  This work addresses the classic machine learning problem of online prediction
with expert advice. We consider the finite-horizon version of this zero-sum,
two-person game. Using verification arguments from optimal control theory, we
view the task of finding better lower and upper bounds on the value of the game
(regret) as the problem of finding better sub- and supersolutions of certain
partial differential equations (PDEs). These sub- and supersolutions serve as
the potentials for player and adversary strategies, which lead to the
corresponding bounds. To get explicit bounds, we use closed-form solutions of
specific PDEs. Our bounds hold for any given number of experts and horizon; in
certain regimes (which we identify) they improve upon the previous state of the
art. For two and three experts, our bounds provide the optimal leading order
term.
