  We consider the classical problem of prediction with expert advice. In the
fixed-time setting, where the time horizon is known in advance, algorithms that
achieve the optimal regret are known when there are two, three, or four experts
or when the number of experts is large. Much less is known about the problem in
the anytime setting, where the time horizon is not known in advance. No minimax
optimal algorithm was previously known in the anytime setting, regardless of
the number of experts. Even for the case of two experts, Luo and Schapire have
left open the problem of determining the optimal algorithm.
  We design the first minimax optimal algorithm for minimizing regret in the
anytime setting. We consider the case of two experts, and prove that the
optimal regret is $\gamma \sqrt{t} / 2$ at all time steps $t$, where $\gamma$
is a natural constant that arose 35 years ago in studying fundamental
properties of Brownian motion. The algorithm is designed by considering a
continuous analogue of the regret problem, which is solved using ideas from
stochastic calculus.
