  We consider the online convex optimization problem. In the setting of
arbitrary sequences and finite set of parameters, we establish a new fast-rate
quantile regret bound. Then we investigate the optimization into the L1-ball by
discretizing the parameter space. Our algorithm is projection free and we
propose an efficient solution by restarting the algorithm on adaptive
discretization grids. In the adversarial setting, we develop an algorithm that
achieves several rates of convergence with different dependencies on the
sparsity of the objective. In the i.i.d. setting, we establish new risk bounds
that are adaptive to the sparsity of the problem and to the regularity of the
risk (ranging from a rate 1 / $\sqrt T$ for general convex risk to 1 /T for
strongly convex risk). These results generalize previous works on sparse online
learning. They are obtained under a weak assumption on the risk
({\L}ojasiewicz's assumption) that allows multiple optima which is crucial when
dealing with degenerate situations.
