Patent Application: US-93112907-A

Abstract:
most recent research of scalable inductive learning on very large streaming dataset focuses on eliminating memory constraints and reducing the number of sequential data scans . however , state - of - the - art algorithms still require multiple scans over the data set and use sophisticated control mechanisms and data structures . there is discussed herein a general inductive learning framework that scans the dataset exactly once . then , there is proposed an extension based on hoeffding &# 39 ; s inequality that scans the dataset less than once . the proposed frameworks are applicable to a wide range of inductive learners .

Description:
we first describe a strawman algorithm that scans the data set exactly once , then propose the extension that scans the data set less than once . the strawman algorithm is based on probabilistic modeling . suppose p ( l i | x ) is the probability that x is an instance of class l i . in addition , we have a benefit matrix b [ l i , l i ] that records the benefit received by predicting an example of class l i ′ to be an instance of class l i . for traditional accuracy - based problems , ∀ i , b [ i , l i ]= 1 and { i ′≠ i , b [ l i , l i ]= 0 . for cost - sensitive application such as credit card fraud detection , assume that the overhead to investigate a fraud is $ 90 and y ( x ) is the transaction amount , then b [ fraud , fraud ]= y ( x )−$ 90 and b [ fraud , fraud ]=−$ 90 . using benefit matrix and probability , the expected benefit received by predicting x to be an instance of class l i is expected ⁢ ⁢ benefit ⁢ : ⁢ ⁢ e ⁡ ( ℓ i | x ) = ∑ ℓ i ′ ⁢ b ⁡ [ ℓ i ′ , ℓ i ] · p ⁡ ( ℓ i ′ | x ) ( 1 ) based on optimal decision policy , the best decision is the label with the highest expected benefit : assuming that l ( x ) is the true label of x , the accuracy of the decision tree on a test data set st is accuracy ⁢ : ⁢ ⁢ a = ∑ x ∈ st ⁢ b ⁡ [ ℓ ⁡ ( x ) , l max ] ( 3 ) for traditional accuracy - based problems , a is always normalized by dividing | st |; for cost - sensitive problems , a is usually represented in some measure of benefits such as dollar amount . for cost - sensitive problems , we sometimes use “ total benefits ” to mean accuracy . the strawman algorithm is based on averaging ensemble [ 3 ]. assume that a data set s is partitioned into k disjoint subsets s j with equal size . a base level model c j is trained from each s j . given an example x , each classifier outputs individual expected benefit based on probability p j ( l i ′ | x ) e j ⁡ ( ℓ i | x ) = ∑ ℓ i ′ ⁢ b ⁡ [ ℓ i ′ , ℓ i ] · p j ⁡ ( ℓ i ′ | x ) ( 4 ) e k ⁡ ( ℓ i | x ) = ∑ 1 k ⁢ e j ⁡ ( ℓ i | x ) k ( 5 ) we then predict the class label with the highest expected return as in eq [ 2 ]. optimal decision : l k = arg max l i e k ( l i | x ) ( 6 ) a clear advantage is that the strawman algorithm scans the dataset exactly once as compared to two scans by meta - learning and multiple scans by bagging and boosting . in previous research [ 3 ], the accuracy by the strawman algorithm is also significantly higher than both meta - learning and bagging . [ 3 ] explains the statistical reason why the averaging ensemble is also more likely to have higher accuracy that a single classifier trained from the same dataset . a “ less - than - one - scan ” algorithm , in accordance with at least one presently preferred embodiment of the present invention , returns the current ensemble with k (& lt ; k ) number of classifiers when the accuracy of current ensemble is the same as the complete ensemble with high confidence . for a random variable y in the range of r = a − b with observed mean of y after n observations , without any assumption about the distribution of y , hoeffding &# 39 ; s inequality states that with probability ≧ p , the error of y to the true mean is at most ɛ n = r ⁡ ( 1 - f 2 ⁢ n ⁢ ln ⁡ ( 1 1 - p ) ) 1 2 ( 7 ) ɛ n = r ⁡ ( 1 - f 2 ⁢ n ⁢ ln ⁡ ( 1 1 - p ) ) 1 2 ⁢ ⁢ where ⁢ ⁢ f = n n ( 8 ) the range r of expected benefit for class label l i can be found from the index to the data , or predefined . when k base models are constructed , the hoeffding error ε k can be computed by using eq [ 8 ]. for data example x , assume that e ( l a | x ) is the highest expected benefit and e ( l b | x ) is the second highest , ε k ( l a ) and ε k ( l b ) are the hoeffding errors . if e ( l a | x )− ε k ( l a )& gt ; e ( l b | x )+ ε k ( l b ) or e ( l a | x )− e ( l b | x )& gt ; ε k ( l a )+ ε k ( l b ), with confidence ≧ p , the prediction on x by the complete multiple model and the current multiple model is the same . otherwise , more base models will be trained . the algorithm is summarized in algorithm 1 ( all algorithms appear in the appendix hereto ). if an example x satisfies the confidence p when k classifiers are computed , there is no utility to check its satisfaction when more classifiers are computed . this is because that an ensemble with more classifiers is likely to be a more accurate model . in practice , we can only read and keep one example x from the validation set in memory at one time . we only read a new instance from the validation set if the current set of classifiers satisfy the confidence test . in addition , we keep only the predictions on one example at any given time . this guarantees that the algorithm scans the validation dataset once with nearly no memory requirement . the extra overhead of the hoeffding - based less than one scan algorithm is the cost for the base classifiers to predict on the validation set and calculate the statistics . all these can be done in main memory . as discussed above , we can predict on one example from the validation set at any given time . assume that we have k classifiers at the end and n is the size of the validation set , the total number of predictions is approximately on average . the calculation of both averaging and standard deviation can be done incrementally . we only need to keep σx i and σx i 2 for just one example at anytime and calculate as follows : x _ = ∑ x i k ( 9 ) σ 2 ⁡ ( x ) = ∑ x i 2 - k · x _ 2 k - 1 ( 10 ) the problem that the proposed algorithm solves is one in which the training set is very large and the i / o cost of data scan is the major overhead . when i / o cost is the bottle neck , the extra cost of prediction and statistical analysis is minimum . to illustrate the effectiveness of at least one embodiment of the present invention by way of experimentation , we first compare the accuracy of the complete multiple model ( one scan as well as less than one scan ) and the accuracy of the single model trained from the same data set . we then evaluate the amount of data scan and accuracy of the less than one scan algorithm as compared to the one scan models . additionally , we generate a dataset with biased distribution and study the results of the less than one scan algorithm . the first one is the famous donation data set that first appeared in kddcup &# 39 ; 98 competition ( the 1998 knowledge discovery and data mining cup competition ). suppose that the cost of requesting a charitable donation from an individual x is $ 0 . 68 , and the best estimate of the amount that x will donate is y ( x ) y ( x ). its benefit matrix is : as a cost - sensitive problem , the total benefit is the total amount of received charity minus the cost of mailing . the data has already been divided into a training set and a test set . the training set consists of 95 , 412 records for which it is known whether or not the person made a donation and how much the donation was . the test set contains 96 , 367 records for which similar donation information was not published we used the standard training / test set splits to compare with previous results . the feature subsets were based on the kddcup &# 39 ; 98 winning submission . to estimate the donation amount , we employed the multiple linear regression method . the second data set is a credit card fraud detection problem . assuming that there is an overhead $ 90 to dispute and investigate a fraud and y ( x ) is the transaction amount , the following is the benefit matrix : as a cost - sensitive problem , the total benefit is the sum of recovered frauds minus investigation costs . the data set was sampled from a one year period and contains a total of 5 million transaction records . we use data of the last month as test data ( 40 , 038 examples ) and data of previous months as training data ( 406 , 009 examples ). the third data set is the adult data set from uci repository . for cost - sensitive studies , we artificially associate a benefit of $ 2 to class label f and a benefit of $ 1 to class label n , as summarized below : we use the natural split of training and test sets , so the results can be easily duplicated . the training set contains 32 , 561 entries and the test set contains 16 , 281 records . by way of experimental setup , there were selected three learning algorithms , decision tree learner c4 . 5 , rule builder ripper , and a naive bayes learner . ( these three algorithms are described in detail in the following publications , respectively : quinlan , r ., “ c4 . 5 : programs for machine learning ”, morgan kaufman , 1993 ; cohen , w ., “ a fast rule induction algorithm ”, proceeedings of 1995 international conferences on machine learning ; and mitchell , t ., “ machine learning ”, mcgraw hill , 1997 .) we have chosen a wide range of partitions , k ∈{ 8 , 16 , 32 , 64 , 128 , 256 }. the validation dataset sv is the complete training set . all reported accuracy results were run on the test dataset . in tables 1 and 2 ( all tables appear in the appendix hereto ), we compare the results of the single classifier ( which is trained from the complete dataset as a whole ), one scan algorithm , and the less than one scan algorithm . we use the original “ natural order ” of the dataset . later on , we use a biased distribution . each data set under study is treated both as a traditional and cost - sensitive problem . the less than one scan algorithm is run with confidence p = 99 . 7 %. the baseline traditional accuracy and total benefits of the single model are shown in the two columns under “ single ” in tables 1 and 2 . these results are the baseline that the one scan and less than one scan algorithms should achieve . for the one scan and less than one scan algorithm , each reported result is the average of different multiple models with k ranging from 2 to 256 . in tables 1 and 2 , the results are shown in two columns under accuracy and benefit as we compare the respective results in tables 1 and 2 , the multiple model either significantly beat the accuracy of the single model or have very similar results . the most significant increase in both accuracy and total benefits is for the credit card data set . the total benefits have been increased by approximately $ 7 , 000 ˜$ 10 , 000 ; the accuracy has been increased by approximately 1 %˜ 3 %. for the kddcup &# 39 ; 98 donation data set , the total benefit has been increased by $ 1400 for c4 . 5 and $ 250 for nb . we next study the trends of accuracy when the number of partitions k increases . in fig1 a , 1b and 1 c , we plot the accuracy and total benefits for the credit card data sets , and the total benefits for the donation data set with increasing number of partitions k . c4 . 5 was the base learner for this study . as we can see clearly that for the credit card data set , the multiple model consistently and significantly improve both the accuracy and total benefits over the single model by at least 1 % in accuracy and $ 40000 in total benefits for all choices of k . for the donation data set , the multiple model boosts the total benefits by at least $ 1400 . nonetheless , when k increases , both the accuracy and total benefits show a slow decreasing trend . it would be expected that when k is extremely large , the results will eventually fall below the baseline . another important observation is that the accuracy and total benefit of the less than one scan algorithm are very close to the one scan algorithm . their results are nearly identical . in both tables 1 and 2 , we show the amount of data scanned for the less than one scan algorithm . it ranges from 40 % ( 0 . 4 ) to about 70 % ( 0 . 7 ). the adult dataset has the most amount of data scanned since the training set is the smallest and it requires more data partitions to compute an accurate model . c4 . 5 scans more data than both ripper and nb . this is because we generate the completely unpruned tree for c4 . 5 , and there are wide variations among different models . in table 3 , we compare the differences in accuracy and amount of training data when the validation set is either read completely by every classifier ( under “ batch ”) or sequentially only by newly computed base classifiers ( under “ seq ”) ( as discussed in section 3 ). our empirical studies have found that “ batch ” mode usually scans approximately 1 % to 2 % more training data , and the models computed by both methods are nearly identical in accuracy . the extra training data from the “ batch ” method is due to the fact that some examples satisfied by previously learned classifiers have high probability , but may not necessarily be satisfied by more base classifiers . however , our empirical studies have shown that the difference in how the validation set is handled doesn &# 39 ; t significantly influence the final model accuracy . when a data is biased in its distribution , the less than one scan algorithm needs to scan more data than in uniform distribution to produce an accurate model . with the same amount of datascan , it may not have the same accuracy as uniform distribution . we have created a “ trap ” using the credit card dataset . we sorted the training data with increasing transaction amount . the detailed results are shown in table 4 ( a ) and ( b ). the accuracy ( and total benefits ) in table 4 ( a ) are nearly identical to the results of “ natural distribution ” as reported in tables 1 and 2 . however , the amount of datascan by the less than one scan algorithm is over 0 . 9 as compared to approximately 0 . 6 for natural distribution . as shown in table 4 ( b ), when the datascan is less than 0 . 9 ( the confidence is not satisfied and less one scan will continue to compute more model ), the total benefits are much lower . when distribution is biased , the variations in base classifiers &# 39 ; prediction are wider . it requires more data to compute an accurate model and the less than one scan algorithm is performing in the correct way . in connection with training efficiency , we recorded both the training time of the batch mode single model , and the training time of both the one scan algorithm and less than one scan algorithm plus the time to classify the validation set multiple times and statistical estimation . we than computed serial improvement , which is the ratio that the one scan and less than one scan algorithm are faster than training the single model . in fig2 , we plot results for the credit card dataset using c4 . 5 . our training data can fit into the main memory of the machine . any single classifier algorithm that reduces the number of data scan [ 9 , 5 , 6 , 8 ] will not have training time less than this result . as shown in fig2 , both one scan and less than one scan algorithm are significantly faster than the single classifier , and the less than one scan algorithm is faster than the one scan algorithm . one of the biggest suspicions people have towards multiple models or ensembles is accuracy . previous experiments have shown than ensemble of classifiers is less accurate than a single model . there is no explanation of why an ensemble works and when it will fail . however , statistical reasons using the “ smoothing effect ” can be pointed to . in addition , the accuracy estimation by the random - distribution method also predicts when the method will fail . even none of the existing single model methods can predict when it will fail . by way of recapitulation , there are proposed herein , in accordance with at least one presently preferred embodiment of the present invention , two scalable inductive learning algorithms . the strawman multiple model algorithm scans the data set exactly once . there is then proposed a less than one scan extension based on hoeffding &# 39 ; s inequality . it returns a partial multiple model when its accuracy is the same as the complete multiple model with confidence ≧ p . since the hoeffding inequality makes no assumption about the data distribution , the advantage of this method is that the data items can be retrieved sequentially . there has also been discussed herein a manner of sequentially reading the validation set exactly once using minimal memory . we have evaluated these methods on several data sets as both traditional accuracy - based and cost - sensitive problems using decision tree , rule and naive bayes learners . we have found that the accuracy of all our methods are the same or far higher than the single model . the amount of data scan by the less than one scan algorithms range from 0 . 45 to 0 . 7 for the original natural distribution of data . for a significantly biased dataset , the amount of datascan by the less than one scan algorithm is over 0 . 9 . it needs extra data to resolve the bias in data distribution in order to compute an accurate model . there were also empirically measured herein the efficiency of both one scan and less than one scan algorithms . in addition , our empirical studies have shown that both methods are significantly faster than computing a single model even when the training data can be held in main memory , and the less than one scan algorithm is faster than the one scan algorithm . the best known scalable decision tree algorithm scans the data set twice . our algorithms can be applied to many inductive learners , including decision trees . it is to be understood that the present invention , in accordance with at least one presently preferred embodiment , includes an arrangement for scanning at least a portion of an input large data set , which may be implemented on at least one general - purpose computer running suitable software programs . these may also be implemented on at least one integrated circuit or part of at least one integrated circuit . thus , it is to be understood that the invention may be implemented in hardware , software , or a combination of both . if not otherwise stated herein , it is to be assumed that all patents , patent applications , patent publications and other publications ( including web - based publications ) mentioned and cited herein are hereby fully incorporated by reference herein as if set forth in their entirety herein . although illustrative embodiments of the present invention have been described herein with reference to the accompanying drawings , it is to be understood that the invention is not limited to those precise embodiments , and that various other changes and modifications may be affected therein by one skilled in the art without departing from the scope or spirit of the invention . 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[ 9 ] j shafer , ramesh agrawl , and m mehta . sprint : a scalable parallel classifier for data mining . in proceedings of twenty - second international conference on very large databases ( vldb - 96 ), pages 544 - 555 , san francisco , calif ., 1996 . morgan kaufmann . | partition s into k disjoint subsets of equal size { s 1 , . . . , s k }; | | ∀ l i , compute hoeffding error ε k ( l i ) ( eq [ 8 ]); | | | e ( l a | x ) is the highest and e ( l b | x ) is the second highest ; | | | if e ( l a | x ) + e ( l b | x ) ≦ ε k ( l a ) + ε k ( l b ) then