This thesis takes place within the machine learning theory. In particular it focuses on three sub-domains, stochastic optimization, online learning and clustering. These subjects exist for decades, but all have been recently studied under a new perspective. For instance, bandits games now offer a unified framework for stochastic optimization and online learning. This point of view results in many new extensions of the basic game. In the first part of this thesis, we focus on the mathematical study of these extensions (as well as the classical game). On the other hand, in the second part we discuss two important theoretical concepts for clustering, namely the consistency of algorithms and the stability as a tool for model selection.
