  This work addresses the classic machine learning problem of online prediction
with expert advice. A new potential-based framework for the fixed horizon
version of this problem has been recently developed using verification
arguments from optimal control theory. This paper extends this framework to the
random (geometric) stopping version. To obtain explicit bounds, we construct
potentials for the geometric version from potentials used for the fixed horizon
version of the problem. This construction leads to new explicit lower and upper
bounds associated with specific adversary and player strategies. While there
are several known lower bounds in the fixed horizon setting, our lower bounds
appear to be the first such results in the geometric stopping setting with an
arbitrary number of experts. Our framework also leads in some cases to improved
upper bounds. For two and three experts, our bounds are optimal to leading
order.
