In machine learning, we try to use a collection of training data—for example, a large collection…

In machine learning, we try to use a collection of training data—for example, a large collection of himage, letteri pairs of images of handwritten letters and the English letter that they represent—to compute a predictor that will do well on predicting answers on a set of novel test data. One danger in such a system is overfitting: we might build a predictor that’s overly affected by idiosyncrasies of the training data. One way to address the risk of overfitting is a technique called cross-validation: we divide the training data into several subsets, and then, for each subset S, train our predictor based on ∼S and test it on S. We might then average the parameters of our predictor across the subsets S. In ten-fold cross-validation on a n-element training set, we would split our n training examples into disjoint sets S1, S2, . . . , S10 where |Si | = n 10 . How many ways are there to split an n-element set into disjoint subsets S1, S2, . . . , S10 of size n 10 each? (Note the order of the subsets themselves doesn’t matter, nor does the order of the elements within a subset.)


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