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Pac-Bayes bounds are among the most accurate generalization bounds forclassifiers learned from independently and identically distributed (IID) data,and it is particularly so for margin classifiers: there have been recentcontributions showing how practical these bounds can be either to perform modelselection (Ambroladze et al., 2007) or even to directly guide the learning oflinear classifiers (Germain et al., 2009). However, there are many practicalsituations where the training data show some dependencies and where thetraditional IID assumption does not hold. Stating generalization bounds forsuch frameworks is therefore of the utmost interest, both from theoretical andpractical standpoints. In this work, we propose the first - to the best of ourknowledge - Pac-Bayes generalization bounds for classifiers trained on dataexhibiting interdependencies. The approach undertaken to establish our resultsis based on the decomposition of a so-called dependency graph that encodes thedependencies within the data, in sets of independent data, thanks to graphfractional covers. Our bounds are very general, since being able to find anupper bound on the fractional chromatic number of the dependency graph issufficient to get new Pac-Bayes bounds for specific settings. We show how ourresults can be used to derive bounds for ranking statistics (such as Auc) andclassifiers trained on data distributed according to a stationary {\ss}-mixingprocess. In the way, we show how our approach seemlessly allows us to deal withU-processes. As a side note, we also provide a Pac-Bayes generalization boundfor classifiers learned on data from stationary $\varphi$-mixing distributions.

Chromatic PAC-Bayes Bounds for Non-IID Data: Applications to Ranking and Stationary $\beta$-Mixing Processes