cvpr cvpr2013 cvpr2013-143 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Steve Branson, Oscar Beijbom, Serge Belongie
Abstract: unkown-abstract
Reference: text
sentIndex sentText sentNum sentScore
1 111888000644 up being faster by heuristically randomly sampling bounding boxes without absorbing the cost of firing the detector. [sent-1, score-0.329]
2 In this paper, we propose two changes that allow structured SVMs to be at least as fast as their binary SVM counterparts for problems such as object detection, deformable part models, and multiclass classification. [sent-2, score-0.526]
3 Second, we allow problem-specific knowledge to be injected into the optimization algorithm by incorporat- ing a user-defined importance sampling function. [sent-4, score-0.113]
4 Here, our optimization algorithm takes an update step with respect to a set of intelligently selected output labels (e. [sent-5, score-0.131]
5 , a set of bounding boxes in a particular image that have high training error with respect to the current detector). [sent-7, score-0.219]
6 This change allows the method to be a superset of techniques used by conventional structured learning methods and commonly used heuristics for mapping problems into binary classification problems, with additional parameters to explore to tailor optimization to a particular application. [sent-8, score-0.324]
7 Our main contributions are as follows: 1) We introduce SVM-IS, a fast structured SVM solver that is easy to apply to novel problems. [sent-9, score-0.354]
8 2) We show how our solver can be used to create faster learning algorithms for cost-sensitive multiclass SVMs, object detection, and deformable part models. [sent-10, score-0.5]
9 Background and Related Work Structured SVMs: Structured SVMs [23, 25, 24] provide a method for training a system to predict a multidimensional structured output, such as a bounding box, set of part locations, or segmentation. [sent-13, score-0.441]
10 They minimize a convex upper bound on a customizable loss function. [sent-14, score-0.207]
11 Structured SVMs are a superset of SVM-based learning algorithms, which includes many of the most popular and highest performing algorithms for object detection [12, 7, 3], and multiclass classification [17, 9]. [sent-15, score-0.255]
12 They have been applied to training object detectors using more appropriate loss functions [2], multiclass SVMs with customizable class confusion costs [5], and deformable part models [27, 4, 20]. [sent-16, score-0.489]
13 Whereas non-linear kernel SVMs train in time that is at least quadratic in the number of training examples, solvers such as Liblinear [11] and [14] train in linear time, or in time that does not even depend on the size of the training set (at least in expectation) [21]. [sent-18, score-0.215]
14 Among fast linear SVM solvers, online sub-gradient algorithms [21] and sequential dual algorithms [11] are faster than methods that train in batch. [sent-19, score-0.565]
15 Fast Solvers For Large Scale Structured SVMs: The basic convex optimization methods used by the above algorithms are general to many convex optimization problems such as structured SVMs [19, 6, 25]; however, there is a comparative scarcity of publicly available fast methods for structured SVMs. [sent-20, score-0.628]
16 Kakade and Shalev-Shwartz provided a template algorithm for developing fast optimization algorithms for novel strongly convex optimization problems and a theoretical framework for studying their statistical conver- SVMperf gence properties [15]. [sent-21, score-0.15]
17 [13] recently introduced a cutting-plane-based structured SVM solver that incorporates a similar idea to our importance sampling routine. [sent-24, score-0.395]
18 Our method is different in that it is based on extending sequential dual and sub-gradient algorithms (which are often much faster than cutting plane algorithms). [sent-25, score-0.449]
19 yO be a multidimensional structured output defining its ground truth label. [sent-33, score-0.239]
20 For example, for deformable part models, X is an image and each yp ∈ Y defines the location of a part in the image. [sent-34, score-0.237]
21 A structured prediction function predicts the label arg maxY s(X, Y ) with highest score: s(X, Y ) = hw, Ψ(X, Y )i (1) where s(X, Y ) is a score measuring the goodness of a particular label Y , Ψ(X, Y ) is a feature space extracted with respect to label Y (e. [sent-35, score-0.389]
22 Let ∆(Y, Yi) be a customizable loss function associated with predicting label Y when the true label is Yi. [sent-41, score-0.245]
23 Structured SVM learning minimizes the training error: n Fn(w) =Xf(w;Zi) (2) iX= X1 f(w;Zi) =2λkwk2+ ℓ(w,Zi) (3) ℓ(w, Zi) = mYax (s(Xi, Y ) + ∆(Y, Yi)) − s(Xi, Yi) (4) where each Zi = (Xi, Yi) is a training example, λ is a regularization constant. [sent-43, score-0.128]
24 Let Y¯i be the value of Y that maximizes ℓ(w, Zi) : Y¯i = argmYax (s(Xi, Y ) + ∆(Y, Yi)) (5) Solving Eq 5 resembles a prediction problem (Eq 1) and is the primary computation for many structured learning optimization algorithms1 . [sent-45, score-0.274]
25 The set of problems that are appropriate for structured SVMs is limited to problems and loss functions for which Eq 5 is efficiently solvable. [sent-46, score-0.322]
26 Dual Problem: Eq 2 can be represented by its equivalent dual problem max Dn (α1 . [sent-47, score-0.171]
27 α w 1 (6) n The dual objective is often useful for deriving optimization algorithms and theoretical guarantees. [sent-53, score-0.249]
28 For example, for sliding window based object detection, vi (Y ) is a feature vector for every possible bounding box Y in image Xi. [sent-76, score-0.3]
29 The bounding box feature vectors vi (Y ) that have non-zero weights αi (Y ) correspond to bounding boxes in the margin (e. [sent-77, score-0.401]
30 The relationship between the dual parameters α and primal parameters w is w(α) = −λ1nXi,Yαi(Y )vi(Y ) (9) 3. [sent-81, score-0.239]
31 Fast algorithms for linear SVMs such as Pegasos and Liblinear can be understood as variants ofthese algorithms; whereas our algorithm has additional choices that make it more appropriate for tweaking optimization speed for a particular application. [sent-84, score-0.078]
32 In each timestep, the algorithm solves for weights on each sample, with the objective of maximally increasing the online dual objective ∆Dt(αt) = Dt(α1 , . [sent-90, score-0.346]
33 , αtK] are vectors, c is a constant, Q is a K K matrix with elements Qjk = αtj, vtj, Dvjtλ,vtktE, ℓtj c =λ(t − 1)2ktwt−1k2 (13) and and are shorthand for per sample weights αt(Y¯tj), features vt(Y¯tj), and loss hwt−1, vtji +∆(Y¯tj, Yt), respectively. [sent-104, score-0.222]
34 Using Eq 9, this will result in an update of the model weights wt according to: wt←t −t 1wt−1−λ1tjX=K1αtjvtj (14) Let R be a bound on the magnitude of the feature space kΨ(X, Y ) k R: ≤ Theorem 3. [sent-105, score-0.326]
35 1 Algorithm 1 obtains an ǫ-accurate solution in at most iterations in expectation, for any subroutine that includes Y¯t1 = Y¯i (Eq 5) and any approximate solver on line 7 that is at least as good as the one obtained by setting αt1 = 1. [sent-106, score-0.078]
36 Note that this bound does not depend directly on the size of the training set. [sent-108, score-0.096]
37 The proof is based on the observation that choosing αt1 = 1increases Dt by a predictable amount (see supplementary material). [sent-109, score-0.087]
38 Y¯tK ← IMPORTANCESAMPLE(Xi, Yi, wt−1) 5: ∀j, vtj ← Ψ(Xi, Y¯ij) Ψ(Xi, Yi) ← hwt−1, + Yi) 6: Define ∆Dt := −21αTQα + ℓtTα + c − ℓtj vtji ∆(Y¯ij, Qjk 7: 8: := Approx. [sent-116, score-0.115]
39 solve wt ← hvjtλ,vtkti, αt ← t−t1wt−1 c = λ(t−1)k2twt−1k2 arg maxα ∆Dt (α) s. [sent-117, score-0.205]
40 αtK) will converge at least as quickly, with a larger increase yielding faster convergence. [sent-122, score-0.129]
41 Due to space limitations, we discuss two efficient algorithms for solving Eq 12 in the supplementary material. [sent-123, score-0.082]
42 The runtime for both such algorithms is roughly equal to the time needed to compute one dot product hwt−1 , vt per sample j. [sent-124, score-0.183]
43 The first algorithm is a general purpose one-pass approximate algo- (Y¯tj)i rithm, whereas the 2nd is an exact algorithm that applies to multiclass classification problems. [sent-125, score-0.13]
44 Heuristics for choosing samples: Examining Eq 12, given a sample set the utility of a new sample Y¯t1. [sent-126, score-0.128]
45 t) is high (the loss associated with sample Y¯tj) and each Qij is low. [sent-134, score-0.123]
46 Thus in general we should favor samples with high loss that are as uncorrelated as possible (Q is as close to a diagonal matrix as possible). [sent-135, score-0.083]
47 Detecting Convergence: Convergence is detected based on the primal dual gap F(w) D(α) 2 is less than ǫ. [sent-136, score-0.239]
48 Rather than explicitly compute we approximate it by the accumulated loss − Fn(w), Pst=1f(wts−1;Zi(s)), which exceeds jective tends to be decreasing. [sent-138, score-0.083]
49 By the arguments presented in [22], it shares the same worst case convergence rate as Algorithm 1, typically with faster convergence when making multiple passes through the data. [sent-176, score-0.129]
50 Comparison to Other Algorithms In this section, we briefly mention various possible optimization algorithms for structured SVMs (all of which are evaluated in our experimental results). [sent-179, score-0.317]
51 Cutting Plane Algorithm: SVMstruct–a popular software package for structured SVMs–optimizes Eq 2 using a cutting plane algorithm. [sent-180, score-0.333]
52 An n-slack variety of SVMstruct solves Eq 5 for all n training examples before solving a QP problem, whereas a 1-slack variety solves a QP problem after each time Eq 5 is solved for a particular example. [sent-181, score-0.142]
53 The n-slack method is asymptotically slower (by a factor of n) than the 1-slack method and our algorithm in terms of the number of times the inference algorithm will be invoked. [sent-182, score-0.08]
54 SGD iterates over each example sequentially, taking an update step: wt ← wt−1 − ηt∇f(wt−1; Zt) (16) where ∇f(wt−1 ; Zt) is the sub-gradient of Eq 3. [sent-185, score-0.256]
55 A Pegasos-like update [21] uses a decaying step size ηt = λ1t followed by a projective step3. [sent-186, score-0.096]
56 One can verify that SGD is equivalent to Algorithm 1 with K = 1, Y¯t1 = Y¯ , and choosing an update αt1 = 1, meeting the criteria of Theorem 3. [sent-188, score-0.144]
57 Applications In this section, we show how Algorithm 1-2 can be applied to a variety of popular learning problems, including cost-sensitive multiclass SVMs, object detection, and deformable part models. [sent-192, score-0.282]
58 Implement a routine to solve Eq 5: Y¯t = arg maxY (s(Xt , Y ) + ∆(Y, Yt)), which is similar to an inference problem 3. [sent-195, score-0.119]
59 Favor samples with high loss s(Xt, Y¯tj)+∆(Y¯tj, Yt), and where hvt(Y¯ti), vt(Y¯tj)i tends to be small for i j = 4. [sent-200, score-0.083]
60 C is a class label, a cost-sensitive multiclass SVM solves the optimization problem: FnC(w) = n 2λkwk2+n1iX=n1ǫi! [sent-205, score-0.204]
61 , wC] concatenates the weights for all classes, and ∆C (c, Yi) is a confusion cost associated with predicting class c when the true label is Yi. [sent-211, score-0.113]
62 As in [25], this problem is solvable using a structured SVM, where Ψ(X, Y ) concatenates features for each class: ΨC(X, Y ) ψc(X,Y ) = [ψ1(X, Y ), . [sent-212, score-0.279]
63 In the supplementary material, we derive a fast exact solver for Eq 12. [sent-219, score-0.154]
64 The time of this update step is roughly equal to the time to evaluate one dot product hw, ΨC(X, Y )i . [sent-220, score-0.135]
65 Sliding Window Object Detection A sliding window object detector can be trained using a structured SVM [2], where Y = {x, y, scale} encodes a bounding box and ΨB (X, Y ) is a vector of features extracted at Y . [sent-223, score-0.484]
66 Let ∆B (Y, Yi) be an arbitrary loss function associated with predicting bounding box Y when the true bounding box is Yi. [sent-224, score-0.389]
67 The sliding window detector can be trained by optimizing the structured SVM objective (Eq 2). [sent-225, score-0.331]
68 Similarly, let Mi be an array of sliding window responses, such that Mi [Y ] = hw, Ψ(Xi , Y )i, and let Li = Mi + ∆iB. [sent-227, score-0.092]
69 Note that Y¯i =argmYax (hw, Ψ(Xi, Y )i + ∆(Y, Yi)) = arg mYa′x Li[Y′] (20) Choosing samples: We choose a sparse sample set of bounding boxes Y¯t1. [sent-228, score-0.24]
70 This method greedily selects the bounding box with highest loss Y¯j = arg maxY Li [Y ], then suppresses all overlapping bounding boxes by setting Li [Y ] ← −∞ for all Y that overlap with Y¯j . [sent-232, score-0.436]
71 The motivation behind this sampling technique is that overlapping bounding boxes will tend to have more correlatedfeatures, suchthat hΨ(Xt, Y¯tj), Ψ(Xt, Y¯ti)i is more likely to be high. [sent-233, score-0.2]
72 Deformable Part Model Based Detection A Felzenszwalb-like deformable part model [12] can be trained using a structured SVM, where Y = y1, . [sent-236, score-0.359]
73 , yP encodes a set of part locations, each of which is represented using a 4-tuple yp = {xp, yp, scalep, aspectp}, where aspectp defines a mixture component index (e. [sent-239, score-0.178]
74 Details of the mapping into structured SVMs are presented in [27, 4]. [sent-243, score-0.239]
75 These methods concatenate appearance features and spatial offsets for all part-aspect pairs into a structured feature space ΨP (X, Y ). [sent-244, score-0.239]
76 The label Y¯i can be efficiently solved for using standard dynamic programming algorithms for pictorial structure inference. [sent-245, score-0.078]
77 Similar to the method described in the previous section, this is implemented using a modified unary detection score Lip = Mip + ∆ip for each part p, where ∆ip [yp] encodes the loss associated with placing part p at location yp. [sent-246, score-0.163]
78 The first is the same bounding box non-maximal suppression technique described in the previous section, applied to the score for the root of the part tree after running dynamic programming. [sent-248, score-0.243]
79 Wecreate amodifiedversion of the greedy non-maximal suppression method that favors Y¯tj 111888001088 111888 010 919 (a) Results on 400 image training set (b) Results on 5794 image training set (c) Effect of Parameter K Figure 5794, 3. [sent-250, score-0.178]
80 (a-b) Results on training sets of size 400 and Our method SVM-IS converged orders of magnitude faster than popular learning methods, including the 1-slack and n- SV Mstruct and a method based on mining hard examples and training a binary classifier using Liblinear. [sent-254, score-0.324]
81 Our proposed sampling method outperforms one based on non-maximal suppression (bbox) by a factor of two. [sent-258, score-0.13]
82 4(a) shows that the runtime of our algorithm compares favorably to state-of-the-art specialized solvers for linear SVMs (e. [sent-263, score-0.087]
83 It also achieves stronger final results by optimizing the structured loss directly We compare runtimes of cost-sensitive SVM methods in Fig. [sent-266, score-0.322]
84 4(c)) and show that SVM-IS converges faster for higher values of K. [sent-269, score-0.17]
85 This is expected because the update for K = 200 takes roughly the same amount of time as for K = 1, and allows us to update weights for all classes (as opposed to weights for a single class 4. [sent-270, score-0.307]
86 batch algorithms: Methods that employ updates in batches the size of the training set are slower by an asymptotic factor that scales with training set size. [sent-276, score-0.208]
87 [8] used a method for making a cost insensitive linear SVM cost sensitive by estimating posterior probabilities from the svm decision values. [sent-281, score-0.083]
88 multi sample: The multi-sample update reduces training time by a factor of approximately 550 compared to single sample online updates (PA-I [6] and SGD [19]) in the experiments that we have considered. [sent-284, score-0.333]
89 This is to be expected as the level of improvement is dependent on structural properties of the problem: the more complex the output space is, the more important the importance sampling routine becomes. [sent-285, score-0.152]
90 3a-b, we draw a vertical dotted red line indicating the time it takes to make one pass through the training set (where training roughly equals test time). [sent-287, score-0.167]
91 We see that for training sets of size n = 400 and n = 5000, test error is close to saturation point with only 1-2 passes through the training set. [sent-288, score-0.128]
92 Memory usage: An additional benefit of online/sequential algorithms is that only one training example needs to be loaded into memory at the same time. [sent-292, score-0.107]
93 , it excludes potentially faster methods based on boosting or random forests). [sent-296, score-0.129]
94 Conclusion We introduced a fast structured SVM solver that is shown to be significantly faster than existing methods based 111888111200 (a) Comparison to popular fast linear SVM-solvers (b) Other methods for cost-sensitive SVMs (c) Effect of Parameter K Figure 4. [sent-299, score-0.552]
95 Results on ImageNet: (a) Our method converges at least as quickly as existing specialized solvers for linear SVMs, while incorporation ofcost-sensitive learning allows us to obtain lower hierarchical loss. [sent-300, score-0.128]
96 (b) We implemented cost-sensitive versions of Stochastic Gradient Descent with a Pegasos-like update [21] and the Online Passive Aggressive Update (PA-I) method from [6]. [sent-301, score-0.096]
97 The proposed method converges noticeably faster than [21, 6], and is a full order of magnitude faster than the publicaly availible SVMstruct . [sent-302, score-0.36]
98 (c) A parameter sweep of the K parameter of the importance sampling routine illustrate that it’s best to update model parameters for all classes (K=200). [sent-303, score-0.307]
99 It reduces train time by a factor of 20-1000 for cost-sensitive multiclass learning and deformable part model training on Imagenet and CUB-200-201 1. [sent-305, score-0.349]
100 Faster training of structural svms with diverse m-best cutting-planes. [sent-394, score-0.342]
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