A pac-bayes risk bound for general loss functions

We provide a PAC-Bayesian bound for the expected loss of convex combinations of classifiers under a wide class of loss functions (which includes the exponential loss and the logistic loss). Our numerical experiments with Adaboost indicate that the proposed upper bound, computed on the training set, behaves very similarly as the true loss estimated on the testing set. 1 Intoduction The PAC-Bayes approach [1, 2, 3, 4, 5] has been very effective at providing tight risk bounds for large-margin classifiers such as the SVM [4, 6]. Within this approach, we consider a prior distribu-tion P over a space of classifiers that characterizes our prior belief about good classifiers (before the observation of the data) and a posterior distribution Q (over the same space of classifiers) that takes into account the additional information provided by the training data. A remarkable result that came out from this line of research, known as the “PAC-Bayes theorem”, provides a tight upper bound on the risk of a stochastic classifier (defined on the posterior Q) called the Gibbs classifier. In the context of binary classification, the Q-weighted majority vote classifier (related to this stochasti