  We propose a general framework for studying adaptive regret bounds in the
online learning framework, including model selection bounds and data-dependent
bounds. Given a data- or model-dependent bound we ask, "Does there exist some
algorithm achieving this bound?" We show that modifications to recently
introduced sequential complexity measures can be used to answer this question
by providing sufficient conditions under which adaptive rates can be achieved.
In particular each adaptive rate induces a set of so-called offset complexity
measures, and obtaining small upper bounds on these quantities is sufficient to
demonstrate achievability. A cornerstone of our analysis technique is the use
of one-sided tail inequalities to bound suprema of offset random processes.
  Our framework recovers and improves a wide variety of adaptive bounds
including quantile bounds, second-order data-dependent bounds, and small loss
bounds. In addition we derive a new type of adaptive bound for online linear
optimization based on the spectral norm, as well as a new online PAC-Bayes
theorem that holds for countably infinite sets.
