Algorithmic luckiness

Classical statistical learning theory studies the generalisation performance of machine learning algorithms rather indirectly. One of the main detours is that algorithms are studied in terms of the hypothesis class that they draw their hypotheses from. In this paper, motivated by the luckiness framework of Shawe-Taylor et al. (1998), we study learning algorithms more directly and in a way that allows us to exploit the serendipity of the training sample. The main difference to previous approaches lies in the complexity measure; rather than covering all hypotheses in a given hypothesis space it is only necessary to cover the functions which could have been learned using the fixed learning algorithm. We show how the resulting framework relates to the VC, luckiness and compression frameworks. Finally, we present an application of this framework to the maximum margin algorithm for linear classifiers which results in a bound that exploits the margin, the sparsity of the resultant weight vector, and the degree of clustering of the training data in feature space.