  We study various discrete nonlinear combinatorial optimization problems in an
online learning framework. In the first part, we address the question of
whether there are negative results showing that getting a vanishing (or even
vanishing approximate) regret is computational hard. We provide a general
reduction showing that many (min-max) polynomial time solvable problems not
only do not have a vanishing regret, but also no vanishing approximation
$\alpha$-regret, for some $\alpha$ (unless $NP=BPP$). Then, we focus on a
particular min-max problem, the min-max version of the vertex cover problem
which is solvable in polynomial time in the offline case. The previous
reduction proves that there is no $(2-\epsilon)$-regret online algorithm,
unless Unique Game is in $BPP$; we prove a matching upper bound providing an
online algorithm based on the online gradient descent method. Then, we turn our
attention to online learning algorithms that are based on an offline
optimization oracle that, given a set of instances of the problem, is able to
compute the optimum static solution. We show that for different nonlinear
discrete optimization problems, it is strongly $NP$-hard to solve the offline
optimization oracle, even for problems that can be solved in polynomial time in
the static case (e.g. min-max vertex cover, min-max perfect matching, etc.). On
the positive side, we present an online algorithm with vanishing regret that is
based on the follow the perturbed leader algorithm for a generalized knapsack
problem.
