  In the framework of prediction of individual sequences, sequential prediction
methods are to be constructed that perform nearly as well as the best expert
from a given class. We consider prediction strategies that compete with the
class of switching strategies that can segment a given sequence into several
blocks, and follow the advice of a different "base" expert in each block. As
usual, the performance of the algorithm is measured by the regret defined as
the excess loss relative to the best switching strategy selected in hindsight
for the particular sequence to be predicted. In this paper we construct
prediction strategies of low computational cost for the case where the set of
base experts is large. In particular we provide a method that can transform any
prediction algorithm $\A$ that is designed for the base class into a tracking
algorithm. The resulting tracking algorithm can take advantage of the
prediction performance and potential computational efficiency of $\A$ in the
sense that it can be implemented with time and space complexity only
$O(n^{\gamma} \ln n)$ times larger than that of $\A$, where $n$ is the time
horizon and $\gamma \ge 0$ is a parameter of the algorithm. With $\A$ properly
chosen, our algorithm achieves a regret bound of optimal order for $\gamma>0$,
and only $O(\ln n)$ times larger than the optimal order for $\gamma=0$ for all
typical regret bound types we examined. For example, for predicting binary
sequences with switching parameters under the logarithmic loss, our method
achieves the optimal $O(\ln n)$ regret rate with time complexity
$O(n^{1+\gamma}\ln n)$ for any $\gamma\in (0,1)$.
