Online Convex Optimization

A convex repeated game is a two players game that is performed in a sequence of consecutive rounds. On round t of the repeated game, the first player chooses a vector wt from a convex set A. Next, the second player responds with a convex function gt : A → R. Finally, the first player suffers an instantaneous loss gt(wt). We study the game from the viewpoint of the first player. In offline convex optimization, the goal is to find a vector w within a convex set A that minimizes a convex objective function, g : A→ R. In online convex optimization, the set A is known in advance, but the objective function may change along the online process. The goal of the online optimizer, which we call the learner, is to minimize the averaged objective value 1 T ∑T t=1 gt(wt), where T is the total number of rounds.