Playing non-linear games with linear oracles

Linear optimization is many times algorithmically simpler than non-linear convex optimization. Linear optimization over matroid polytopes, matching polytopes and path polytopes are example of problems for which we have efficient combinatorial algorithms, but whose non-linear convex counterpart is harder and admit significantly less efficient algorithms. This motivates the computational model of online decision making and optimization using a linear optimization oracle. In this computational model we give the first efficient decision making algorithm with optimal regret guarantees, answering an open question of [1], [2], in case the decision set is a polytope. We also give an extension of the algorithm for the partial information setting, i.e. the "bandit" model. Our method is based on a novel variant of the conditional gradient method, or Frank-Wolfe algorithm, that reduces the task of minimizing a smooth convex function over a domain to that of minimizing a linear objective. Whereas previous variants of this method give rise to approximation algorithms, we give such algorithm that converges exponentially faster and thus runs in polynomial-time for a large class of convex optimization problems over polyhedral sets, a result of independent interest.