PAC-Bayes risk bounds for stochastic averages and majority votes of sample-compressed classifiers

We propose a PAC-Bayes theorem for the sample-compression setting where each classifier is described by a compression subset of the training data and a message string of additional information. This setting, which is the appropriate one to describe many learning algorithms, strictly generalizes the usual data-independent setting where classifiers are represented only by data-independent message strings (or parameters taken from a continuous set). The proposed PAC-Bayes theorem for the sample-compression setting reduces to the PAC-Bayes theorem of Seeger (2002) and Langford (2005) when the compression subset of each classifier vanishes. For posteriors having all their weights on a single sample-compressed classifier, the general risk bound reduces to a bound similar to the tight sample-compression bound proposed in Laviolette et al. (2005). Finally, we extend our results to the case where each sample-compressed classifier of a data-dependent ensemble may abstain of predicting a class label.