Abstract
In game-theoretic formulations of prediction problems, a strategy makes a decision, observes an outcome and pays a loss. The aim is to minimize the regret, which is the amount by which the total loss incurred exceeds the total loss of the best decision in hindsight. This talk will focus on the minimax optimal strategy, which minimizes the regret, in three settings: prediction with log loss (a formulation of sequential probability density estimation that is closely related to sequential compression, coding, gambling and investment problems), sequential least squares (where decisions and outcomes lie in a subset of a Hilbert space, and loss is squared distance), and fixed-design linear regression (where the aim is to predict real-valued labels as well as the best linear function). We obtain the minimax optimal strategy for these problems under appropriate conditions, and show that it can be efficiently computed.