Efficient Online Linear Optimization with Approximation Algorithms.

We revisit the problem of online linear optimization in case the set of feasible actions is accessible through an approximated linear optimization oracle with a factor alpha multiplicative approximation guarantee. This setting is in particular interesting since it captures natural online extensions of well-studied offline linear optimization problems which are NP-hard, yet admit efficient approximation algorithms. The goal here is to minimize the alpha-regret which is the natural extension of the standard regret in online learning to this setting. We present new algorithms with significantly improved oracle complexity for both the full information and bandit variants of the problem. Mainly, for both variants, we present alpha-regret bounds of O(T-1/3), were T is the number of prediction rounds, using only O(log(T)) calls to the approximation oracle per iteration, on average. These are the first results to obtain both average oracle complexity of O(log(T)) (or even poly-logarithmic in T) and alpha-regret bound O(T-c) for a constant c > 0, for both variants.