Online Learning: Stochastic and Constrained Adversaries
  Learning theory has largely focused on two main learning scenarios. The first is the classical statistical setting where instances are drawn i.i.d. from a fixed distribution and the second scenario is the online learning, completely adversarial scenario where adversary at every time step picks the worst instance to provide the learner with. It can be argued that in the real world neither of these assumptions are reasonable. It is therefore important to study problems with a range of assumptions on data. Unfortunately, theoretical results in this area are scarce, possibly due to absence of general tools for analysis. Focusing on the regret formulation, we define the minimax value of a game where the adversary is restricted in his moves. The framework captures stochastic and non-stochastic assumptions on data. Building on the sequential symmetrization approach, we define a notion of distribution-dependent Rademacher complexity for the spectrum of problems ranging from i.i.d. to worst-case. The bounds let us immediately deduce variation-type bounds. We then consider the i.i.d. adversary and show equivalence of online and batch learnability. In the supervised setting, we consider various hybrid assumptions on the way that x and y variables are chosen. Finally, we consider smoothed learning problems and show that half-spaces are online learnable in the smoothed model. In fact, exponentially small noise added to adversary's decisions turns this problem with infinite Littlestone's dimension into a learnable problem. 
Alexander Rakhlin, Karthik Sridharan, Ambuj Tewari
stat.ML cs.GT cs.LG