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We develop a new tool for data-dependent analysis of theexploration-exploitation trade-off in learning under limited feedback. Our toolis based on two main ingredients. The first ingredient is a new concentrationinequality that makes it possible to control the concentration of weightedaverages of multiple (possibly uncountably many) simultaneously evolving andinterdependent martingales. The second ingredient is an application of thisinequality to the exploration-exploitation trade-off via importance weightedsampling. We apply the new tool to the stochastic multiarmed bandit problem,however, the main importance of this paper is the development and understandingof the new tool rather than improvement of existing algorithms for stochasticmultiarmed bandits. In the follow-up work we demonstrate that the new tool canimprove over state-of-the-art in structurally richer problems, such asstochastic multiarmed bandits with side information (Seldin et al., 2011a).

PAC-Bayes-Bernstein Inequality for Martingales and its Application to Multiarmed Bandits