is known as logistic regression. 2.3 Estimation/Learning Problem Given the training data D consisting of N instances with annotations from R annotators, that is, D = {xi , y1 , . . . , yR }N , the task is to estimate the weight vector w and also the sensitivity α = i i i=1 [α1 , . . . , αR ] and the speciﬁcity β = [β1 , . . . , βR ] of the R annotators. It is also of interest to get an estimate of the unknown gold standard y1 , . . . , yN . 2.4 Maximum Likelihood Estimator Assuming the training instances are independently sampled, the likelihood function of the parameters θ = {w, α, β} given the observations D can be factored as N Pr[D |θ] = ∏ Pr[y1 , . . . , yR |xi , θ]. i i i=1 8. While this is a reasonable assumption, it is not entirely true. It is known that some radiologists are good at detecting certain kinds of malignant lesions based on their training and experience. 1301 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY j Conditioning on the true label yi , and also using the assumption yi is conditionally independent (of everything else) given α j , β j and yi , the likelihood can be decomposed as N Pr[D |θ] = ∏ Pr[y1 , . . . , yR |yi = 1, α]Pr[yi = 1|xi, w] i i i=1 + Pr[y1 , . . . , yR |yi = 0, β]Pr[yi = 0|xi, w] . i i Given the true label yi , we assume that y1 , . . . , yR are independent, that is, the annotators make their i i decisions independently.9 Hence, R R Pr[y1 , . . . , yR |yi = 1, α] = ∏ Pr[yi |yi = 1, α j ] = ∏ [α j ]yi [1 − α j ]1−yi . i i j j=1 j j j=1 Similarly, we have R Pr[y1 , . . . , yR |yi = 0, β] = ∏ [β j ]1−yi [1 − β j ]yi . i i j j j=1 Hence the likelihood can be written as N Pr[D |θ] = ∏ ai pi + bi (1 − pi ) , i=1 where we have deﬁned pi := σ(w⊤ xi ). R ai := ∏[α j ]y [1 − α j ]1−y . j i j i j=1 R bi := ∏[β j ]1−y [1 − β j ]y . j i j i j=1 The maximum-likelihood estimator is found by maximizing the log-likelihood, that is, ˆ θML = {α, β, w} = arg max{ln Pr[D |θ]}. ˆ ˆ ˆ θ 2.5 The EM Algorithm This maximization problem can be simpliﬁed a lot if we use the Expectation-Maximization (EM) algorithm (Dempster et al., 1977). The EM algorithm is an efﬁcient iterative procedure to compute the maximum-likelihood solution in presence of missing/hidden data. We will use the unknown hidden true label yi as the missing data. If we know the missing data y = [y1 , . . . , yN ] then the complete likelihood can be written as N ln Pr[D , y|θ] = ∑ yi ln pi ai + (1 − yi ) ln(1 − pi )bi . i=1 9. This assumption is not true in general and there is some correlations among the labels assigned by multiple annotators. For example in the CAD domain if the cancer is in advanced stage (which is very easy to detect) almost all the radiologists assign the same label. 1302 L EARNING F ROM C ROWDS Each iteration of the EM algorithm consists of two steps: an Expectation(E)-step and a Maximization(M)step. The M-step involves maximization of a lower bound on the log-likelihood that is reﬁned in each iteration by the E-step. 1. E-step. Given the observation D and the current estimate of the model parameters θ, the conditional expectation (which is a lower bound on the true likelihood) is computed as N E {ln Pr[D , y|θ]} = ∑ µi ln pi ai + (1 − µi ) ln(1 − pi )bi , (3) i=1 where the expectation is with respect to Pr[y|D , θ], and µi = Pr[yi = 1|y1 , . . . , yR , xi , θ]. Using i i Bayes’ theorem we can compute µi ∝ Pr[y1 , . . . , yR |yi = 1, θ] · Pr[yi = 1|xi, θ] i i ai pi = . ai pi + bi (1 − pi ) 2. M-step. Based on the current estimate µi and the observations D , the model parameters θ are then estimated by maximizing the conditional expectation. By equating the gradient of (3) to zero we obtain the following estimates for the sensitivity and speciﬁcity: j j αj = ∑N µi yi i=1 , ∑N µi i=1 βj = ∑N (1 − µi )(1 − yi ) i=1 . ∑N (1 − µi ) i=1 Due to the non-linearity of the sigmoid, we do not have a closed form solution for w and we have to use gradient ascent based optimization methods. We use the Newton-Raphson update given by wt+1 = wt − ηH −1 g, where g is the gradient vector, H is the Hessian matrix, and η is the step length. The gradient vector is given by N g(w) = ∑ µi − σ(w⊤ xi ) xi . i=1 The Hessian matrix is given by N H(w) = − ∑ σ(w⊤ xi ) 1 − σ(w⊤ xi ) xi x⊤ . i i=1 Essentially, we are estimating a logistic regression model with probabilistic labels µi . These two steps (the E- and the M-step) can be iterated till convergence. The log-likelihood increases monotonically after every iteration, which in practice implies convergence to a local maximum. The EM algorithm is only guaranteed to converge to a local maximum. In practice multiple restarts with different initializations can potentially mitigate the local maximum problem. In this j paper we use majority voting µi = 1/R ∑R yi as the initialization for µi to start the EM-algorithm. j=1 1303 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 2.6 A Bayesian Approach In some applications we may want to trust a particular expert more than the others. One way to achieve this is by imposing priors on the sensitivity and speciﬁcity of the experts. Since α j and β j represent the probability of a binary event, a natural choice of prior is the beta prior. The beta prior is also conjugate to the binomial distribution. For any a > 0, b > 0, and δ ∈ [0, 1] the beta distribution is given by δa−1 (1 − δ)b−1 , Beta(δ|a, b) = B(a, b) where B(a, b) = 01 δa−1 (1 − δ)b−1 dδ is the beta function.We assume a beta prior10 for both the sensitivity and the speciﬁcity as R j j j j j j j j Pr[α j |a1 , a2 ] = Beta(α j |a1 , a2 ). Pr[β j |b1 , b2 ] = Beta(β j |b1 , b2 ). For sake of completeness we also assume a zero mean Gaussian prior on the weights w with inverse covariance matrix Γ, that is, Pr[w] = N (w|0, Γ−1 ). Assuming that {α j }, {β j }, and w have independent priors, the maximum-a-posteriori (MAP) estimator is found by maximizing the logposterior, that is, ˆ θMAP = arg max{ln Pr[D |θ] + ln Pr[θ]}. θ An EM algorithm can be derived in a similar fashion for MAP estimation by relying on the interpretation of Neal and Hinton (1998). The ﬁnal algorithm is summarized below: j 1. Initialize µi = (1/R) ∑R yi based on majority voting. j=1 2. Given µi , estimate the sensitivity and speciﬁcity of each annotator/expert as follows. j αj = j a1 − 1 + ∑N µi yi i=1 j j a1 + a2 − 2 + ∑N µi i=1 j β j = . j b1 − 1 + ∑N (1 − µi )(1 − yi ) i=1 j j b1 + b2 − 2 + ∑N (1 − µi ) i=1 . (4) The Newton-Raphson update for optimizing w is given by wt+1 = wt − ηH −1 g, with step length η, gradient vector N g(w) = ∑ µi − σ(w⊤ xi ) xi − Γw, i=1 and Hessian matrix N H(w) = − ∑ σ(w⊤ xi ) 1 − σ(w⊤ xi ) xi x⊤ − Γ. i i=1 10. It may be convenient to specify a prior in terms of the mean µ and variance σ2 . The mean and the variance for a beta prior are given by µ = a/(a + b) and σ2 = ab/((a + b)2 (a + b + 1)). Solving for a and b we get a = (−µ3 + µ2 − µσ2 )/σ2 and b = a(1 − µ)/µ. 1304 L EARNING F ROM C ROWDS 3. Given the sensitivity and speciﬁcity of each annotator and the model parameters, update µi as µi = ai pi , ai pi + bi (1 − pi ) (5) where pi = σ(w⊤ xi ). R ai = ∏[α j ]y [1 − α j ]1−y . j i j i j=1 R bi = ∏[β j ]1−y [1 − β j ]y . j i j i (6) j=1 Iterate (2) and (3) till convergence. 2.7 Discussions 1. Estimate of the gold standard The value of the posterior probability µi is a soft probabilistic estimate of the actual ground truth yi , that is, µi = Pr[yi = 1|y1 , . . . , yR , xi , θ]. The actual i i hidden label yi can be estimated by applying a threshold on µi , that is, yi = 1 if µi ≥ γ and zero otherwise. We can use γ = 0.5 as the threshold. By varying γ we can change the misclassiﬁcation costs and obtain a ground truth with large sensitivity or large speciﬁcity. Because of this in our experimental validation we can actually draw an ROC curve for the estimated ground truth. 2. Log-odds of µ A particularly revealing insight can be obtained in terms of the log-odds or the logit of the posterior probability µi . From (5) the logit of µi can be written as logit(µi ) = ln Pr[yi = 1|y1 , . . . , yR , xi , θ] µi i i = ln 1 − µi Pr[yi = 0|y1 , . . . , yR , xi , θ] i i R = w⊤ xi + c + ∑ yi [logit(α j ) + logit(β j )]. j j=1 j where c = ∑R log 1−α is a constant term which does not depend on i. This indicates that j=1 βj the estimated ground truth (in the logit form of the posterior probability) is a weighted linear combination of the labels from all the experts. The weight of each expert is the sum of the logit of the sensitivity and speciﬁcity. 3. Using any other classiﬁer For ease of exposition we used logistic regression. However, the proposed algorithm can be used with any generalized linear model or in fact with any classiﬁer that can be trained with soft probabilistic labels. In each step of the EM-algorithm, the classiﬁer is trained with instances sampled from µi . This modiﬁcation is easy for most probabilistic classiﬁers. For general black-box classiﬁers where we cannot tweak the training algorithm an alternate approach is to replicate the training examples according to the soft label. For example a probabilistic label µi = 0.8 can be effectively simulated by adding 8 training examples with deterministic label 1 and 2 examples with label 0. 1305 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 4. Obtaining ground truth with no features In some scenarios we may not have features xi and we wish to obtain an estimate of the actual ground truth based only on the labels from multiple annotators. Here instead of learning a classiﬁer we estimate p which is the prevalence of the positive class, that is, p = Pr[yi = 1]. We further assume a beta prior for the prevalence, that is, Beta(p|p1 , p2 ). The algorithm simpliﬁes as follows. j (a) Initialize µi = (1/R) ∑R yi based on majority voting. j=1 (b) Given µi , estimate the sensitivity and speciﬁcity of each annotator using (4). The prevalence of the positive class is estimated as follows. p = p1 − 1 + ∑N µi i=1 . p1 + p2 − 2 + N (c) Given the sensitivity and speciﬁcity of each annotator and prevalence, reﬁne µi as follows. ai p µi = . ai p + bi (1 − p) Iterate (2) and (3) till convergence. This algorithm is similar to the one proposed by Dawid and Skene (1979) and Smyth et al. (1995). 5. Handling missing labels The proposed approach can easily handle missing labels, that is, when the labels from some experts are missing for some instances. Let Ri be the number of radiologists labeling the ith instance, and let N j be the number of instances labeled by the jth radiologist. Then in the EM algorithm, we just need to replace N by N j for estimating the sensitivity and speciﬁcity in (4), and replace R by Ri for updating µi in (6). 6. Evaluating a classiﬁer We can use the probability scores µi directly to evaluate classiﬁers. If zi are the labels obtained from any other classiﬁer, then sensitivity and speciﬁcity can be estimated as ∑N (1 − µi )(1 − zi ) ∑N µi zi , β = i=1 N . α = i=1 ∑N µi ∑i=1 (1 − µi ) i=1 7. Posterior approximation At the end of each EM iteration a crude approximation to the posterior is obtained as N N α j ∼ Beta α j |a1 + ∑ µi yi , a2 + ∑ µi (1 − yi ) , j j i=1 N j j i=1 N β j ∼ Beta β j |b1 + ∑ (1 − µi )(1 − yi ), b2 + ∑ (1 − µi )yi j j i=1 j j . i=1 3. Multi-class Classiﬁcation In this section we describe how the proposed approach for binary classiﬁcation can be extended to categorical data. Suppose there are K ≥ 2 categories. An example for categorical data from the CAD domain is in LungCAD, where the radiologist needs to label whether a nodule (known to be precursors of cancer) is a solid, a part-solid, or a ground glass opacity—which are three 1306 L EARNING F ROM C ROWDS different kinds on nodules. We can extend the previous model and introduce a vector of multinomial j j j parameters αc = (αc1 , . . . , αcK ) for each annotator, where αck := Pr[y j = k|y = c] j and ∑K αck = 1. Here αck denotes the probability that the annotator j assigns class k to an instance k=1 j j given the true class is c. When K = 2, α11 and α00 are sensitivity and speciﬁcity, respectively. A similar EM algorithm can be derived. In the E-step, we estimate j j R K Pr[yi = c|Y , α] ∝ Pr[yi = c|xi ] ∏ ∏ (αck )δ(yi ,k) , j j j=1 k=1 where δ(u, v) = 1 if u = v and 0 otherwise and in the M-step we learn a multi-class classiﬁer and update the multinomial parameter as j αck = j ∑N Pr[yi = c|Y , α]δ(yi , k) i=1 . ∑N Pr[yi = c|Y , α] i=1 One can also assign a Dirichlet prior for the multinomial parameters, and this results in a smoothing term in the above updates in the MAP estimate. 4. Ordinal Regression We now consider the situation where the outputs are categorical and have an ordering among the labels. In the CAD domain the radiologist often gives a score (for example, 1 to 5 from lowest to highest) to indicate how likely she thinks it is malignant. This is different from a multi-class setting in which we do not have any preference among the multiple class labels. j Let yi ∈ {1, . . . , K} be the label assigned to the ith instance by the jth expert. Note that there is an ordering in the labels 1 < . . . < K. A simple approach is to convert the ordinal data into a series of binary data (Frank and Hall, 2001). Speciﬁcally the K class ordinal labels are transformed into K − 1 binary class labels as follows: jc yi = j 1 if yi > c 0 otherwise c = 1, . . . , K − 1. Applying the same procedure used for binary labels we can estimate Pr[yi > c] for c = 1, . . . , K − 1. The probability of the actual class values can then be obtained as Pr[yi = c] = Pr[yi > c − 1 and yi ≤ c] = Pr[yi > c − 1] − Pr[yi > c]. The class with the maximum probability is assigned to the instance. 5. Regression In this section we develop a similar algorithm to learn a regression function using annotations from multiple experts. In the CAD domain as a part of the annotation process a common task for a radiologist is to measure the diameter of a suspicious lesion. 1307 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 5.1 Model for Annotators j Let yi ∈ R be the continuous target value assigned to the ith instance by the jth annotator. Our model is that the annotator provides a noisy version of the actual true value yi . For the jth annotator we will assume a Gaussian noise model with mean yi (the true unknown value) and inverse-variance (precision) τ j , that is, j j Pr[yi |yi , τ j ] = N (yi |yi , 1/τ j ), (7) where the Gaussian distribution is deﬁned as N (z|m, σ2 ) = (2πσ2 )−1/2 exp(−(z − m)2 /2σ2 ). The unknown inverse-variance τ j measures the accuracy of each annotator—the larger the value of τ j the more accurate the annotator. We have assumed that τ j does not depend on the instance xi . For example, in the CAD domain, this means that the radiologist’s accuracy does not depend on the nodule she is measuring. While this a practical assumption, it is not entirely true. It is known that some nodules are harder to measure than others. 5.2 Linear Regression Model for Features As before we consider the family of linear regression functions: F = { fw}, where for any x, w ∈ Rd , fw(x) = w⊤ x. We assume that the actual target response yi is given by the deterministic regression function fw with additive Gaussian noise, that is, yi = w⊤ xi + ε, where ε is a zero-mean Gaussian random variable with inverse-variance (precision) γ. Hence Pr[yi |xi , w, γ] = N (yi |w⊤ xi , 1/γ). (8) 5.3 Combined Model Combining both the annotator (7) and the regressor (8) model we have Pr[yi |xi , w, τ j , γ] = j Z Pr[yi |yi , τ j ]Pr[yi |xi , w, γ]dyi = N (yi |w⊤ xi , 1/γ + 1/τ j ). j j Since the two precision terms (γ and τ j ) are grouped together they are not uniquely identiﬁable. Hence we will deﬁne a new precision term λ j as 1 1 1 = + j. j λ γ τ So we have the following model Pr[yi |xi , w, λ j ] = N (yi |w⊤ xi , 1/λ j ). j j (9) 5.4 Estimation/Learning Problem Given the training data D consisting of N instances with annotations from R experts, that is, D = {xi , y1 , . . . , yR }N , the task is to estimate the weight vector w and the precision λ = [λ1 , . . . , λR ] of i i=1 i all the annotators. 1308 L EARNING F ROM C ROWDS 5.5 Maximum-likelihood Estimator Assuming the instances are independent the likelihood of the parameters θ = {w, λ} given the observations D can be factored as N Pr[D |θ] = ∏ Pr[y1 , . . . , yR |xi , θ]. i i i=1 Conditional on the instance xi we assume that y1 , . . . , yR are independent, that is, the annotators i i provide their responses independently. Hence from (9) the likelihood can be written as N R Pr[D |θ] = ∏ ∏ N (yi |w⊤ xi , 1/λ j ). j i=1 j=1 The maximum-likelihood estimator is found by maximizing the log-likelihood θML = {λ, w} = arg max{ln Pr[D |θ]}. θ By equating the gradient of the log-likelihood to zero we obtain the following update equations for the precision and the weight vector. 1 λj = 1 N ∑ yij − w⊤ xi N i=1 N w = ∑ −1 2 . (10) N ∑R λ j yi j=1 i=1 ∑R λ j j=1 ∑ xi xi x⊤ i i=1 j . (11) As the parameters w and λ are coupled together we iterate these two steps till convergence. 5.6 Discussions 1. Is this standard least-squares? Deﬁne the design matrix X = [x1 , . . . , xN ]⊤ and the rej j sponse vector for each annotator as y j = [y1 , . . . , yN ]⊤ . Using matrix notation Equation 11 can be written as w = (X ⊤ X)−1 X ⊤ y where y = ∑R λ j y j j=1 ∑R λ j j=1 . (12) Equation 12 is essentially the solution to a standard linear regression model, except that we are training a linear regression model with y as the ground truth, which is a precision weighted mean of the response vectors from all the annotators. The variance of each annotator is estimated using (10). The ﬁnal algorithm iteratively establishes a particular gold standard (y), measures the performance of the annotators and learns a regressor given that gold standard, and reﬁnes the gold standard based on the performance measures. 2. Are we better than the best annotator? If we assume λ is ﬁxed (i.e., we ignore the variability and assume that it is well estimated) then w is an unbiased estimator of w and the ˆ covariance matrix is given by Cov(w) = Cov(y) X ⊤ X ˆ 1309 −1 = 1 ∑R λ j j=1 X ⊤X −1 . R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY Since ∑R λ j > max j (λ j ) the proposed method has a lower variance than the regressor learnt j=1 with the best annotator (i.e., the one with the minimum variance). 3. Are we better than the average? For a ﬁxed X the error in w depends only on the variance of y j . If we know the true λ j then yi is the best linear unbiased estimator for yi which minij mizes the variance. To see this consider any linear estimator of the form yi = ∑ j a j (yi − b j ). The variance is given by Var[yi ] = ∑ j (a j )2 /λ j . Since E[yi ] = yi ∑ j a j , for the bias of this estimator to be zero we require that ∑ j a j = 1. Solving the constrained minimization problem we see that a j = λ j / ∑ j λ j minimizes the variance. 4. Obtaining a consensus without features When no features are available the same algorithm can be simpliﬁed to get a consensus estimate of the actual ground truth and also evaluate the annotators. Essentially we have to iterate the following two updates till convergence ∑R λ j yi j=1 1 ∑R λ j j=1 λj j yi = = 1 N ∑ y j − yi N i=1 i 2 . 6. Experimental Validation We now experimentally validate the proposed algorithms on both simulated and real data. 6.1 Classiﬁcation Experiments We use two CAD and one text data set in our experiments. The CAD data sets include a digital mammography data set and a breast MRI data set, both of which are biopsy proven, that is, the gold standard is available. For the digital mammography data set we simulate the radiologists in order to validate our methods. The breast MRI data has annotations from four radiologists. We also report results on a Recognizing Textual Entailment data collected by Snow et al. (2008) using the Amazon’s Mechanical Turk which has annotations from 164 annotators. 6.1.1 D IGITAL M AMMOGRAPHY WITH S IMULATED R ADIOLOGISTS Mammograms are used as a screening tool to detect early breast cancer. CAD systems search for abnormal areas (lesions) in a digitized mammographic image. These lesions generally indicate the presence of malignant cancer. The CAD system then highlights these areas on the images, alerting the radiologist to the need for a further diagnostic mammogram or a biopsy. In classiﬁcation terms, given a set of descriptive morphological features for a region on a image, the task is to predict whether it is potentially malignant (1) or not (0). In order to train such a classiﬁer, a set of mammograms is collected from hospitals. The ground truth (whether it is cancer or not) is obtained from biopsy. Since biopsy is an expensive, tedious, and an invasive process, very often CAD systems are built from labels collected from multiple expert radiologists who visually examine the mammograms and mark the lesion locations—this constitutes our ground truth (multiple labels) for learning. In this experiment we use a proprietary biopsy-proven data set (Krishnapuram et al., 2008) containing 497 positive and 1618 negative examples. Each instance is described by a set of 27 morphological features. In order to validate our proposed algorithm, we simulate multiple radiologists according to the two-coin model described in § 2.1. Based on the labels from multiple radiologists, 1310 L EARNING F ROM C ROWDS we can simultaneously (1) learn a logistic-regression classiﬁer, (2) estimate the sensitivity and speciﬁcity of each radiologist, and (3) estimate the golden ground truth. We compare the results with the classiﬁer trained using the biopsy proved ground truth as well as the majority-voting baseline. For the ﬁrst set of experiments we use 5 radiologists with sensitivity α = [0.90 0.80 0.57 0.60 0.55] and speciﬁcity β = [0.95 0.85 0.62 0.65 0.58]. This corresponds to a scenario where the ﬁrst two radiologists are experts and the last three are novices. Figure 1 summarizes the results. We compare on three different aspects: (1) How good is the learnt classiﬁer? (2) How well can we estimate the sensitivity and speciﬁcity of each radiologist? (3) How good is the estimated ground truth? The following observations can be made. 1. Classiﬁer performance Figure 1(a) plots the ROC curve of the learnt classiﬁer on the training set. The dotted (black) line is the ROC curve for the classiﬁer learnt using the actual ground truth. The solid (red) line is the ROC curve for the proposed algorithm and the dashed (blue) line is for the classiﬁer learnt using the majority-voting scheme. The classiﬁer learnt using the proposed method is as good as the one learnt using the golden ground truth. The area under the ROC curve (AUC) for the proposed algorithm is around 3.5% greater than that learnt using the majority-voting scheme. 2. Radiologist performance The actual sensitivity and speciﬁcity of each radiologist is marked as a black × in Figure 1(b). The end of the solid red line shows the estimates of the sensitivity and speciﬁcity from the proposed method. We used a uniform prior on all the parameters. The ellipse plots the contour of one standard deviation as obtained from the beta posterior estimates. The end of the dashed blue line shows the estimate obtained from the majorityvoting algorithm. We see that the proposed method is much closer to the actual values of sensitivity and speciﬁcity. 3. Actual ground truth Since the estimates of the actual ground truth are probabilistic scores, we can also plot the ROC curves of the estimated ground truth. From Figure 1(b) we can see that the ROC curve for the proposed method dominates the majority voting ROC curve. Furthermore, the area under the ROC curve (AUC) is around 3% higher. The estimate obtained by majority voting is closer to the novices since they form a majority (3/5). It does not have an idea of who is an expert and who is a novice. The proposed algorithm appropriately weights each radiologist based on their estimated sensitivity and speciﬁcity. The improvement obtained is quite large in Figure 2 which corresponds a situation where we have only one expert and 7 novices. 4. Joint Estimation To learn a classiﬁer, Smyth et al. (1995) proposed to ﬁrst estimate the golden ground truth and then use the probabilistic ground truth to learn a classiﬁer. In contrast, our proposed algorithm learns the classiﬁer and the ground truth jointly as a part of the EM algorithm. Figure 3 shows that the classiﬁer and the ground truth learnt obtained by the proposed algorithm is superior than that obtained by other procedures which ﬁrst estimates the ground truth and then learns the classiﬁer. 6.1.2 B REAST MRI In this example, each radiologist reviews the breast MRI data and assesses the malignancy of each lesion on a BIRADS scale of 1 to 5. The BIRADS scale is deﬁned as follows: 1 Negative, 2 Benign, 1311 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY Majority Voting Estimated 1 Estimated 2 Estimated 3 Estimated 4 Estimated 5 EM algorithm Estimated 1 Estimated 2 Estimated 3 Estimated 4 Estimated 5 True 1 True 2 True 3 True 4 True 5 x x x x x 0.0217 0.5869 0.2391 0.1521 0.0000 0 0 0 1 0 x x x x x 0.0000 0.1785 0.1071 0.2500 0.4642 True 1 True 2 True 3 True 4 True 5 x x x x x 0.0000 0.6957 0.1304 0.1739 0.0000 0 0 0 1 0 x x x x x 0.0000 0.1428 0.0000 0.3214 0.5357 Table 1: The confusion matrix for the estimate obtained using majority voting and the proposed EM algorithm. The x indicates that there was no such category in the true labels (the gold standard). The gold-standard is obtained by the biopsy which can conﬁrm whether it is benign (BIRADS=2) or malignant (BIRADS=5). 3 Probably Benign, 4 Suspicious abnormality, and 5 Highly suggestive of malignancy. Our data set comprises of 75 lesions with annotations from four radiologists, and the true labels from biopsy. Based on eight morphological features, we have to predict whether a lesion is malignant or not. For the ﬁrst experiment we reduce the BIRADS scale to a binary one: any lesion with a BIRADS > 3 is considered malignant and benign otherwise. The set included 28 malignant and 47 benign lesions. Figure 4 summarizes the results. We show the leave-one-out cross validated ROC for the classiﬁer. The cross-validated AUC of the proposed method is approximately 6% better than the majority voting baseline. We also consider the BIRADS labels as a set of ordinal measurements since there is an ordering among the BIRADS label. The confusion matrix in Table 1 shows that the EM algorithm is signiﬁcantly superior than the majority voting in estimating the true BIRADS. 6.1.3 R ECOGNIZING T EXTUAL E NTAILMENT Finally we report results on Recognizing Textual Entailment data collected by Snow et al. (2008) using the Amazon’s Mechanical Turk. In this task, the annotator is presented with two sentences and given a choice of whether the second sentence can be inferred from the ﬁrst. The data has 800 tasks and 164 distinct readers, with 10 annotations per task along with the golden ground truth. The majority of the entries (94 %) in the 800x164 matrix are missing. There is one annotator who has labeled all the tasks. We use this data set to obtain an estimate of the actual ground truth. Figure 5 plots the accuracy of the estimated ground truth as a function of the number of annotators. The proposed EM algorithm achieves a higher accuracy than majority voting. In other words to achieve a desired accuracy the proposed algorithm needs fewer annotators than the majority voting scheme. 1312 L EARNING F ROM C ROWDS ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Golden ground truth AUC=0.915 Proposed EM algorithm AUC=0.913 Majority voting baseline AUC=0.882 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Proposed EM algorithm AUC=0.991 Majority voting baseline AUC=0.962 0.2 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 1: Results for the digital mammography data set with annotations from 5 simulated radiologists. (a) The ROC curve of the learnt classiﬁer using the golden ground truth (dotted black line), the majority voting scheme (dashed blue line), and the proposed EM algorithm (solid red line). (b) The ROC curve for the estimated ground truth. The actual sensitivity and speciﬁcity of each of the radiologists is marked as a ×. The end of the dashed blue line shows the estimates of the sensitivity and speciﬁcity obtained from the majority voting algorithm. The end of the solid red line shows the estimates from the proposed method. The ellipse plots the contour of one standard deviation. 1313 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Golden ground truth AUC=0.915 Proposed EM algorithm AUC=0.906 Majority voting baseline AUC=0.884 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Proposed EM algorithm AUC=0.967 Majority voting baseline AUC=0.872 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 2: Same as Figure 1 except with 8 different radiologist annotations. 1314 L EARNING F ROM C ROWDS ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Proposed EM algorithm [Joint Estimation] AUC=0.905 Decoupled Estimation AUC=0.884 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Proposed EM algorithm [Joint Estimation] AUC=0.972 Decoupled Estimation AUC=0.921 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 3: ROC curves comparing the proposed algorithm (solid red line) with the Decoupled Estimation procedure (dotted blue line), which refers to the algorithm where the ground truth is ﬁrst estimated using just the labels from the ﬁve radiologists and then a logistic regression classiﬁer is trained using the soft probabilistic labels. In contrast the proposed EM algorithm estimates the ground truth and learns the classiﬁer simultaneously during the EM algorithm. 1315 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY Leave−One−Out ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Golden ground truth AUC=0.909 Majority voting baseline AUC=0.828 Proposed EM algorithm AUC=0.879 0.2 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 0.9 True Positive Rate (sensitivity) 0.8 0.7 0.6 0.5 0.4 Proposed EM algorithm AUC=0.944 Majority voting baseline AUC=0.937 0.3 0.2 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 4: Breast MRI results. (a) The leave-one-out cross validated ROC. (b) ROC for the estimated ground truth. 1316 L EARNING F ROM C ROWDS 0.95 0.9 0.85 Accuracy 0.8 0.75 0.7 0.65 Majority Voting EM Algorithm 0.6 0.55 0.5 20 40 60 80 100 120 Number of Annotators 140 160 Figure 5: The mean and the one standard deviation error bars for the accuracy of the estimated ground truth for the Recognizing Textual Entailment task as a function of the number of annotators. The plot was generated by randomly sampling the annotators 100 times. 6.2 Regression Experiments We ﬁrst illustrate the algorithm on a toy dataset and then present a case study for automated polyp measurements. 6.2.1 I LLUSTRATION Figure 6 illustrates the the proposed algorithm for regression on a one-dimensional toy data set with three annotators. The actual regression model (shown as a blue dotted line) is given by y = 5x − 2. We simulate 20 samples from three annotators with precisions 0.01, 0.1, and 1.0. The data are shown by the annotators’s number. While we can ﬁt a regression model using each annotators’s response, we see that only the model for annotator three (with highest precision) is close to the true regression model. The green dashed line shows the model learnt using the average response from all the three annotators. The red line shows the model learnt by the proposed algorithm. 6.2.2 AUTOMATED P OLYP M EASUREMENTS Colorectal polyps are small colonic ﬁndings that may develop into cancer at a later stage. The diameter of the polyp is one of the key factors which decides the malignancy of a suspicious polyp. Hence accurate size estimation is crucial to decide the action to be taken on a polyp. We have developed various algorithms to segment a polyp. Multiple segmentation algorithms give rise to a set of features which are correlated with the diameter of the polyp. We want to learn a regression function which can predict the diameter of a polyp as a function of these features. In order to learn a regression function we collect our ground truth by asking many radiologists to manually measure the the diameter of the polyps from the three-dimensional images. In practice there is a lot of disagreement among the radiologists as to the actual size of the polyp. 1317 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 2 1 N=50 examples R=3 annotators 2 2 5 Actual regression model Model with annotator 1 (Precision=0.01) 2 1 Model with annotator 2 (Precision=0.10) 2 Model with annotator 3 (Precision=1.00) 3 2 3 2 Model with average response 2 Proposed algorithm 2 3 1 1 4 3 2 2 y (response) 2 2 1 3 2 3 3 2 3 1 −1 3 3 2 2 3 3 1 2 2 2 3 33 3 1 1 3 3 3 3 2 3 3 2 33 33 2 2 2 3 3 3 32 1 2 3 2 1 3 3 2 1 2 1 2 2 1 1 2 3 2 3 2 3 1 3 3 −3 2 3 3 2 3 3 3 2 3 3 2 1 3 3 2 0 −2 2 2 3 2 3 2 3 1 2 2 1 2 3 −4 1 2 2 1 −5 0 2 0.1 0.2 1 0.3 0.4 0.5 0.6 10.8 2 0.7 x (feature) 0.9 1 1 Figure 6: Illustration of the proposed algorithm on a one-dimensional toy data set. The actual regression model (shown as a blue dotted line) is given by y = 5x − 2. We simulate 50 samples from three annotators with precisions 0.01, 0.1, and 1.0. The data are shown by the annotators’s number. While we can ﬁt a regression model using each annotators’s response, we see that only the model for annotator three (with highest precision) is close to the true regression model. The green dashed line shows the model learnt using the average response from all the three annotators. The red line shows the model learnt by the proposed algorithm. 5 5 10 Actual Polyp Diameter (mm) (a) Gold standard model 15 10 5 0 0 Pearson Correlation Coefficient=0.554966 RMSE=2.887740 15 Estimated Polyp Diameter (mm) 10 0 0 Pearson Correlation Coefficient=0.706558 RMSE=1.991576 15 Estimated Polyp Diameter (mm) Estimated Polyp Diameter (mm) Pearson Correlation Coefficient=0.714720 RMSE=1.970815 15 5 10 Actual Polyp Diameter (mm) (b) Proposed model 15 10 5 0 0 5 10 Actual Polyp Diameter (mm) 15 (c) Average model Figure 7: Scatter plot of the actual polyp diameter vs the diameter predicted by the models learnt using (a) the actual gold standard, (b) the proposed algorithm with annotations from ﬁve radiologists, and (c) the average of the radiologist’s annotations. (See § 6.2.2 for a description of the experimental setup.) 1318 L EARNING F ROM C ROWDS We use a proprietary data set containing 393 examples (which point to 285 distinct polyps— the segmentation algorithms generally return multiple marks on the same polyp.) along with the measured diameter (ranging from 2mm to 15mm) as our training set. Each example is described by a set of 60 morphological features which are correlated to the diameter of the polyp. In order to validate the feasibility of our proposed algorithm, we simulate ﬁve radiologists according to the noisy model described in § 5.1 with τ = [0.001 0.01 0.1 1 10]. This corresponds to a situation where the ﬁrst three radiologists are extremely noisy and the last two are quite accurate. Based on the measurements from multiple radiologists, we can simultaneously (1) learn a linear regressor and (2) estimate the precision of each radiologist. We compare the results with the classiﬁer trained using the actual golden ground truth as well as the regressor learnt using the average of the radiologists measurements. The results are validated on an independent test set containing 397 examples (which point to 298 distinct polyps). Figure 7 shows the scatter plot of the actual polyp diameter vs the diameter predicted by the three different models. We compare the performance based on the root mean squared error (RMSE) and also the Pearson’s correlation coefﬁcient. The regressor learnt using the proposed iterative algorithm (Figure 7(b)) is almost as good as the one learnt using the golden ground truth (Figure 7(a)). The correlation coefﬁcient for the proposed algorithm is signiﬁcantly larger than that learnt using the average of the radiologists response. The estimate obtained by averaging is closer to the novices since they form a majority (3/5). The proposed algorithm appropriately weights each radiologist based on their estimated precisions. 7. Conclusions and Future Work In this paper we proposed a probabilistic framework for supervised learning with multiple annotators providing labels but no absolute gold standard. The proposed algorithm iteratively establishes a particular gold standard, measures the performance of the annotators given that gold standard, and then reﬁnes the gold standard based on the performance measures. We speciﬁcally discussed binary/categorical/ordinal classiﬁcation and regression problems. We made two key assumptions: (1) the performance of each annotator does not depend on the feature vector for a given instance and (2) conditional on the truth the experts are independent, that is, they make their errors independently. As we pointed out earlier these assumptions are not true in practice. The annotator performance depends on the instance he is labeling and there is some degree of correlation among the annotators. We brieﬂy discuss some strategies to relax these two assumptions. 7.1 Instance Difﬁculty One drawback of the current model is that it doesn’t estimate difﬁculty of items. It is often observed that for the easy instances all the annotators agree on the labels—thus violating our conditional independence assumption. The difﬁculty of annotating an item can be captured by another latent variable γi for each instance—which modulates the annotators performance. Models for this have been developed in the area of item-response theory (Baker and Kim, 2004) and also in epidemiology (Uebersax and Grove, 1993)—see also Whitehill et al. (2009) for a recent paper in the machine learning community. While these models do not take into account the available features our pro1319 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY posed model for sensitivity and speciﬁcity can be extended as follows (in place of (1) and (2)): α j (γi ) := Pr[yi = 1|yi = 1, γi ] = σ(a j1 + b j1 γi ). j β j (γi ) := Pr[yi = 0|yi = 0, γi ] = σ(a j0 + b j0 γi ). j Here the parameters a j1 and a j0 are related to the sensitivity and speciﬁcity of the jth annotator, while the latent term γi captures the difﬁculty of the instance. The key assumption here is that the annotators are independent conditional on both yi and γi . Various assumptions can be made on two parameters b j1 and b j0 to simplify these models further—for example we could set b j1 = b1 and b j0 = b0 for all the annotators. 7.2 Annotators Actually Look at the Data In our model we made the assumption that the sensitivity α j and the speciﬁcity β j of the jth annotator does not depend on the feature vector xi . For example, in the CAD domain, this meant that the radiologist’s performance is consistent across different sub-groups of data—which is not entirely true. It is known that some radiologists are good at detecting certain kinds of malignant lesions based on their training and experience. We can extend the previous model such that the sensitivity and the speciﬁcity depends on the feature vector xi explicitly as follows j⊤ α j (γi , xi ) := Pr[yi = 1|yi = 1, γi , xi ] = σ(a j1 + b j1 γi + wα xi ). j j⊤ α j (γi , xi ) := Pr[yi = 0|yi = 0, γi , xi ] = σ(a j0 + b j0 γi + wβ xi ). j However this change increases the number of parameters to be learned. References P. S. Albert and L. E. Dodd. A cautionary note on the robustness of latent class models for estimating diagnostic error without a gold standard. Biometrics, 60:427–435, 2004. F. B. Baker and S. Kim. Item Response Theory: Parameter Estimation Techniques. CRC Press, 2 edition, 2004. B. Carpenter. 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