cvpr cvpr2013 cvpr2013-168 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Raphael Sznitman, Carlos Becker, François Fleuret, Pascal Fua
Abstract: Cascade-style approaches to implementing ensemble classifiers can deliver significant speed-ups at test time. While highly effective, they remain challenging to tune and their overall performance depends on the availability of large validation sets to estimate rejection thresholds. These characteristics are often prohibitive and thus limit their applicability. We introduce an alternative approach to speeding-up classifier evaluation which overcomes these limitations. It involves maintaining a probability estimate of the class label at each intermediary response and stopping when the corresponding uncertainty becomes small enough. As a result, the evaluation terminates early based on the sequence of responses observed. Furthermore, it does so independently of the type of ensemble classifier used or the way it was trained. We show through extensive experimentation that our method provides 2 to 10 fold speed-ups, over existing state-of-the-art methods, at almost no loss in accuracy on a number of object classification tasks.
Reference: text
sentIndex sentText sentNum sentScore
1 While highly effective, they remain challenging to tune and their overall performance depends on the availability of large validation sets to estimate rejection thresholds. [sent-6, score-0.343]
2 It involves maintaining a probability estimate of the class label at each intermediary response and stopping when the corresponding uncertainty becomes small enough. [sent-9, score-0.579]
3 Typically, using an ensemble classifier requires evaluating the many individual classifiers they are made of. [sent-15, score-0.248]
4 Perhaps the most famous demonstration of this trade-off is found in the seminal face detection paper [26], where a “hard” cascade of classifiers was used to filter out and reject non-faces while only mildly decreasing the overall classifier accuracy. [sent-18, score-0.314]
5 For example, a relaxation of the hard cascade to a soft one was introduced to more effectively reject non-target candidates and to alleviate many of the difficulties encountered when training hard cascades [4]. [sent-20, score-0.196]
6 Similarly, some have used rejection criteria based on Walds’ Sequential Probability Ratio Test [23] or using more empir- ical observations [29]. [sent-21, score-0.214]
7 Unlike previous methods that need multiple stage-specific thresholds, the class label uncertainty in our method only requires a single threshold, common to all stages, to specify when the uncertainty is small enough. [sent-27, score-0.192]
8 We show experimentally that our method provides significant gains in speed or accuracy, and often both, over state-of-the-art early stopping methods on a number of object classification tasks when using different ensemble classifiers. [sent-28, score-0.65]
9 Among methods designed for this purpose and that attempt to reject negatives early, one ofthe earliest and most influential works is the hard cascade of classifiers proposed in [26] for face detection. [sent-40, score-0.227]
10 In that work, rejection thresholds on a set of distinct classifiers were used to conservatively filter out non-faces during classification. [sent-41, score-0.355]
11 Perhaps most relevant to this work is that of [4], where the hard cascade was replaced by a single soft cascade classifier with stage-wise rejection thresholds that were computed using a cascade calibration procedure (CCAL). [sent-43, score-0.632]
12 More specifically, thresholds on the sum of stage scores were found by adjusting a performance vector such that each stage of the soft cascade rejected at most a fixed proportion of positive targets from a validation set. [sent-44, score-1.042]
13 Furthermore, the final classifier accuracy is closely linked to the quantity and variability of positive samples in the validation set used to calibrate the cascade. [sent-46, score-0.31]
14 Along the same lines, a simple Direct Backward Pruning algorithm (DBP) was introduced in [29] which sets rejection thresholds on the sum of stage scores. [sent-47, score-0.568]
15 This was achieved by taking the threshold at each stage to be the minimum sum of stage scores observed over a validation set or subset. [sent-48, score-0.757]
16 While effectively removing the need for the performance vector of [4], this strategy can still only perform well for validation subsets large enough so that thresholds generalize to the test set. [sent-49, score-0.306]
17 A different approach to setting rejection thresholds is that of [23], where stage-wise thresholds are based on the Sequential Probability Ratio Test (SPRT). [sent-50, score-0.434]
18 This test is shown to be optimal when individual observations at each stage are i. [sent-51, score-0.237]
19 However, in practice, the ratio test used relies on the sum of stage scores, hence strongly correlating observations and thus violating a number of assumptions. [sent-54, score-0.302]
20 In short, all the above-mentioned approaches to early termination of a classifier evaluation critically depend either on large validation sets or on several strong assumptions on the behavior of weak classifiers. [sent-55, score-0.358]
21 Method As in cascade approaches [26, 23, 4, 29], given an ensemble classifier that sequentially evaluates stages, our goal is to reduce the computational burden by using as few resources as possible on easy-to-classify cases. [sent-58, score-0.238]
22 We differ from earlier approaches in that we track the class label in a Bayesian way by modeling the individual stage computations, and dynamically infer when additional classification stages are necessary at run-time. [sent-59, score-0.431]
23 In some sense, our method normalizes each stage individually and accumulates their normalized evidence. [sent-60, score-0.263]
24 , (1) where gk : RD → R, (2) is a stage score at stage index k. [sent-71, score-1.035]
25 In BS classifiers [9, 13], a stage is defined as gk (x) = αkhk (x) where hk is a weak learner and αk its weight. [sent-72, score-0.866]
26 Similarly, in BT and RF classifiers [5], gk is a decision tree. [sent-73, score-0.606]
27 1 depicts this sum for a few examples from a validation set V = 333222667199 Table 1. [sent-84, score-0.237]
28 Summary of Notation x ∈ RDSample feature vector Yx ∈∈ R{−1, 1} VY VN φ kx∗ gk (x) Gk (x) fk (x) F(x) ? [sent-85, score-0.699]
29 In this figure, each curve shows the sum {of( stage scores of a BS face classifier for an example face (green) or non-face (red). [sent-88, score-0.51]
30 Note that we consider the validation set to be disjoint from the classifier training set. [sent-89, score-0.247]
31 Framework For a given test sample x, we want to evaluate as few stages as possible for a classifier of the form given in Eq. [sent-92, score-0.181]
32 ,gk(x)) ≥ 0},(4) where the φ function is a stopping criterion. [sent-100, score-0.42]
33 In fact, in both [4] and [29], the stopping criteria are of the form φ (g1(x), . [sent-103, score-0.492]
34 9] selects each threshold by computing θk={n:yn=1m,fiKn(xn)>τ}fk(xn), (6) where τ is a user specified threshold on the final sum fK(x). [sent-111, score-0.185]
35 n where |P| is the total number of positive examples in the vwahleidraet |ioPn| set, pk tios a user specified proportion amnpdl epsre idn tihs a function that returns one if fk (xn) ≤ r and zero otherwise. [sent-114, score-0.271]
36 Similarly, the SPRT stopping c)ri ≤ter ri aonn din z e[2ro3] o, hise rowf itshee. [sent-115, score-0.42]
37 Example of the sum of stage scores produced by a face classifier as function of the stage index. [sent-120, score-0.695]
38 Each green and red line corresponds to a face and non-face sample from a validation set. [sent-121, score-0.254]
39 The jittery black line shows a typical set of rejection thresholds produced by the CCAL procedure. [sent-122, score-0.308]
40 Examples with sum of scores that fall below the black line at any stage k are rejected early. [sent-123, score-0.385]
41 Clearly, the performances of these stopping criteria are fk θk strongly dependent on the quality and representativity of the validation set used, as their threshold values are explicitly selected from examples. [sent-128, score-0.898]
42 Entropy Driven Evaluation We define the stopping criterion φ in a significantly different way. [sent-131, score-0.494]
43 After each stage, we observe the value, gk (x), which we also treat as a random variable and for which we can evaluate P(gk (x) |Y = 1) and P(gk (x) |Y = −1). [sent-135, score-0.539]
44 In addition, we assume that gk (x) is conditionally independent from gj (x) ,j < k given the class label, which leads to ? [sent-136, score-0.568]
45 = 1 Unlike [23] which also assumes conditional independence given the class label, this is our only assumption on the behavior of the stage scores. [sent-140, score-0.304]
46 Given this, we take our stopping criteria to be φ (g1(x) , . [sent-141, score-0.492]
47 , gk (x)) = γ 333222777200 − H(Y |Gk (x)) , (10) Stage Index k Stage Index k Figure 2. [sent-144, score-0.539]
48 (top) Sum of stage scores, fk (x) for a positive sample x0 (green) and negative samples x1 and x2 (red), as function of the stage index k. [sent-146, score-0.77]
49 The dotted line depicts a chosen entropy threshold γ, converted to its corresponding probability thresholds ? [sent-148, score-0.363]
50 n nI no bthseisr context, nthde γ γco ∈n (d0it,io1n)a isl entropy provides a measure of uncertainty on the class label, and hence we look for kx∗ such that it reduces the uncertainty below a specified level γ. [sent-152, score-0.274]
51 As in [23], we can estimate the conditional likelihoods, P(gk (x) |Y = 1) and P(gk (x) |Y = −1) using the validation s(ext. [sent-165, score-0.198]
52 2, let two samples, x1 and x2 have sum of scores fk∗ (x1) = fk∗ (x2), where k∗ = 5 is the first stage where rejection takes place for both x1 and x2, when using a method from Sec. [sent-205, score-0.461]
53 In this case, both x1 and x2 are rejected since rejection solely depends on fk∗ (·). [sent-208, score-0.208]
54 In EDE however, early stopping depends on the posterior distribution, which depends on the sequence Gk (x) and not fk (·). [sent-209, score-0.763]
55 That is, early stopping of our method depends on the progression sof, evaarlluyes st oatp peaincgh stage, manetdh noodt d on etnhed sum hoef stage scores. [sent-210, score-0.808]
56 Furthermore, our method normalizes each stage individually, and estimates the reliability of individual stages. [sent-212, score-0.263]
57 To build some intuition about this fact, consider an extreme example where a given stage computes a random response that is completely unrelated to x. [sent-213, score-0.237]
58 Such a stage would strongly impact previous methods in a negative way, as it would force the threshold at that stage to be unreliable. [sent-214, score-0.566]
59 EDE on the other hand, would effectively ignore the stage from the overall estimation, as it would weigh the information at this stage very poorly, since P(gk (x) |Y = 1) and P(gk (x) |Y = −1) would be similar. [sent-215, score-0.474]
60 Block Evaluation While in [23, 4, 29] the stopping criteria are evaluated at each stage of the prediction procedure, this may not be necessary in some cases. [sent-218, score-0.79]
61 For this reason, we propose to evaluate the stopping criterion at specific intervals of stages. [sent-219, score-0.494]
62 , K − 1} be a block offset, then we can update δthe ∈ posterior Kdis −tr 1ib}ut bioen a a bnldo cekva olfufsaetet, tthhee stopping criterion at every δ stages. [sent-223, score-0.583]
63 own in our experiments, this block evaluation of the stopping criterion is not only beneficial as it reduces the need to update the posterior at each stage, but also makes EDE less sensitive to noise contained in each stage observation. [sent-233, score-0.82]
64 (x) |Y = y), the stopping threshold γ and the block offset δ. [sent-242, score-0.522]
65 5) In general, the only difference between our algorithm and the evaluation of a typical ensemble classifier is the update of the posterior distribution (line 4). [sent-255, score-0.216]
66 Experiments We now demonstrate the efficiency of our proposed EDE stopping criterion for three different tasks: face detection, image classification and structure recognition1 . [sent-259, score-0.586]
67 As noted in [23, 29], comparing published results of competing early stopping methods is difficult because they are produced by pipelines that depend on training data, specific features being used, approach to non-maximal suppression, and parameter settings among many other things. [sent-262, score-0.508]
68 Therefore, to compare our early stopping approach against others as fairly as possible, we reimplemented the CCAL, DBP and SPRT stopping criteria and evaluate each approach using the same classifier and validation set for each task mentioned above. [sent-263, score-1.247]
69 5 )of a thned parameter space, a,n wde s peelercfoterdm tehde a parameters t sheaatr required the smallest number of stage evaluations, and for which at most 1% of the classification accuracy was incorrect when compared to non-early-stopping prediction. [sent-275, score-0.299]
70 For each triple, this process was repeated 5 times over the validation set and the best triple was selected. [sent-276, score-0.188]
71 In general, we are interested in observing how a method performs in terms of the number of stage evaluations and prediction accuracy. [sent-277, score-0.377]
72 The validation set was comprised of 4000 positives and 6000 negatives. [sent-284, score-0.182]
73 This set was used to compute both the CCAL, DBP and SPRT rejection thresholds and the conditional probabilities P(gk (x) |Y = y) of Section 3. [sent-285, score-0.326]
74 For this experiment, we adjusted the number of stage evaluations, or stumps, required by each early stopping × method to be approximately the same. [sent-287, score-0.767]
75 We then compared the accuracy performance of the four stopping criteria on the MIT+CMU dataset [19], which consists of 130 images containing a total of 507 labelled faces. [sent-288, score-0.492]
76 3, we report the accuracy and the distribution of stage evaluations required by each method. [sent-292, score-0.361]
77 (top) Face detection ROC curves for each early stopping method evaluated and for non-early stopping on the MIT+CMU face dataset. [sent-295, score-1.003]
78 Each stopping criterion was re-implemented and evaluated using the same Boosted Stumps (BS) classifier, training set and validation set. [sent-296, score-0.677]
79 (bottom) Distribution of the number of stage evaluations of test sample for each method. [sent-297, score-0.361]
80 In this case, each stage consists in the evaluation of a single stump. [sent-298, score-0.237]
81 Since a validation set is required to estimate the rejection thresholds, we only evaluate object categories for which enough positive samples are available to form large enough training and validation sets. [sent-303, score-0.547]
82 Average number of stage evaluations and best F-score (in brackets) on object classification tasks for three Caltech-256 categories either without an early stopping criterion or with one of the four discussed in this paper. [sent-341, score-0.961]
83 Our EDE criterion, in bold, not only provides the least number of stage evaluations but also the best F-score when compared to other early stopping methods. [sent-342, score-0.847]
84 For each classifier and stopping criterion pair, we report in Table 2 both the average number of stage evaluations required and the best F-score (in brackets) for each classification task. [sent-346, score-0.982]
85 In all evaluated cases, the EDE stopping criterion outperforms the other three methods on both speed and accu- racy. [sent-349, score-0.545]
86 4 depicts the best F-scores of each early stopping method as a function of the average number of evaluations when using a BT classifier on the Airplane classification task. [sent-352, score-0.771]
87 Best F-score as function of number of stage evaluations required for a Boosted Tree (BT) classifier with different stopping criteria for Caltech-256 Airplane classification. [sent-355, score-0.94]
88 Each method uses the same classifier and validation set and each point shows the average performance for a stopping method when used with a specific set of parameters. [sent-356, score-0.667]
89 We constructed our training and validation set by computing the gradient histogram features described in [25] for 5000 positive and 5000 negative randomly selected samples from two different volumes3. [sent-368, score-0.252]
90 From the training and validation set, half the samples were randomly selected to train a BS classifier with 500 stumps and the other half to learn the early stopping criteria. [sent-370, score-0.887]
91 We repeated this 10 times and evaluated the resulting classifiers and corresponding stopping criteria on the test set. [sent-371, score-0.582]
92 (bottom) Best F-score as a function of the number of stage evaluations required for a Boosted Stump (BS) classifier on the above classification task. [sent-380, score-0.488]
93 Each point shows the average performance for a stopping method when used with different sets of parameters values. [sent-381, score-0.42]
94 To this end, we trained a BS classifiers with 500 stumps and re-learned the thresholds and probability distributions using ever smaller validation sets. [sent-385, score-0.505]
95 Even by the time the number of positives in the validation set has dropped from 2500 to 200, the performance of EDE is barely affected, while that of the others have degraded substantially. [sent-386, score-0.182]
96 Conclusion This paper addressed the problem of speeding up the prediction of binary classification when using ensemble classifiers. [sent-388, score-0.2]
97 Our proposed solution uses Bayesian inference to establish and track the probability of a samples class label as stage evaluations are computed. [sent-389, score-0.465]
98 Our early stopping criteria is to terminate the prediction process when the uncertainty of the class label, measured by its Shannon entropy, falls below a chosen level. [sent-390, score-0.712]
99 We showed through extensive experimentation on several classification tasks that our approach provides significant accuracy and speed improvements over state-of-the-art early stopping methods. [sent-391, score-0.604]
100 (top) Best F-score and (bottom) number of stage evaluations as a function of the number of positives used in the validation set on a logarithmic scale. [sent-395, score-0.521]
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