Online Bagging and Boosting

Transcription

1 Abstract Bagging and boosting are two of the ost well-known enseble learning ethods due to their theoretical perforance guarantees and strong experiental results. However, these algoriths have been used ainly in batch ode, i.e., they require the entire training set to be available at once and, in soe cases, require rando access to the data. In this paper, we present online versions of bagging and boosting that require only one pass through the training data. We build on previously presented work by presenting soe theoretical results. We also copare the online and batch algoriths experientally in ters of accuracy and running tie. Keywords: Bagging, boosting, enseble learning, online learning. 1 Introduction Traditional supervised learning algoriths generate a single odel such as a Naïve Bayes classifier or ultilayer perceptron () and use it to classify exaples. 1 Enseble learning algoriths cobine the predictions of ultiple base odels, each of which is learned using a traditional algorith. Bagging [3] and Boosting [4] are well-known enseble learning algoriths that have been shown to be very effective in iproving generalization perforance copared to the individual base odels. Theoretical analysis of boosting's perforance supports these results [4]. In previous work [1][2], we developed online versions of bagging and boosting. Online learning algoriths process each training exaple once ``on arrival'' without the need for storage and reprocessing, and aintain a current odel that reflects all the training exaples seen so far. Such algoriths have advantages over typical batch algoriths in situations where data arrive continuously. They are also useful with very large data sets on secondary storage, for which the ultiple passes through the training set required by ost batch algoriths are prohibitively expensive. In Sections 2 and 3, we describe our online bagging and online boosting algoriths, respectively. In particular, we describe how we irror the ethods that the batch bagging and boosting algoriths use to generate Online Bagging and Boosting 1 In this paper, we only deal with the classification proble. Nikunj C. Oza Intelligent Systes Division NASA Aes Research Center Mail Stop Moffett Field, CA, USA oza@eail.arc.nasa.gov distinct base odels, which are known to help enseble perforance. In our previous work, we also discussed soe preliinary theoretical results and soe epirical coparisons of the classification accuracies of our online algoriths and the corresponding batch algoriths on any datasets of varying size. In Sections 2 and 3, we give a brief description of soe additional theoretical results. In Section 4, we review the experiental results in our previous work deonstrating the perforance of our online algoriths relative to their batch counterparts. In this paper, we expand upon these results by coparing and studying their running ties. We run our online bagging and boosting algoriths with two different base odels: Naïve Bayes classifiers and s. We use a lossless online learning algorith for Naïve Bayes classifiers. For a given training set, a lossless online learning algorith returns a odel identical to that returned by the corresponding batch algorith. For s, we are forced to use a lossy online learning algorith. In particular, we do not allow the s backpropagation algorith to cycle through the training set in ultiple epochs the way backpropagation is norally allowed to do. Overall, our online bagging and boosting algoriths perfor coparably to their batch counterparts in ters of classification accuracy when using Naïve Bayes base odels. The loss experienced by online s relative to batch s leads to a significant loss for online bagging and boosting relative to the batch versions. Online bagging often does iprove significantly upon online s; however, online boosting never perfored significantly better than single online s. We also copare the running ties of the batch and online algoriths. If the online base odel learning algorith is not significantly slower than the corresponding batch algorith, then the bagging and online bagging algoriths do not have a large difference in their running tie in our tests. On the other hand, our online boosting algorith runs significantly faster than batch boosting. For exaple, on our largest dataset, batch boosting ran twice as long as online boosting to achieve coparable classification accuracy. 2 Online Bagging Given a training dataset T of size N, standard batch bagging creates M base odels. Each odel is trained by calling the batch learning algorith L b on a bootstrap saple of size N created by drawing rando saples with

2 replaceent fro the original training set. Figure 1 gives the pseudocode for bagging. Bagging(T,L b,m) For each { 1,2,K, M}, T = Saple_With_Replaceent(T, T ) h = L b ( T ) { } Return h 1,h 2,K,h M Figure 1: Bagging Algorith Each base odel's training set contains K copies of each of the original training exaples where ( ) = N k P K = k 1 k N 1 1 N N k which is the Binoial distribution. As N, the distribution of K tends to a Poisson(1) distribution: P(K = k) ~ exp( 1) /k!. As discussed in [1] [2], we can perfor bagging online as follows: as each training exaple d=(x,y) is presented to our algorith, for each base odel, choose the exaple K ~ Poisson(1) ties and update the base odel accordingly using the online base odel learning algorith L o (see Figure 2). New exaples are classified the sae way in online and batch bagging---by unweighted voting of the M base odels. OnlineBagging(h,L o,d) For each base odel h h, { 1,2,K,M}, Set k according to Poisson(1). Do k ties h = L o ( h,d). Figure 2: Online Bagging Algorith Online bagging is a good approxiation to batch bagging to the extent that their base odel learning algoriths produce siilar odels when trained with siilar distributions of training exaples. In past work [1][5], we proved that if the sae original training set is supplied to the two bagging algoriths, then the distributions over the training sets supplied to the base odels in batch and online bagging converge as the size of that original training set grows to infinity. We have also proven that the classifiers returned by bagging and online bagging converge to the sae classifier given the sae training set as the nuber of odels and training exaples tends to infinity under two conditions. The first is that the base odel learning algoriths return classifiers that converge toward the sae classifier as the nuber of training exaples grows. The second is that, given a fixed training set T, the online and batch base odel learning algoriths return the sae classifier for any nuber of copies of T that are presented to the learning algorith. For exaple, doubling the training set by repeating every exaple in T yields the sae classifier as T would yield. For exaple, this condition is true of decision trees and Naïve Bayes classifiers, but is not true of s, since doubling the training set effectively doubles the nuber of epochs in backpropagation training. More foral details are presented in [5]. 3 Online Boosting Our online boosting algorith is designed to correspond to the batch boosting algorith, AdaBoost.M1 [4] (the pseudocode is given in Figure 3). AdaBoost generates a sequence of base odels h 1,h 2,K,h M using weighted training sets (weighted by D 1,D 2,K,D M ) such that the training exaples isclassified by odel h 1 are given half the total weight when generating odel h and the correctly classified exaples are given the reaining half of the weight. When the base odel learning algorith cannot learn with weighted training sets, one can generate saples with replaceent according to D. AdaBoost( {( x 1,y 1 ),( x 2,y 2 ),K,( x N,y N )},L b,m) Initialize D 1 ( n) =1/N for all n { 1,2,K,N}. For all { 1,2,K,M}, h = L b ({( x 1,y 1 ),( x 2,y 2 ),K,( x N,y N )},D ). N ε = D ( n)i( h ( x n ) y n ). n=1 If ε 1/2 then set M = -1 and abort loop. For all n { 1,2,K,N}, D +1 ( n) = D ( n) Return h fin ( x) = argax y Y M = ε 1 2ε ( ) if h x ( n) = y n log 1 ε I h ε Figure 3: AdaBoost algorith otherwise ( ( x n ) = y n ) Our online boosting algorith (Figure 4) siulates sapling with replaceent using the Poisson distribution just like online bagging does. The only difference is that

3 when a base odel isclassifies a training exaple, the Poisson distribution paraeter λ associated with that exaple is increased when presented to the next base odel; otherwise it is decreased. Just as in AdaBoost, our algorith gives the exaples isclassified by one stage half the total weight in the next stage; the correctly classified exaples are given the reaining half of the weight. This is done by keeping track of the total weights of each base odel s correctly classified and isclassified training exaples ( λ sc and λ sw, respectively) and using these to update each base odel s error ε. At this point, a training exaple s weight is updated the sae way as in AdaBoost. One area of concern is that, in AdaBoost, an exaple's weight is adjusted based on the perforance of a base odel on the entire training set while in online boosting, the weight adjustent is based on the base odel's perforance only on the exaples seen earlier. To see why this ay be an issue, consider running AdaBoost and online boosting on a training set of size In AdaBoost, the first base odel h 1 is trained on all exaples before being tested on, say, the tenth training exaple. In online boosting, h 1 is trained on only the first ten exaples before being tested on the tenth exaple. Clearly, at the oent when the tenth training exaple is being tested, we ay expect the two h 1 's to be very different; therefore, h 2 in AdaBoost and h 2 in online boosting ay be presented with different weights for the tenth training exaple. This ay, in turn, lead to different weights for the tenth exaple when generating h 3 in each algorith, and so on. Intuitively, we want online boosting to get a good ix of training exaples so that the base odels and their noralized errors in online boosting quickly converge to what they are in AdaBoost. The ore rapidly this convergence occurs, the ore siilar the training exaples' weight adjustents will be and the ore siilar their perforances will be. We have proven [5] that for Naive Bayes base odels, the online and batch boosting algoriths converge to the sae classifier as the nuber of odels and training exaples tends to infinity. 4 Experiental Results In this section, we discuss results on several different datasets, whose naes and nubers of training exaples, test exaples, inputs, and classes are given in Table 1. The Soybean-Large and Census Incoe datasets coe with fixed training and test sets, which we use in our experients. For the reaining datasets, we give the training and test set sizes that result fro using 5-fold cross-validation. We discuss the results on soe sall datasets to show that the online algoriths can often achieve perforance coparable to batch algoriths even when given a sall nuber of data points. We discuss results on several larger datasets in ore detail since online algoriths are ost useful for such datasets. All but three of the datasets were drawn fro the UCI KDD repository [6]. The reaining three are synthetic datasets that were chosen because the perforance of a single Naive Bayes classifier varies significantly across these three datasets. These datasets allow us to copare the perforances of the online and batch enseble algoriths on datasets of varying difficulty. Table 1: The datasets used in our experients. Data Set Train Test Input Class Set Set Prooters Balance Soybean-Large Breast Car Chess Mushroo Nursery Connect Synthetic Synthetic Synthetic Census Incoe Accuracy For both bagging and boosting, we present results using two different base odel types : Naive Bayes classifiers and ultilayer perceptrons (s). Both bagging algoriths generated 100 base odels. Both boosting algoriths were allowed to generate up to 100 base odels. All the results shown are based on 10 runs of 5-fold cross validation (except on the Soybean-Large and Census Incoe, where we used the supplied training and test sets). All the online algoriths were run five ties for every one tie the batch algorith was run, with different rando orders of the training set. This was done to account for the effect that the order of the training exaples can have on the perforance of an online learning algorith. Tables 2-5 show the results of online bagging and boosting copared to their batch counterparts and single base odels using Naive Bayes classifiers and s. The online was trained by using backpropagation to update the with each training exaple ten ties upon arrival; however, the algorith only ran through the entire training set once in the order in which it was presented. The batch was trained by using backpropagation to update the in ten epochs (ten cycles through the entire training set). All coparisons between algoriths were ade using a paired t-test (α=0.05). Table 2 shows the results of running bagging with Naive Bayes classifiers. Entries in boldface/italics indicate that the enseble algorith perfored significantly better/worse than a single Naive Bayes classifier. The bagging algoriths perfored coparably to each other and ostly perfored coparably to the batch Naive Bayes algorith. This is expected due to the stability of Naive

4 Bayes classifiers [3]. That is, the Naïve Bayes classifiers in a bagged enseble tend to classify new exaples the sae way (we obtained at least 90% agreeent on all test exaples) in spite of the differences in the training sets. Initial conditions : For all { 1,2,K, M}, λ sc = 0,λ sw = 0. OnlineBoosting(h,L o,(x,y)) Set λ =1. For each base odel h h, { 1,2,K,M}, Set k according to Poisson(λ). Do k ties h = L o ( h,(x,y)). If y = h ( x) λ sc λ sc + λ ε λ sw λ sc sw + λ 1 λ λ 2( 1- ε ) else λ sw λ sw + λ ε λ sw λ sc sw + λ λ λ 1 2ε end To classify a new exaple with input x, return : h fin M ( x) = argax y Y log 1 ε I( h ( x) = y). =1 ε Figure 4: Online Boosting Algorith Table 3 shows the results of running the boosting algoriths with Naive Bayes classifiers. In the «Online Boosting» colun, any entry with a + or - after it indicates that online boosting perfored significantly better or worse than batch boosting, respectively. Batch boosting significantly outperfors online boosting in any cases--- especially the saller datasets. However, the perforances of boosting and online boosting relative to a single Naive Bayes classifier agree to a rearkable extent. That is, when one of the is significantly better or worse than a single Naive Bayes classifier, the other tends to be the sae way. Table 2: Bagging vs. Online Bagging, Naive Bayes Dataset Naive Bayes Bagging Online Bagging Prooters Balance Breast Geran Car Chess Mushroo Nursery Connect Synthetic Synthetic Synthetic Census Incoe Table 3: Boosting vs. Online Boosting, Naive Bayes Dataset Naive Bayes Boosting Online Boosting Prooters Balance Breast Geran Car Chess Mushroo Nursery Connect Synthetic Synthetic Synthetic Census Incoe Table 4 shows the results of running bagging with s. The entries for bagging shown in boldface/italics indicate that bagging significantly outperfored/underperfored relative to the batch. The entries for online bagging shown in boldface/italics indicate that online bagging significantly outperfored/underperfored relative to the online. The entries for online bagging with a - after the indicate ties when it perfored significantly worse than batch bagging. The online always perfored significantly worse than the batch ; therefore, it is not surprising that online bagging often perfored significantly worse than batch bagging. However, online bagging did significantly outperfor online s ost of the tie. Table 5 gives the results of running boosting with s. Entries in the online and boosting colun that are given in boldface/italics indicate that it significantly outperfored/underperfored relative to batch s. Entries in the online boosting colun given in boldface/italics indicate ties when it significantly outperfored/underperfored relative to the online. Entries with a - after the indicate ties when online boosting perfored significantly worse than batch

5 boosting. Clearly, the significant loss in using an online instead of a batch has rendered the online boosting algorith significantly worse than batch boosting. Table 4: Bagging vs. Online Bagging, s Dataset Online Bagging Online Bagging Prooters Balance Breast Geran Car Chess Mushroo Nursery Connect Synthetic Synthetic Synthetic Census Incoe Table 5: Boosting vs. Online Boosting, s Dataset Online Boosting Online Boosting Prooters Balance Breast Geran Car Chess Mushroo Nursery Connect Synthetic Synthetic Synthetic Census Incoe Running Tie In this section, we report and analyze the running ties of the batch and online algoriths that we experiented with. There are several factors that affect the difference between the running ties of an online learning algorith and its batch counterpart. Online learning algoriths ain advantage over batch learning algoriths is the ability to increentally update their odels with new training exaples---batch algoriths often have to throw away the previously learned odel and learn a new odel after adding the new exaples to the training set. This is clearly very wasteful coputationally and is ipossible when there are ore data than can be stored. Additionally, batch bagging ust cycle through the dataset at least MT ties, where M is the nuber of base odels and T is the nuber of ties the base odel learning algorith ust cycle through the training set to construct one odel. Therefore, each training exaple is exained MT ties. On the other hand, online bagging only needs to sweep through the training set once, which eans that each training exaple is exained only M ties (once to update each base odel s paraeters). Online algoriths do not require storing the entire training set. However, for a fixed training set (i.e., one to which new training exaples are not continually added), batch algoriths soeties run faster than the corresponding online algoriths. This is because batch algoriths can often set their odel paraeters once and for all by exaining the entire training set at once while online algoriths have to update their paraeters once per training exaple. Table 6: Running ties for Naive Bayes and Ensebles. Dataset Naive Bayes Bag Online Bag Boost Online Boost Prooters Balance Breast Geran Car Chess Mushroo Nursery Connect Synthetic Synthetic Synthetic Census Incoe The coparison between batch and online boosting has the additional factor of the nuber of base odels. Batch boosting, when called with the upper liit of M base odels, can choose to generate fewer odels---recall that if a odel s error is greater than 0.5, then boosting will discard that odel and return the enseble generated so far. Online boosting does not have this luxury because it does not know what the final error rates will be for each base odel. This difference can lead to lower training ties for batch boosting. However, batch boosting needs to cycle through the training set M(T+1) ties---each of the M base odels requires T cycles through the training set to learn the odel and one cycle to calculate the error on the

6 training set. Online boosting only requires one sweep through the training set. Table 7: Running ties for s and Bagging. Dataset Online Bag Online Bag Prooters Balance Breast Geran Car Chess Mushroo Nursery Connect Synthetic Synthetic Synthetic Census Incoe Table 8: Running ties for and Boosting. Dataset Online Boost Online Boost Prooters Balance Breast Geran Car Chess Mushroo Nursery Connect Synthetic Synthetic Synthetic Census Incoe Table 6 shows the running ties for Naive Bayes as well as all the enseble learning algoriths using Naive Bayes classifiers as base odels. The running tie for online bagging is generally soewhat greater than for batch bagging. The total nuber of ties each training exaple is exained is the sae for both batch and online bagging with Naive Bayes classifiers. However, online bagging requires a greater nuber of procedure calls to the learning algorith (MT as opposed to M), which ay explain the running tie difference. On the other hand, online boosting has a clear running tie advantage over batch boosting. Online boosting s fewer sweeps through the dataset clearly outweigh any reduction in the nuber of base odels returned by batch boosting. Tables 7-8 give the running ties for s and the batch and online enseble algoriths. This tie, both online bagging and online boosting are faster than their batch counterparts. The batch algoriths are slowed down because each requires ten cycles through the dataset. 5 Conclusions In this paper, we discussed online versions of bagging and boosting and gave both theoretical and experiental evidence that they can perfor coparably to their batch counterparts while running uch faster. In this paper, we experiented only with batch datasets, i.e., one is not concerned with concept drift. Online algoriths are useful for batch datasets that cannot be loaded into eory in their entirety. We plan to experient with online doains---doains where data arrive continually and where a prediction ust be generated for each data point upon arrival. In these situations, the learner ay be given iediate feedback (such as a calendar assistant which ay suggest a eeting tie which the user can either select or change) or ay obtain feedback periodically. The tievarying nature of such datasets ake the ore difficult to deal with but ore needy of online enseble learning algoriths. References [1] Nikunj C. Oza and Stuart Russell, Online Bagging and Boosting, in Artificial Intelligence and Statistics 2001, Key West, FL, USA, pp January [2] Nikunj C. Oza and Stuart Russell, Experiental Coparisons of Online and Batch Versions of Bagging and Boosting, The Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, pp , August [3] Leo Breian, Bagging Predictors, Machine Learning, Vol. 24, No. 2, pp , [4] Yoav Freund and Robert Schapire, A Decision- Theoretic Generalization of On-line Learning and an Application to Boosting, Journal of Coputer Syste Sciences, Vol. 55, No. 1, pp , [5] Nikunj C. Oza, Online Enseble Learning, Ph.D. thesis, Departent of Electrical Engineering and Coputer Science, University of California, Berkeley, [6] Stephen D. Bay, The UCI KDD Archive, Departent of Inforation and Coputer Sciences, University of California, Irvine