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56 COMPARING LEARNING ALGORITHMS
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Often we are interested in comparing the performance of two learning algorithms L A and L B , rather than two specific hypotheses What is an appropriate test for comparing learning algorithms, and how can we determine whether an observed difference between the algorithms is statistically significant Although there is active debate within the machine-learning research community regarding the best method for comparison, we present here one reasonable approach A discussion of alternative methods is given by Dietterich (1996) As usual, we begin by specifying the parameter we wish to estimate Suppose we wish to determine which of LA and LB is the better learning method on average for learning some particular target function f A reasonable way to define "on average" is to consider the relative performance of these two algorithms averaged over all the training sets of size n that might be drawn from the underlying instance distribution VIn other words, we wish to estimate the expected value
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of the difference in their errors
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where L(S) denotes the hypothesis output by learning method L when given the sample S of training data and where the subscript S c V indicates that the expected value is taken over samples S drawn according to the underlying instance distribution V The above expression describes the expected value of the difference in errors between learning methods L A and L B Of course in practice we have only a limited sample Do of data when comparing learning methods In such cases, one obvious approach to estimating the above quantity is to divide Do into a training set So and a disjoint test set To The training data can be used to train both LA and LB, and the test data can be used to compare the accuracy of the two learned hypotheses In other words, we measure the quantity
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Notice two key differences between this estimator and the quantity in Equation (514) First, we are using errorTo(h)to approximate errorv(h) Second, we are only measuring the difference in errors for one training set So rather than taking the expected value of this difference over all samples S that might be drawn ) from the distribution 2 One way to improve on the estimator given by Equation (515) is to repeatedly partition the data Do into disjoint training and test sets and to take the mean of the test set errors for these different experiments This leads to the procedure shown in Table 55 for estimating the difference between errors of two learning methods, based on a fixed sample Do of available data This procedure first partitions the data into k disjoint subsets of equal size, where this size is at least 30 It then trains and tests the learning algorithms k times, using each of the k subsets in turn as the test set, and using all remaining data as the training set In this way, the learning algorithms are tested on k independent test sets, 'and the mean difference in errors 8 is returned as an estimate of the difference between the two learning algorithms The quantity 8 returned by the procedure of Table 55 can be taken as an estimate of the desired quantity from Equation 514 More appropriately, we can view 8 as an estimate of the quantity
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ID01 drawn uniformly from Do where S represents a random sample of size The only difference between this expression and our original expression in Equation (514) is that this new expression takes the expected value over subsets of the available data Do, rather than over subsets drawn from the full instance dis) tribution 2
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1 Partition the available data Do into k disjoint subsets T I , T2, ,Tk of equal size, where this size is at least 30 2 For i from 1 to k, do use Ti for the test set, and the remaining data for training set Si 0 Si c {Do - Ti}
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hA C LA(Si) h~ + L ~ ( s i ) Si t errorq ( h A )- errorz ( h B )
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