Efficient online algorithms for fast-rate regret bounds under sparsity
This work provides improved theoretical guarantees for sparse online learning, generalizing previous results with broader applicability to degenerate optimization scenarios.
The paper tackles online convex optimization problems by developing algorithms that achieve fast-rate regret bounds under sparsity conditions, establishing new quantile regret bounds in arbitrary sequences and adaptive risk bounds ranging from 1/√T to 1/T depending on convexity.
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 (Łojasiewicz's assumption) that allows multiple optima which is crucial when dealing with degenerate situations.