Minimax Time Series Prediction

We consider an adversarial formulation of the problem ofpredicting a time series with square loss. The aim is to predictan arbitrary sequence of vectors almost as well as the bestsmooth comparator sequence in retrospect. Our approach allowsnatural measures of smoothness such as the squared norm ofincrements. More generally, we consider a linear time seriesmodel and penalize the comparator sequence through the energy ofthe implied driving noise terms. We derive the minimax strategyfor all problems of this type and show that it can be implementedefficiently.