Comprehensive lower bounds on sequential prediction

We study the problem of sequential prediction of real-valued sequences under the squared error loss function. While refraining from any statistical and structural assumptions on the underlying sequence, we introduce a competitive approach to this problem and compare the performance of a sequential algorithm with respect to the large and continuous class of parametric predictors. We define the performance difference between a sequential algorithm and the best parametric predictor as regret, and introduce a guaranteed worst-case lower bounds to this relative performance measure. In particular, we prove that for any sequential algorithm, there always exists a sequence for which this regret is lower bounded by zero. We then extend this result by showing that the prediction problem can be transformed into a parameter estimation problem if the class of parametric predictors satisfy a certain property, and provide a comprehensive lower bound to this case.