Tighter Perceptron with Improved Dual Use of Cached Data for Model Representation and Validation

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1 Proceedngs of Internatonal Jont Conference on Neural Networks, Atlanta, Georga, USA, June 49, 29 Tghter Perceptron wth Improved Dual Use of Cached Data for Model Representaton and Valdaton Zhuang Wang and Slobodan Vucetc Abstract Kernel Perceptrons are represented by a subset of tranng ponts, called the support vectors, and ther assocated weghts. To address the ssue of unlmted growth n model sze durng tranng, budget kernel perceptrons mantan the fxed number of support vectors and thus acheve the constant update tme and space complexty. In ths paper, a new kernel perceptron algorthm for onlne learnng on a budget s proposed. Followng the dea of Tghter Perceptron, upon exceedng the budget, the algorthm removes the support vector wth the mnmal mpact on classfcaton accuracy. To optmze memory use, nstead on mantanng a separate valdaton data set for accuracy estmaton, the proposed algorthm only uses the support vectors for both model representaton and valdaton. Ths s acheved by estmatng posteror class probablty of each support vector and usng ths nformaton n valdaton. The expermental results on benchmark data sets ndcate that the proposed algorthm s sgnfcantly more accurate than the competng budget kernel perceptrons and that t has comparable accuracy to the resource unbounded perceptrons, ncludng the orgnal kernel perceptron and the Tghter Perceptron that uses whole tranng data set for valdaton. T I. INTRODUCTION HE nventon of the Support Vector Machnes [2] attracted a lot of nterest n adaptng the kernel methods for both batch and onlne learnng. Kernel perceptrons [5, 7, 8, 9] are a popular class of algorthms for onlne learnng. They are represented by a subset of observed examples, called the support vectors, and ther weghts. The baselne kernel perceptron algorthm s smple t observes a new example and, f t s msclassfed by the current model, adds t to the model as a new support vector. The popularty of kernel perceptrons s due to ther ease of mplementaton, the ablty to acheve qute compettve classfcaton accuracy to the batch mode alternatves, and the exstence of theoretcal results characterzng ther behavor. In addton to ther appealng propertes, kernel perceptrons often suffer from an unbounded growth n the number of support vectors wth tranng data sze. Ths, n turn, causes unbounded growth n tranng tme and space needed to store the classfer. Such behavor s unacceptable n many practcal onlne learnng applcatons. To address the ssue of unbounded growth n computatonal resources, a Ths work was supported by the U.S. Natonal Scence Foundaton under Grant IIS Zhuang Wang and Slobodan Vucetc are wth the Center for Informaton Scence and Technology, Department of Computer and Informaton Scences, Temple Unversty, Phladelpha, PA 922, USA, emal: zhuang@temple.edu, vucetc@st.temple.edu. class of onlne kernel perceptron algorthms on a fxed budget has been developed. To mantan the budget, the proposed algorthms typcally decde to dscard one of the support vectors when the budget s exceeded upon addton of a new support vector. Whle there are theoretcal guarantees for convergence of several budget kernel perceptron algorthms, ther actual performance s often poor on nosy classfcaton problems. Among budget kernel perceptrons, the Tghter Perceptron algorthm [3] s one of the most successful n practce. It removes the support vector that has the mnmal postve mpact on the classfcaton accuracy. To estmate the accuracy, the algorthm requres mantenance of an addtonal valdaton data set. When the valdaton set s large, Tghter Perceptron s able to acheve respectable classfcaton accuracy. However, the exstence of valdaton set also puts an ncreasng burden on tranng speed and memory. When, n order to decrease the budget, the support vector set s used both for model representaton and valdaton, the accuracy decreases dramatcally. The man reason for such behavor s that support vectors are a based sample of tranng examples that were ncorrectly classfed. Therefore, support vectors are lkely to contan an overwhelmng amount of nosy tranng examples and could therefore provde qute msleadng accuracy estmates. In ths paper, we propose a new algorthm, called here for convenence the Tghtest Perceptron, that manages to obtan hghly accurate estmates of classfcaton accuracy usng exclusvely the support vectors. To acheve ths, a smple data summary s mantaned along each support vector to estmate the posteror class probablty for each support vector. Durng the valdaton, the expectaton of the valdaton hnge loss s calculated based on the estmated class probabltes. The expermental results show that the proposed algorthm has mpressve performance on benchmark data sets. II. PROBLEM SETTING AND PREVIOUS WORK We study the onlne learnng for bnary classfcaton. Onlne learnng s performed n a sequence of consecutve rounds. On each round, the algorthm observes an example from the tranng set. An example s a par (x, y), where x s an M-dmensonal attrbute vector and y + {, } s the assocated bnary label. The ndependent and dentcally dstrbuted tranng set D s a sequence of examples (x, y ),...,(x N, y N ) and can only be observed n a sngle pass. The classcal perceptron algorthm [] has the followng tranng procedure. Intally, the predcton model f (x) s set to zero, f (x) =. In round t, the new example (x t, y t ) s observed /9/$ IEEE 3297

2 and ts label s predcted by the current model f (x) as sgn( f (x t ) ). If the margn of ths example, defned as the product y t f (x t ), s below threshold β = (.e. yf t ( xt) β ), then weght α t = s assgned to ths example and the model s updated as f ( x) = f ( x) + αtytxt x. Each example added to the model s called the Support Vector (SV). If the example s correctly classfed, ts weght s set to α t =, and the model s thus not updated. Alternatvely, the algorthm can also modfy the current hypothess by multplyng t wth scalar φ t ( ft( x) = φt ft( x) ). The standard parameter values of β =, α t =, and φ t = can be chosen dfferently, as was done n ALMA [7], ROMMA [9], NORMA [8] and PA [5] algorthms. In ths paper, we wll consder kernel perceptrons wth the standard parameter values. The classcal perceptron mples a lnear decson functon. It could be made nonlnear by usng Φ(x) as attrbutes nstead of x, where Φ s a nonlnear mappng of the orgnal attrbute space nto the feature space. If there exsts a kernel functon k such that Φ( x) Φ ( x) = k( x, x) [], the model f (x) can be represented as f ( x N N ) = α y ( ) ( ) (, ) Φ x Φ x = α yk x = = x and s denoted as the kernel perceptron. It s mportant to note that the kernel functon k allows us to express the model n terms of the orgnal attrbutes and avod explctly workng n the potentally hgh (or nfnte) dmensonal feature space. A. Budget Perceptron Algorthms In spte of the powerful performance, kernel methods often suffer from an unbounded growth n the number of support vectors wth tranng data. Ths creates serous problems n both tranng and testng phase because the tme needed to compute f (x) and the space needed to store the model scales lnearly wth the number of SVs. In many practcal onlne applcatons where a short feedback tme and bounded space s a requrement, the unbounded growth mentoned above s not acceptable. Ths fact motvated work n developng onlne algorthms on a fxed budget. ) Fxed Budget Perceptron: The poneerng work was done n [4] to address the problem. There, a standard kernel perceptron was modfed by addng a support vector removal procedure to keep the budget. Let us denote I t as the set of support vectors at round t of the kernel perceptron algorthm. If the number of support vectors exceeds the predefned budget B n round t (.e. I t >B), support vector wth the largest margn, arg max y f( x ) α y k( x, x ), j It { ( j j j j j j s removed. Whle ths algorthm acheves respectable accuracy on relatvely nose-free data t s less successful on nosy data. Ths s because n the nosy case ths algorthm tends to remove well-classfed ponts and accumulate the nosy examples, resultng n degradaton of accuracy. 2) Random Perceptron: The smplest removal procedure s to remove a randomly selected support vector. Despte ts smplcty, ths algorthm often has satsfactory performance. )} In addton, the algorthm s convergence has also been proven [2]. 3) Forgetron: A more advanced removal procedure was developed n [6] by ntroducng a forgettng factor. After each update step, forgettng factor < φ t < s used to scale the current model (and all ts support vectors). The oldest support vector (wth the smallest weght) s removed f budget s exceeded. The algorthm s convergence has also been proven. 4) Tghter Perceptron: [3] proposed to remove the support vector that has the smallest postve nfluence on accuracy. To allow accuracy estmaton, an addtonal valdaton set composed of the prevously observed tranng examples s mantaned. Specfcally, on the t-th round where I t >B, the algorthm removes j-th support vector wth arg mn l ( yk,sgn ( f( xk) α jyjk( xj, xk) )), j It k Vt where V t s the valdaton set and l denotes the classfcaton loss. From the perspectve of accuracy estmaton, t s deal to use all the prevously seen tranng examples for valdaton. However, the use of valdaton set puts an addtonal burden to the memory budget and the tranng tme. Due to practcal consderatons, the sze of valdaton data set should be restrcted. There are several varants of the Tghter algorthm dependng on the sze of the valdaton set: Tghter Full uses all tranng examples for valdaton, Tghter A uses selected A examples that are dsjont from the support vectors, and Tghter uses support vectors. Whle Tghter A and Tghter are budget algorthms, ther accuracy estmates are less relable. That s especally the case for Tghter because the support vector set s a based sample from tranng data that s lkely to contan dsproportonally large fracton of nosy examples. In the followng secton, a statstcally-based method s proposed to mprove accuracy estmaton usng only the support vector set. III. THE PROPOSED ALGORITHM The man property of the proposed algorthm s an mproved accuracy valdaton usng only the support vector set. The valdaton mprovement s possble when posteror class probabltes of support vectors are used nstead of ther actual labels (as s done n Tghter ). The open problem wth ths approach s that posteror class probabltes of support vectors are unknown and should be estmated. Our dea s that a hgh-qualty class probablty estmate for each support vector can be obtaned by lookng at labels of tranng examples n ts neghborhood. We call the resultng algorthm the Tghtest Perceptron or Tghtest, n short. The proposed algorthm s sketched n Fgure. Instead of smply dscardng the selected support vector or the new tranng pont that does not become the support vector, we are usng ts class nformaton to mprove the class probablty estmate of ts nearest support vector. To mplement the dea, the -th SV s represented by tuple (x, y, c +, c ) contanng ts 3298

3 Input: (x, y ),...,(x N, y N ), budget B Intalzaton: f(x) =, S = Output: f(x) for = to N f yf ( x) f(x) = f(x) + y k(x,x) f y = S = S {( x, y,,)} else S = S {( x, y,,)} f S > B r = arg mn j S loss( f( x) yjk( xj, x)) f(x) = f(x) y r k(x r,x) + S = S {( xr, yr, cr, cr )} UpdateSummary(S, x r, c + r, c r ) else f y = UpdateSummary(S, x,, ) else UpdateSummary(S, x,, ) Subroutne UpdateSummary(S, x, c +, c ) k = arg mn j S xj x c + k = c + k + c + k (x, x k ) c k = c k + c k (x, x k ) Fg. The pseudo code for Tghtest attrbute values x, the orgnal label y, and counts c + and c that represent the total number of postve and negatve tranng examples observed n ts close neghborhood. As wll be descred n III.A, these counts are used to estmate the posteror class dstrbuton at x. We denote the set of support vectors augmented by the counts wth S. After ntalzng f(x) to zero and settng the augmented support set S to empty, examples from the tranng data are read sequentally. If the observed example s well classfed, the current model s retaned. Before dscardng the example, UpdateSummary subroutne s used to update count of ts nearest support vector. Instead of ncrementng the count by one, we use the soft ncrement that s a functon of kernel dstance. In ths way, larger weght s gven to the labels of tranng examples closest to the support vectors. If a tranng example s msclassfed, t s added to the current model, and S s updated accordngly. When the number of support vectors exceeds the budget B, S > B, the algorthm evaluates removal of each SV, selects the one whose removal ntroduces the least valdaton loss, and updates the model by removng t. Detals of the selecton are gven n III.A. Before dscardng the support vector, ts counts are used to update the counts of ts nearest support vector. A. Accuracy Estmaton Gven the support vector set S, the best support vector for removal s determned as the one wth ndex r = arg mn j S loss( f ( x) y jk( x j, x)), where loss s defned as the expected accuracy loss on the support vector set, + + loss ( f ( x)) = ( p l + p l ), () S S where p + = P(y = x ) and p = P(y = x ) are the posteror probabltes that x s labeled as postve and negatve, respectvely. Quantty l + (or l ) denotes the accuracy loss at x assumng ts class label s actually postve (or negatve). One choce of accuracy loss s the tradtonal loss defned as l + = f f(x ) <, and l + = otherwse. A slght problem wth loss s that t could not dstngush between large and small errors, whch can be mportant when valdaton data sze s small. The alternatve choce, mplemented n our algorthm, s to use the hnge loss defned as l + = max(, (+) f(x )) and l = max(, ( ) f(x )). We observe that, usng the ntroduced notaton, n Tghter algorthm p + = and p = f y = +, and p + = and p = f y =, and that loss s used for l + and l. The remanng ssue s estmatng value of p + (observe that p = p + ) based on counts c + and c mantaned by the algorthm. The maxmum lkelhood estmate p + = c + /(c + + c ) s unrelable when counts are small. Instead, we use the Bayesan approach where p + s treated as a random varable whose pror has Beta dstrbuton Beta(a +,a ), where a + and a are some postve values (typcally set to ). In ths case, the posteror dstrbuton of p + has Beta dstrbuton + Beta(c + a +, c + a ). Snce we are treatng p + and p as random varables, we need to modfy the accuracy loss n equaton () to loss ( f ( x)) = ( w l + ( w ) l ), (2) S S where w + s calculated as w = Beta( x c + a, c + a ) dx..5 B. Complexty The space requrement of the proposed Tghtest perceptron s constant n tranng sze and scales as O(B) wth the budget B, because only B support vectors are mantaned n the memory. Let us now consder the tme complexty. The predcton for the new comng example takes O(B) runtme. Wth some bookkeepng (predctons of the current perceptron on each support vector should be mantaned), the evaluaton of accuracy loss after removal of a sngle SV requres O(B) tme, and there are B+ such evaluatons. Fndng the nearest neghbor n UpdateSummary subroutne costs another O(B). Therefore, the total runtme for an update s O(B 2 ) and the total tranng tme for a data set of sze N s O(NB 2 ). 3299

4 IV. EXPERIMENTS In ths secton, we present results of detaled evaluaton of the proposed Tghtest perceptron on a number of benchmark datasets. A. Data sets Propertes of benchmark data sets for bnary classfcaton are summarzed n Table. The mult-class data sets were converted to two-class sets as follows. For the dgt datasets Pendgts and USPS we converted the orgnal -class problems to bnary by representng dgts, 2, 4, 5, 7 (non-round dgts) as negatve class and dgts 3, 6, 8, 9, (round dgts) as postve class. For Letter dataset, negatve class was created from the frst 3 letters of the alphabet and postve class from the remanng 3. The -class MNIST data set was smplfed to bnary data by separatng dgt 3 from dgt 8. Class n the 3-class Waveform was treated as negatve and the remanng two as postve. For Covertype TABLE DATA SET AND KERNEL PARAMETER SUMMARIES Data sets Tranng Testng Dm data the class 2 was treated as postve and the remanng 6 classes as negatve. Adult9, Banana, Gauss, and IJCNN were orgnally 2-class data sets. NCheckerboard data was generated as a unformly dstrbuted two-dmensonal 4 4 checkerboard wth alternatng class assgnments where class 2 δ Adult Banana NCheckerboard 5 2. Covertype 54 54/2 Gauss 5 2. IJCNN /2 Letter MNIST /2 Pendgts /2 USPS /2 Waveform TABLE 2 ACCURACY( % ) COMPARISON ON BENCHMARK DATA SETS Data sets (#SVs) Perceptron Tghter Full Tghter B Stoptron Forgetron Random Tghter Tghtest B= B=2(+N) B=2(+2) B=2 B=2 B=2 B=2 B=2 Adult9 (652) 78. ± ± ± ± ± ± ± ±.8 Banana (582) 84.7 ± ± ± ± ± ± ± ±.9 NCheckerb (389) 79.8 ± ± ± ± ± ± ± ± 2.2 Covertype (2856) 72.7 ± ± ± ± ± ± ± ±. Gauss (266) 72.6 ± ± ± ± ± ± ± ±.6 IJCNN (232) 96.2 ± ± ± ± ± ± ± ± 2.6 Letter (25) 95.9 ± ± ± ± ± ± ± ±.8 MNIST (525) 97.4 ± ± ± ± ± ± ± ± 3.5 Pendgts (248) 97.7 ± ± ± ± ± ± ± ± 4.8 USPS (527) 94.5 ±. 73. ± ± ± ± ± ± ±.9 Waveform (482) 86.2 ± ± ± ± ± ± ± ±.5 Average B= B=(+N) B=(+) B= B= B= B= B= Adult9 (652) 78. ± ± ± ± ± ± ± ±.6 Banana (582) 84.7 ± ± ± ± ± ± ± ±.8 NCheckerb (389) 79.8 ± ± ± ± ± ± ± ±.3 Covertype (2856) 72.7 ± ± ± ± ± ± ± ±. Gauss (266) 72.6 ± ± ± ± ± ± ± ±.8 IJCNN (232) 96.2 ± ± ± ± ± ± ± ±.4 Letter (25) 95.9 ± ± ± ± ± ± ± ±.7 MNIST (525) 97.4 ± ± ± ± ± ± ± ±.4 Pendgts (248) 97.7 ± ± ± ± ± ± ± ±.8 USPS (527) 94.5 ±. 85. ± ± ± ± ± ± ±.9 Waveform (482) 86.2 ± ± ± ±.7 8. ± ± ± ±.3 Average B= B=5(+N) B=5(+5) B=5 B=5 B=5 B=5 B=5 Adult9 (652) 78. ± ± ± ± ± ± ± ±.3 Banana (582) 84.7 ± ± ± ± ± ± ± ±. NCheckerb (389) 79.8 ± ± ± ± ± ± ± ±.8 Covertype (2856) 72.7 ± ± ± ± ± ± ± ±.6 Gauss (266) 72.6 ± ± ± ± ± ± ± ±.5 IJCNN (232) 96.2 ± ± ± ± ± ± ± ±.5 Letter (25) 95.9 ± ± ± ± ± ± ± ±.5 MNIST (525) 97.4 ± ± ± ± ± ± ± ±.3 Pendgts (248) 98.2 ± ± ± ± ± ± ± ±.6 USPS (527) 94.5 ± ± ± ± ± ± ± ±.5 Waveform (482) 86.2 ± ± ± ± ± ±. 82. ± ±.6 Average Values n parentheses n the data set column are # of SVs learned by Perceptron. Values n bold n Tghtest column ndcate the hghest accuracy among budget Perceptron algorthms. Values n talcs n Tghtest column ndcate the accuracy s even better than Perceptron. Values n parentheses n Tghter Full and Tghter B columns are the budget sze for the addtonal valdaton set. 33

5 (a) Perceptron soluton (b) Stoptron soluton (c) Forgetron soluton (d) Random soluton (e) Tghter soluton (f) Tghter B soluton length of data stream (g) Tghter Full soluton (h) Tghtest soluton () Computaton tme comparson computaton tme (n seconds) Tghtest Tghter Full (memory unbounded) Fg 2. Solutons of all algorthms on NChecherboard data assgnment was swtched for 5% of the randomly selected examples. For both testng sets, we used the nose-free verson as the test set. In ths way, the hghest reachable accuracy for N-Checkerboard was %. B. Evaluaton Procedure We compared the proposed Tghtest Perceptron algorthm wth four state of the art budget perceptron algorthms: Self-Tuned Forgetron [6], Random Perceptron [2], and Tghter and Tghter A Perceptrons [3], as well as to the baselne algorthm Stoptron where the kernel perceptron termnates once the budget s full. For Tghter A, we use A=B randomly selected examples as the addtonal valdaton set, and denote t as Tghter B. As a reference, we also present results from the orgnal Kernel Perceptron, and the budget unconstraned verson of Tghter Perceptron, Tghter Full [3] (names n talcs are used n Table 2 and Fgure 2). We evaluated three dfferent budgets B = 2,, 5, usng an RBF kernel defned as k(x,y) = exp( x y 2 /2δ 2 ), where δ s the RBF wdth. To keep thngs smple, for Adult9, USPS and Waveform we used the same kernel wdth as n prevous papers [, 3]. For 2-dmensonal data sets, a small kernel wdth of. was used and for all the remanng data sets the kernel wdth was set to δ 2 = M/2 [3], where M s the number of attrbutes. The summary of kernel wdths s shown n Table. Tranng examples were ordered randomly. 33

6 Attrbutes n all data sets were scaled to mean zero and standard devaton one. C. Results In ths secton we summarze performance results on all benchmark data sets. Each result (mean ± std) lsted n Table 2, comparng the alternatve kernel perceptron algorthms at three dfferent budgets, s an average and standard devaton of repeated experments. From Table 2 t can be seen that Tghtest sgnfcantly outperforms all competng budget perceptron algorthms on every data set and for all three budgets. The Tghtest s sgnfcantly more accurate than both Tghter and Tghter B that requre roughly twce larger memory. Ths result confrms that usng the posteror class probablty by the proposed method provdes hghly valuable nformaton for accuracy estmaton. It s worth notng that Tghtest s often better than even the memory unbounded Tghter Full. A part of the explanaton for such behavor s that Tghter Full uses the loss whle Tghtest uses the hnge loss that s more senstve to the errors far from the decson boundary. Therefore, t may be more sutable for removng outlyng nosy support vectors. Comparng Tghtest wth the memory unbounded Kernel Perceptron, we can observe that Tghtest s hghly compettve and sometmes even more accurate than Kernel Perceptron. As seen, the accuracy of Tghtest wth B=5 s better than Perceptron n 8 of data sets, wth a modest budget B= Tghtest s more accurate 5 tmes, and even wth a tny budget of B=2 Tghtest stll beats Perceptron on 3 of the nosest data sets. The success of Tghtest probably les n ts ablty to remove less useful or even harmful support vectors after consultng the accuracy after removal. Of the remanng results, t s nterestng to note that the two theoretcally well behaved algorthms Fogetron and Random had qute poor performance and t was comparable to Tghter. Ther accuracy was often below the smple baselne algorthm Stoptron. Ths behavor s partcularly notceable on the nosest data sets. D. Illustraton on 2D N-Checkerboard In Fgure 2 we llustrate the solutons of varous algorthms on NCheckerboard data. Budget B=5 was used for the budget Perceptron algorthms. In Fgure 2(a-h) magenta and cyan lnes are postve and negatve margns, respectvely. Black lne s the decson boundary, and red and green dots ndcate postve and negatve SVs, respectvely. It can be seen that the decson boundares created by Perceptron, Stoptron, Random, Forgetron and Tghter n Fgure 2(a-f) are not partcularly successful, makng t dffcult to dstngush the underlyng checkerboard. In contrast, Tghter Full and Tghtest solutons are qute successful and t s easy to dstngush the checkerboard pattern. Another nterestng observaton s that the support vectors n the Tghtest soluton le close to the decson boundary. In Fgure 2() the tme comparson between the two optmal soluton algorthms s llustrated. As seen, the memory bounded Tghtest runtme appears lnear whle the memory unbounded Tghter Full runtme appears quadratc, as expected. V. CONCLUSION In ths paper we presented the Tghtest Perceptron algorthm for onlne learnng on a budget. The algorthm acheves constant update runtme and constant space complexty wth the tranng data sze. Expermental results showed that Tghtest sgnfcantly outperforms state-of-the-art budget perceptron algorthms and s often superor to the memory unbounded kernel perceptron, despte usng a rather small budget. Ths hnts at the possblty of buldng accurate perceptron classfers from very large data streams whle operatng under a very lmted memory budgets. Furthermore, Tghtest results n very compact predctors and t drectly addresses a problem often observed n practce where the sze of the support vector set grows wth the tranng data sze. REFERENCES [] M. Azerman, E. Braverman, and L. 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