  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 $\ell^1$-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 $\ell^1$-ball, the input data and the
observations. The minimax regret is shown to exhibit a regime transition around
the point $d = \sqrt{T} U X / (2 Y)$. Furthermore, we present efficient
algorithms that are adaptive, \ie, that do not require the knowledge of $U$,
$X$, $Y$, and $T$, but still achieve nearly optimal regret bounds.
