  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.
