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 Kalai and Vempala, Hazan and Kale, 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.