cvpr cvpr2013 cvpr2013-39 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Samuel Schulter, Paul Wohlhart, Christian Leistner, Amir Saffari, Peter M. Roth, Horst Bischof
Abstract: This paper introduces a novel classification method termed Alternating Decision Forests (ADFs), which formulates the training of Random Forests explicitly as a global loss minimization problem. During training, the losses are minimized via keeping an adaptive weight distribution over the training samples, similar to Boosting methods. In order to keep the method as flexible and general as possible, we adopt the principle of employing gradient descent in function space, which allows to minimize arbitrary losses. Contrary to Boosted Trees, in our method the loss minimization is an inherent part of the tree growing process, thus allowing to keep the benefits ofcommon Random Forests, such as, parallel processing. We derive the new classifier and give a discussion and evaluation on standard machine learning data sets. Furthermore, we show how ADFs can be easily integrated into an object detection application. Compared to both, standard Random Forests and Boosted Trees, ADFs give better performance in our experiments, while yielding more compact models in terms of tree depth.
Reference: text
sentIndex sentText sentNum sentScore
1 org UK Abstract This paper introduces a novel classification method termed Alternating Decision Forests (ADFs), which formulates the training of Random Forests explicitly as a global loss minimization problem. [sent-7, score-0.275]
2 Contrary to Boosted Trees, in our method the loss minimization is an inherent part of the tree growing process, thus allowing to keep the benefits ofcommon Random Forests, such as, parallel processing. [sent-10, score-0.407]
3 Compared to both, standard Random Forests and Boosted Trees, ADFs give better performance in our experiments, while yielding more compact models in terms of tree depth. [sent-13, score-0.196]
4 They all approximate Bayes Decision Rule known to be the optimal classifier via minimizing a margin-based loss function. [sent-21, score-0.237]
5 However, in contrast to other methods, RFs minimize this loss greedily and – – implicitly via recursively reducing the uncertainty of given training samples by using independent base classifiers, i. [sent-22, score-0.273]
6 Although these characteristics result in both, fast and parallel training capabilities, there is no control over an overall classifier loss and its proper minimization. [sent-25, score-0.289]
7 Third, it is hard to extend the learner to special learning tasks, such as, domain adaptation, semi-supervised or multiple instance learning, as this is usually realized via regularizing a global loss function. [sent-29, score-0.24]
8 In this paper, we propose a novel classifier termed Alternating Decision Forests (ADFs) that extends RFs by globally minimizing any given differentiable loss function, however, without losing the main characteristics and benefits of the original RF method as discussed above. [sent-30, score-0.336]
9 We achieve this by borrowing ideas from Boosting methods, where during training a globally tracked weight distribution guides the loss minimization. [sent-31, score-0.26]
10 To allow for incorporating arbitrary differentiable loss functions, we adopt the idea of employing gradient descent in function space [15] to calculate the weight updates. [sent-40, score-0.276]
11 In order to let each stage of the classifier follow the adaptive weight distribution, we replace standard RFs splitting criteria with the weighted entropy, while features are still randomly subsampled. [sent-41, score-0.235]
12 Thus, ADFs alternate between overall weight updates, and parallel randomized tree growing, where the latter one preserves the computational benefits of common Random Forests. [sent-42, score-0.218]
13 Furthermore, we empirically demonstrate that ADFs give more compact models in terms of tree depth as the training of the decision trees is guided by the sample weights, i. [sent-44, score-0.668]
14 Random Forests [1, 5] are ensembles of T binary deci- sion trees Tt(x) : X → RK, where X = RM is the Msdiiomne tnrseieosn Tal f(exa)tu :re X space Rand, R whKe r=e X[0, 1=]K R describes the space osifo nclaalss fe probability d ainstdri Rbutions over the label space Y = {1, . [sent-49, score-0.212]
15 During testing, each decision tree thus rYetu =rns { a c. [sent-53, score-0.284]
16 (1) During training of a RF, the decision trees are provided with a random subset of the training data (i. [sent-58, score-0.497]
17 Training a single decision tree involves recursively splitting each node such that the training data in the newly created child nodes is pure according to class labels. [sent-61, score-0.543]
18 Each tree is grown until some stopping criterion, e. [sent-62, score-0.209]
19 , the maximum tree depth, is reached and the class probability distributions are estimated in the leaf nodes. [sent-64, score-0.174]
20 Lin [23] showed that minimizing so called Fisher-consistent marginbased loss functions automatically leads to good approximations of the unknown Bayes decision rule. [sent-84, score-0.388]
21 As reviewed above, training in RFs is performed via recursively splitting training data into child nodes, where each split locally tries to optimize Eq. [sent-91, score-0.257]
22 ,(5) where lmm (·) is a margin maximizing loss function. [sent-98, score-0.201]
23 Thus, via greedily minimizing such a local score, a decision tree aims at minimizing a Fisher-consistent loss function, and is hence approximating Bayes optimal learners. [sent-100, score-0.497]
24 However, unlike other learners like Boosting or SVM approaches, RFs do not have an explicit global loss function. [sent-101, score-0.244]
25 In turn, the nodes operate locally and trees are grown isolated, disregarding the current state of the entire classifier. [sent-102, score-0.311]
26 Introducing a Global Loss To allow for integrating different, global loss functions into the Random Forests training procedure, we adopt ideas from Boosting [29, 13]. [sent-107, score-0.286]
27 A Boosting classifier F(x) consists of T weak learners ft (x) : X → RK, where each weak learner gives a prediction pt :( Xy|x) → →abo Rut the class confidences for a sample x. [sent-108, score-0.268]
28 There exist several different Boosting variants, most of them differing by the loss function they use, which in turn determines the shape of the weight distribution [17]. [sent-123, score-0.212]
29 1a we illustrate prominent loss functions, such as, Logit, Hinge, Exponential, Savage [25], and Tangent [24]. [sent-125, score-0.173]
30 In more detail, given a labeled training set {xi, yi}iN=1, training a single weak learner can be written as the global loss minimization problem ? [sent-131, score-0.385]
31 As shown in [15], this can be done by training ft (x; Θt) to have high correlation with the negative gradient of the loss, which corresponds to updating the weights wit for each training sample xi in iteration t as wti=? [sent-140, score-0.255]
32 This even allows for incorporating non-convex loss functions, which have been shown to be more robust to label noise [25]. [sent-154, score-0.173]
33 Training Alternating Decision Forests In order to incorporate any differentiable, global loss function into Random Forests, we adopt the idea of the above reviewed Gradient Boosting method, i. [sent-160, score-0.197]
34 While we could easily train Boosted Trees that respect a global loss function, i. [sent-163, score-0.197]
35 , Gradient Boosting having decision trees as weak learners, the goal of Alternating Decision Forests is to explicitly integrate the global loss into the tree growing scheme, thus preserving the parallel training of standard RFs. [sent-165, score-0.89]
36 For that purpose, we need (i) a stage-wise tree growing scheme during which we update the weight distribution and (ii) splitting functions taking these weights into account. [sent-166, score-0.389]
37 To get a stage-wise classifier, we let our trees grow in an iterative, breadth-first manner, contrary to a typical depthfirst scheme in standard RFs. [sent-167, score-0.233]
38 , Dmax, where Dmax is the maximum tree depth of the forest. [sent-173, score-0.213]
39 imate of sample xi returned by tree Ttd, which denotes tree? [sent-175, score-0.164]
40 e d W bey can now use this strong tcrleaess Tifier and a given loss l(·) to update the weights oclfa asslli training samples foosrs t lh(e·) )n teoxt u iptedraatteio tnhe as ielligushttrsa twed in ? [sent-177, score-0.289]
41 tT=1ptd(yi|xi), wid+1 555 010088 Figure 2: Overview of the proposed tree growing principle of Alternating Decision Forests. [sent-178, score-0.185]
42 In the first iteration (d = 1), the weights wid are uniform, and the first split functions are trained in a breadth-first manner. [sent-179, score-0.21]
43 This forest with depth d = 2 can give predictions on the training samples, which are used to calculate weights, based on a global loss function, for the next iteration d = 3. [sent-180, score-0.47]
44 This procedure is repeated until the maximum tree depth d = Dmax is reached. [sent-181, score-0.213]
45 To let the splitting functions consider the sample weights, we change the standard entropy calculation from Eq. [sent-187, score-0.189]
46 This process of alternating bqeutawle teon k training a single stage d and updating the weights for the next stage is repeated until the same stopping criteria as in standard RFs are reached. [sent-193, score-0.312]
47 (8) 7: end for wid+1 Please note that, if each tree would correspond to one weak learner ft (x; Θ), each being fully trained in a single iteration, we would get a version of Boosted Trees [17]. [sent-206, score-0.278]
48 In contrast, we train all trees in parallel, by switching from a depth-first to a breadth-first tree growing scheme. [sent-207, score-0.397]
49 Then, we run Dmax iterations, where in each iteration all trees of the forest are simultaneously grown deeper by one stage. [sent-209, score-0.364]
50 The weighted tree growing scheme in ADFs can be interpreted as a guided training of each single decision tree, but also as an explicit collaboration between all trees in the model. [sent-210, score-0.64]
51 All nodes in the trees concentrate on “hard” training samples and, thus, don’t waste effort on samples that are already learned relatively well by the entire model. [sent-211, score-0.362]
52 However, unlike Boosted Trees, this global loss is now an inherent part of Random Forests. [sent-212, score-0.197]
53 As we show in our experimental evaluations, this property of ADFs leads to more compact models in terms of tree depth. [sent-213, score-0.149]
54 For RFs, an upper bound on the generalization error GE can be defined as GE ≤ , where ρ denotes the mean pair-wise fcionerrde alastio GnE E b ≤etw ρeen trees and s the strength of individual trees [5]. [sent-214, score-0.424]
55 While ADFs explicitly enforce the collaboration of all trees in the forest, the decorrelation ρ can be preserved as in standard RFs, because (i) the splitting functions s(x; Θ) are still drawn completely randomly and (ii) the set of samples falling in common nodes is different for each tree. [sent-216, score-0.454]
56 Relation to Previous Work In [12], Freund and Mason proposed a formulation to represent AdaBoost as a single decision tree. [sent-219, score-0.154]
57 In contrast to standard RFs, the individual trees of ADFs become interdependent or entangled during training. [sent-224, score-0.258]
58 This is similar to the works of [27] and [20] that train decision trees breadth-first or according to a priority queue, in order to incorporate contextual features into the learning process. [sent-225, score-0.366]
59 However, both works only consider the predictions within a single decision tree and use those predictions as additional features for the splitting functions in the next stage. [sent-226, score-0.499]
60 , all decision trees, and uses the predictions in order to minimize a global loss function. [sent-229, score-0.395]
61 Another work that optimizes trees according to a differentiable loss function is that of Jancsary et al. [sent-231, score-0.429]
62 [19], where the leaves of the trees store parameters for a Gaussian Conditional Random Field with different interaction types. [sent-232, score-0.212]
63 Each factor type is only associated with a single tree but they are connected implicitly via the random field. [sent-233, score-0.165]
64 As mentioned before, in BTs, the decision trees are trained sequentially and the weights are updated after training each single tree. [sent-240, score-0.472]
65 Furthermore, during training a single decision tree in BTs, the weights cannot be updated. [sent-242, score-0.37]
66 Contrary, in ADFs, it is exactly this property that allows for learning more compact models as the growing of each tree is somehow guided by the weight updates from the previous depth in the trees. [sent-243, score-0.364]
67 We also investigate different choices of the loss function. [sent-249, score-0.206]
68 For ADFs and BTs we also evaluate 5 different loss functions that can be integrated in the Gradient Boosting formulation: Logit, Hinge, Exponential, Savage [25] and Tangent [24]. [sent-264, score-0.214]
69 As can be seen, the combination of ADFs and the Tangent loss function yields the best results on all 5 data sets. [sent-267, score-0.173]
70 ADFs with Savage loss and Exponential loss are the second best choices on 3, respectively 1data sets. [sent-268, score-0.379]
71 Only for G50c, BTs with Savage loss gives the second best results. [sent-269, score-0.173]
72 It is also worthwhile to investigate the influence of the different loss functions. [sent-273, score-0.192]
73 1a), plays a crucial role for the weight updates and thus the tree growing. [sent-277, score-0.211]
74 For our further experiments, we fix the loss function to be the Tangent loss, as we can expect the best overall performance. [sent-278, score-0.173]
75 We investigate the number of trees T and the maximum tree depth Dmax, which we vary in the ranges [1, 100] and [10, 25], respectively. [sent-503, score-0.425]
76 As mentioned before, we choose the Tangent loss for both ADFs and BTs due to the good overall performance in the last experiment. [sent-504, score-0.173]
77 It is interesting that ADFs improve over RFs and BTs only for larger number of trees (see Fig. [sent-505, score-0.212]
78 However, due to the small number of trees and the fact that the trees are not fully grown yet, these predictions are unreliable. [sent-509, score-0.525]
79 However, as soon as the number of trees increases, also the performance of ADFs increases and outperforms both RFs and BTs. [sent-511, score-0.212]
80 This can be explained by the fact that the weight updates can rely on more stable predictions, as a larger number of trees is available. [sent-517, score-0.293]
81 This experiment also reflects the guidance of the tree growing mentioned in Sec. [sent-518, score-0.185]
82 3, as both RFs and BTs need deeper trees to reach the performance of ADFs. [sent-519, score-0.212]
83 Tre D2e0pthBRATFD25 (a) (b) Figure 3: Influence of (a) the number of trees T and (b) the maximum tree depth Dmax on the three classifiers ADFs, RFs, and BTs on the Char74k data set. [sent-525, score-0.451]
84 Here, we compare several combinations of two different parameters again, the number of trees T and the maximum tree depth Dmax. [sent-529, score-0.425]
85 The performance steadily increases with the number of trees and also the maximum depth of the trees. [sent-533, score-0.295]
86 18% error on the Char74k data set with 300 trees and a maximum – depth of 25. [sent-537, score-0.295]
87 1572 ± ±± ±± ±± ±± 555 111311 Number of trees 10 0. [sent-571, score-0.212]
88 13 Table 4: Different parameter choices of the number of trees T and the maximum tree depth Dmax for Alternating Decision Forests on the Char74k data set. [sent-681, score-0.458]
89 Given those patches Pi, a Random Forest is trained with two different splitting criteria: (i) a standard classification criterion (see Sec. [sent-684, score-0.169]
90 We thus also incorporate the minimization of a global loss function via iterative weight updates in the classification criterion in this framework. [sent-696, score-0.3]
91 Each patch Pi is thus assigned a weight wi, which is always updated after training a single stage of the classifier according to a given global loss function l(·), as described ianc cSoercd. [sent-697, score-0.351]
92 For both data sets, we extract a total number of 12000 patches and train a forest with 10 trees, a maximum depth of 15 and 5000 random tests per node. [sent-712, score-0.202]
93 We also evaluate different loss functions for ADHF and BHF. [sent-713, score-0.214]
94 Interestingly, our evaluations revealed that different loss functions perform better on different data sets. [sent-719, score-0.214]
95 While for TUD-pedestrian both non-convex and noise-robust loss functions, i. [sent-720, score-0.173]
96 For ADHF and BHF, we show the curves of the best performing loss functions as presented above. [sent-727, score-0.214]
97 Conclusion We proposed a novel classifier, termed Alternating Decision Forest (ADF), which illustrates how to formulate the Random Forest training as a global loss minimization problem. [sent-732, score-0.275]
98 In contrast to the local optimization in standard Random Forests, we grow the trees stage-wise and include sample weight updates to minimize the global loss, as in Boosting algorithms. [sent-733, score-0.338]
99 Semantic texton forests for image categorization and segmentation. [sent-921, score-0.243]
100 New multicategory boosting algorithms based on multicategory Fisher-consistent losses. [sent-927, score-0.177]
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