Adaptive and optimal online linear regression on l1-balls

We consider the problem of online linear regression on individual sequences. The goal in this paper is for the forecaster to output sequential predictions which are, after T time rounds, almost as good as the ones output by the best linear predictor in a given l(1)-ball in R-d. We consider both the cases where the dimension d is small and large relative to the time horizon T. We first present regret bounds with optimal dependencies on d. T, and on the sizes U, X and Y of the l(1)-ball, the input data and the observations. The minimax regret is shown to exhibit a regime transition around the point d = root TUX/(2Y). Furthermore, we present efficient algorithms that are adaptive, i.e., that do not require the knowledge of U, X, Y, and T, but still achieve nearly optimal regret bounds. (C) 2013 Elsevier B.V. All rights reserved.