cvpr cvpr2013 cvpr2013-15 knowledge-graph by maker-knowledge-mining

15 cvpr-2013-A Lazy Man's Approach to Benchmarking: Semisupervised Classifier Evaluation and Recalibration


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Author: Peter Welinder, Max Welling, Pietro Perona

Abstract: How many labeled examples are needed to estimate a classifier’s performance on a new dataset? We study the case where data is plentiful, but labels are expensive. We show that by making a few reasonable assumptions on the structure of the data, it is possible to estimate performance curves, with confidence bounds, using a small number of ground truth labels. Our approach, which we call Semisupervised Performance Evaluation (SPE), is based on a generative model for the classifier’s confidence scores. In addition to estimating the performance of classifiers on new datasets, SPE can be used to recalibrate a classifier by reestimating the class-conditional confidence distributions.

Reference: text


Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We show that by making a few reasonable assumptions on the structure of the data, it is possible to estimate performance curves, with confidence bounds, using a small number of ground truth labels. [sent-8, score-0.21]

2 In addition to estimating the performance of classifiers on new datasets, SPE can be used to recalibrate a classifier by reestimating the class-conditional confidence distributions. [sent-10, score-0.423]

3 The pedestrian detector was laboriously trained by a research group who labeled thousands of training and validation examples and publishes good experimental results (see e. [sent-14, score-0.184]

4 Should the urban planner trust the published performance figures and assume that the detector will perform equally well on her images? [sent-17, score-0.246]

5 In order to be sure, the planner needs to compute precision and recall on her dataset. [sent-20, score-0.205]

6 Is it possible at all to obtain reliable bounds on the performance of a detector / classifier without relabeling a new dataset? [sent-26, score-0.323]

7 A pedestrian detector trained on vacation images (the INRIA dataset [5]) performs well on images taken in natural environments (top), and fails miserably on images taken in an urban environment (bottom). [sent-28, score-0.263]

8 Can we estimate the performance of a pretrained classifier / detector on a novel data set? [sent-29, score-0.308]

9 A: Histogram of classifier scores si obtained by running the “ChnFtrs” detector [7] on the INRIA dataset [5]. [sent-34, score-0.435]

10 The red and green curves show the Gamma-Normal mixture model fitting the histogrammed scores with highest likelihood. [sent-35, score-0.348]

11 The red and green bars show the labels of the 10 randomly sampled labels (by chance, the scores for some of the samples are close to each other, thus only 6 bars are shown; the height of the bars has no meaning). [sent-38, score-0.533]

12 B: Precision and recall curves computed from the mixture model in A. [sent-39, score-0.223]

13 C: In black, precision-recall curve computed after all items have been labeled. [sent-40, score-0.297]

14 In red, precision-recall curve estimated using SPE from only 10 labeled examples (with 90% confidence interval shown as the magenta band). [sent-41, score-0.264]

15 We propose a method for achieving minimally supervised evaluation of classifiers, requiring as few as 10 labels to accurately estimate classifier performance. [sent-43, score-0.318]

16 Our method ×× is based on a generative Bayesian model for the confidence scores produced by the classifier, borrowing from the literature on semisupervised learning [16, 20, 2 1]. [sent-44, score-0.27]

17 We show how to use the model to re-calibrate classifiers to new datasets by choosing thresholds to satisfy performance constraints with high likelihood. [sent-45, score-0.184]

18 Modeling the classifier score Let us start with a set of N data items, (xi, yi) ∈ RD {0, 1}, drawn from some unknown distribution p)(x ∈, yR) an×d i{n0d,e1x}e,d d by in r∈o m{ s1,o . [sent-48, score-0.277]

19 e τ is some scalar threshold, has been used to classify all data items into two classes, yˆi ∈ {0, 1}. [sent-56, score-0.18]

20 While the “ground truth” labels yi are assumed t∈o {be0 ,u1n}k. [sent-57, score-0.205]

21 This is an expensive operation and our goal is to ask the oracle for as few labels as possible. [sent-71, score-0.229]

22 The set of items that have been labeled by the oracle at time t is denoted by Lt and its complement, the soerta colef i attem tims feor t w ish dicehn othteed ground truth is unknown, is denoted Ut. [sent-72, score-0.37]

23 A: Estimation error, as measured by the area between the true and predicted precision-recall curves, versus the number of labels sampled, for the ChnFtrs detector on the CPD dataset. [sent-95, score-0.184]

24 The red curve is SPE and the green curve shows the median error of the naive method (RND). [sent-96, score-0.345]

25 B: The performance curve estimated using SPE (red) with 90% confidence intervals (magenta) with 20 known labels. [sent-98, score-0.261]

26 The ground truth performance with all label known is shown as a black curve (GT), and the performance curve computed on 20 labels using the naive method from 5 random samples is shown in green (RND). [sent-99, score-0.567]

27 C–D: same as A–B, but for the logres8 classifier on the DGT dataset (hand-picked as an example where SPE does not work well). [sent-101, score-0.184]

28 E: Comparison of estimation error (area between curves) of SPE and naive method for 20 known labels and different datasets. [sent-102, score-0.199]

29 T anhed component d thenes miti exstu p0 waneidg p1 could be modeled parametrically by Normal distributions, Gamma distributions, or some other probability distributions appropriate for the given classifier (see Section 4. [sent-108, score-0.325]

30 This approach of applying a generative model to score distributions, when all labels are known, has been used in the past to obtain error estimates on classifier performance [ 12, 9, 1 1], and for classifier calibration [ 1]. [sent-110, score-0.566]

31 However, previous approaches require that the all items used to estimate the performance have been labeled. [sent-111, score-0.262]

32 We suggest that it may be possible to estimate classifier performance even when only a fraction of the ground truth labels are known. [sent-112, score-0.386]

33 In this case, the labels for the unlabeled items i ∈ Ut can be marginalized out, p(S, Yt | θ) = ? [sent-113, score-0.38]

34 This allows the model to make use of the scores of unlabeled items in addition to the labeled items, which enables accurate performance estimates with only a handful of labels. [sent-118, score-0.51]

35 Figure 2a shows a histogram of the scores obtained from classifier on a public dataset (see Section 4 for more information about the datasets we use). [sent-122, score-0.37]

36 At first glance, it is difficult to guess the performance of the classifier unless the oracle provides a lot of labels. [sent-123, score-0.337]

37 A–F: Standard parametric distributions py (s | θy) (black solid curve) fitted to the class conditional scores for a few example datasets and classifiers. [sent-128, score-0.47]

38 In all cases, we normalized the scores to be in the interval si ∈ (0, 1], and made the truncation at s = 0 for the truncated distributions. [sent-130, score-0.26]

39 G: Comparison of st∈and (a0r,d1 parametric d thisetri trbuunticoantiso n be astt representing empirical class-conditional score distributions (for a subset of the 78 cases we tried). [sent-133, score-0.254]

40 The distribution families we tried included (with abbreviations used in last three columns in parentheses) the truncated Normal (n), truncated Student’s t (t), Gamma (g), log-normal (ln), left- and right-skewed Gumbel (g-l and g-r), Gompertz (gz), and Frechet right (f-r) distribution. [sent-137, score-0.257]

41 For example, two often used performance measures are the precision P(τ; θ) and recall R(τ; θ) at a particular score threshold τ. [sent-145, score-0.24]

42 We can define these quantities in terms of the conditional distributions py (si | θy). [sent-146, score-0.23]

43 yi = 1, examples that have scores above a given threshold, R(τ;θ) =? [sent-149, score-0.215]

44 The posterior on θ may be used to obtain confidence bounds on the performance of the classifier. [sent-158, score-0.226]

45 For example, for some choice of parameters θ, the precision and recall can be computed for a range of score thresholds τ to obtain a curve (see solid curves in Figure 2b). [sent-159, score-0.398]

46 Similarly, given the posterior on θ, the distribution of P(τ; θ) and R(τ; θ) can be computed for a fixed τ to obtain confidence intervals (shown as colored bands in Figure 2b). [sent-160, score-0.274]

47 The same applies to the precision-recall curve: for some recall r, the distribution of precisions, found using Pr (r; θ) can be used to compute confidence intervals on the curve (see Figure 2c). [sent-161, score-0.345]

48 Since the previous approach views the scores (and the associated labels) as a finite sample from p(S, Y | θ), there will always be uncertainty inp teh fer performance |es θt)im, tahteer. [sent-164, score-0.255]

49 eW whileln aalwl aityesm bse eh uavnebeen labeled by the oracle, the remaining uncertainty in the performance represents the variability in sampling (S, Y ) from p(S, Y | θ). [sent-165, score-0.217]

50 Thus, when the oracle has labeled 333222666533 the whole test set, there should not be any uncertainty in the performance; it can simply be computed directly from (S, Y ). [sent-169, score-0.218]

51 To estimate the sample performance, we need to account for uncertainty in the unlabeled items, i ∈ Ut. [sent-170, score-0.219]

52 On the second line of (6) we rely on the assumption ofa mixture model to factor the joint probability distribution on θ and Yt? [sent-179, score-0.18]

53 and t|h Se scores S to trace out a performance curve (e. [sent-185, score-0.281]

54 The main difference between the sample and population performance estimates will be at the tails of the score distribution, p(S | θ), wtimheartee sin wdiilvli bdeua atl titheem ta lialsbe olfs can choarvee a large impact on tθh)e, performance curve. [sent-193, score-0.233]

55 A more generally applicable method, which we will describe here, is to split the sampling into three steps: (a) sample θ¯ from p(θ | S, Yt), (b) fix the mixture parameters to θ¯ and sample tph(eθ l a| Sbe,lsY Yt? [sent-201, score-0.234]

56 The first step, sampling from the posterior p(θ | S, Yt), can h bee cirasrtri setdep ,o suta using importance sampling (I |S S). [sent-205, score-0.239]

57 In IS, we sample from a proposal distribution q(θ) in order to estimate properties of the desired distribution p(θ | S, Yt). [sent-207, score-0.235]

58 We now have all steps needed to estimate the performance of the classifier, given the scores S and some labels Yt obtained from the oracle: 1. [sent-231, score-0.324]

59 Estimate performance measures using the scores S, labels Y¯t,m = Yt ∪ Y¯t? [sent-249, score-0.28]

60 Datasets We surveyed the literature for published classifier scores with ground truth labels. [sent-254, score-0.314]

61 The blue curve and confidence band show SPE applied to the ChnFtrs detector on the CPD dataset with 100 observed labels (black curve is ground truth). [sent-266, score-0.558]

62 Based on a curve like this, a practitioner can “recalibrate” a pre-trained classifier by picking a threshold for new dataset such that some pre-defined criteria (e. [sent-268, score-0.327]

63 detector scores and ground truth labels are available for a wide variety of detectors [7]. [sent-271, score-0.34]

64 Moreover, the CPD website also has scores and labels available, using the same detectors, for other pedestrian detection datasets, such as the IN- RIA (abbr. [sent-272, score-0.311]

65 We made use of the detections in the CPD and INR datasets as if they were classifier outputs. [sent-274, score-0.218]

66 To complement the pedestrian datasets, we also used a basic linear SVM classifier and a logistic regression classifier on the “optdigits” (abbr. [sent-280, score-0.385]

67 In the figures, the naming convention is as follows: “svm3” is used to mean that the SVM classifier was used with category 3 in the dataset being assigned to the y = 1class, and “logres9” denotes that the logistic regression classifier was used with category 9 being the y = 1 class, and so on. [sent-285, score-0.342]

68 Choosing class conditionals Which distribution families should one use for the class conditional py (s | θy) distributions? [sent-289, score-0.239]

69 To explore this question, we took (thse |c θlassifier scores and split them into two groups, one for yi = 0 and one for yi = 1. [sent-290, score-0.304]

70 We used MLE to fit different families of probability distributions (see Figure 4 for a list of distributions) on 80% of the data (sampled randomly) in each group. [sent-291, score-0.212]

71 We then ranked the distributions by the log likelihood of the remaining 20% of the data (given the MLE-fitted parameters). [sent-292, score-0.185]

72 Figure 4G shows the top-3 distributions that explained the class-conditional scores with highest likelihood for a selection of the datasets and classifiers. [sent-294, score-0.371]

73 We found that the truncated Normal distribution was in the top-3 list for 48/78 dataset class-conditionals, and that the Gamma distribution was in the top-3 list 53/78 times; at least one of the two distributions were always in the top-3 list. [sent-295, score-0.358]

74 In some cases, like Figure 4C, a mixture model would have provided a better fit than the simple distributions we tried. [sent-297, score-0.233]

75 That said, we found that truncated Normal and Gamma distributions were good choices for most of the datasets. [sent-298, score-0.216]

76 As an example, for the truncated Normal distribution, we use a Normal and a Gamma distribution as priors on the mean and standard deviation respectively (since we use sampling for inference, we are not limited to conjugate priors). [sent-301, score-0.209]

77 One heuristic, which we found worked well in our experiments, is to try different combinations of distributions for p0 and p1, and then choose the combination achieving the highest maximum likelihood on the labeled and unlabeled data. [sent-304, score-0.347]

78 While SPE performs as well as the naive method in terms of estimation error, the score distribution is not well explained by the assumptions of the model, so there is a bias in the prediction. [sent-315, score-0.229]

79 Figure 3E compares the estimation error of SPE to the naive method for different datasets, when only 20 labels are known. [sent-318, score-0.199]

80 Classifier recalibration Applying SPE to a test dataset allows us to “recalibrate” the classifier to that dataset. [sent-323, score-0.234]

81 Unlike previous work on classifier calibration [1, 17], SPE does not require all items to be labeled. [sent-324, score-0.338]

82 Similarly, we can c |h So,oYse a threshold τ to use with the classifier h¯(xi; τ) based on some pre-determined criteria. [sent-328, score-0.184]

83 For example, the requirement might be that the classifier performs with recall R(τ) > ˆr and precision P(τ) > pˆ. [sent-329, score-0.273]

84 n Tish esant,i fsofrie eda by calculating teh per expectation ˆ p(C(τ) = 1) = E [C(τ)] over the unlabeled items Yt? [sent-332, score-0.264]

85 Related work Previous approaches for estimating classifier performance with few labels falls into two categories: stratified sampling and active estimation using importance sampling. [sent-337, score-0.531]

86 This work has since been generalized to other classifier performance metrics, such as precision and recall [8]. [sent-339, score-0.311]

87 proposed instead to use importance sampling to focus labeling effort on data items with high classifier uncertainty, and applied it to standard loss functions [19] and Fmeasures [18]. [sent-341, score-0.445]

88 Both of these approaches assume that the classifier threshold τ is fixed (see Section 2) and that a single scalar performance measure is desired. [sent-342, score-0.222]

89 Fitting mixture models to the class-conditional score distributions has been studied in previous work with the goal of obtaining smooth performance curves. [sent-344, score-0.332]

90 This allowed them to provide smooth performance estimates even when the classconditional distributions could not be explained by standard parametric distributions. [sent-350, score-0.266]

91 Similarly, previous work on classifier calibration has involved fitting mixture models to score distributions [1, 17]. [sent-351, score-0.487]

92 In contrast to previous work, which require all data items to be labeled, SPE also makes use of the unlabeled data. [sent-352, score-0.264]

93 This semisupervised approach allows SPE to estimate classifier performance with very few labels, or when the proportions of positive and negative examples are very unbalanced. [sent-353, score-0.313]

94 Discussion We explored the problem of estimating classifier performance from few labeled items. [sent-355, score-0.277]

95 A sampling scheme based on importance sampling enables efficient inference. [sent-360, score-0.181]

96 One disadvantage with using an approach like SPE is that there are no guarantees that the assumption of standard parametric distributions will hold for any dataset, something we hope to address in future work. [sent-361, score-0.193]

97 However, using four public datasets, and multiple classifiers, we showed that classifier score distributions are often well approximated by standard twocomponent mixture models in practice. [sent-363, score-0.452]

98 One possibility in this direction would be to employ importance weighted active sampling techniques [3, 6], so similar in spirit to [19, 18] but for performance curves. [sent-366, score-0.186]

99 That said, as shown by our experiments, SPE already works well for a broad range of classifiers and datasets, and can estimate classifier performance with as few as 10 labels (see Figure 2). [sent-369, score-0.402]

100 Using asymmetric distributions to improve classifier probabilities: A comparison of new and standard parametric methods. [sent-376, score-0.351]


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