  Consider the classical problem of predicting the next bit in a sequence of
bits. A standard performance measure is {\em regret} (loss in payoff) with
respect to a set of experts. For example if we measure performance with respect
to two constant experts one that always predicts 0's and another that always
predicts 1's it is well known that one can get regret $O(\sqrt T)$ with respect
to the best expert by using, say, the weighted majority algorithm. But this
algorithm does not provide performance guarantee in any interval. There are
other algorithms that ensure regret $O(\sqrt {x \log T})$ in any interval of
length $x$. In this paper we show a randomized algorithm that in an amortized
sense gets a regret of $O(\sqrt x)$ for any interval when the sequence is
partitioned into intervals arbitrarily. We empirically estimated the constant
in the $O()$ for $T$ upto 2000 and found it to be small -- around 2.1. We also
experimentally evaluate the efficacy of this algorithm in predicting high
frequency stock data.
