Simpler PAC-Bayesian Bounds for Hostile Data

Learning theory can be traced back to the late 60s and has attracted a great attention since. We refer to the monographs Devroye et al. (1996) and Vapnik (2000) for a survey. Most of the literature addresses the simplified case of i.i.d observations coupled with bounded loss functions. Many bounds on the excess risk holding with large probability were provided - these bounds are refered to as PAC learning bounds since Valiant (1984). In the late 90s, the PAC-Bayesian approach has been pioneered by Shawe-Taylor and Williamson (1997) and McAllester (1998, 1999). It consists in producing PAC bounds for a specific class of Bayesian-flavored estimators. Similarly to classical PAC results, most PAC-Bayesian bounds have been obtained with bounded loss functions (see Catoni, 2007, for some of the most accurate results). Note that Catoni (2004) provides bounds for unbouded loss, but still under very strong exponential moments assumptions. These assumptions were essentially not improved in the most recent works Guedj and Alquier (2013) and Bégin et al. (2016).