We study the fundamental problem of prediction with expert advice and develop
regret lower bounds for a large family of algorithms for this problem. We
develop simple adversarial primitives, that lend themselves to various
combinations leading to sharp lower bounds for many algorithmic families. We
use these primitives to show that the classic Multiplicative Weights Algorithm
(MWA) has a regret of 2Tlnk, there by completely closing
the gap between upper and lower bounds. We further show a regret lower bound of
322Tlnk for a much more general family of
algorithms than MWA, where the learning rate can be arbitrarily varied over
time, or even picked from arbitrary distributions over time. We also use our
primitives to construct adversaries in the geometric horizon setting for MWA to
precisely characterize the regret at δ0.391 for the case
of 2 experts and a lower bound of 212δlnk
for the case of arbitrary number of experts k