Logarithmic Regret Algorithms for Strongly Convex Repeated Games

Many problems arising in machine learning can be cast as a convex optimization problem, in which a sum of a loss term and a regularization term is minimized. For example, in Support Vector Machines the loss term is the average hinge-loss of a vector over a training set of examples and the regularization term is the squared Euclidean norm of this vector. In this paper we study an algorithmic framework for strongly convex repeated games and apply it for solving regularized loss minimization problems. In a convex repeated game, a predictor chooses a sequence of vectors from a convex set. After each vector is chosen, the opponent responds with a convex loss function and the predictor pays for applying the loss function to the vector she chose. The regret of the predictor is the difference between her cumulative loss and the minimal cumulative loss achievable by a fixed vector, even one that is chosen in hindsight. In strongly convex repeated games, the opponent is forced to choose loss functions that are strongly convex. We describe a family of prediction algorithms for strongly convex repeated games that attain logarithmic regret.