  The on-line shortest path problem is considered under various models of
partial monitoring. Given a weighted directed acyclic graph whose edge weights
can change in an arbitrary (adversarial) way, a decision maker has to choose in
each round of a game a path between two distinguished vertices such that the
loss of the chosen path (defined as the sum of the weights of its composing
edges) be as small as possible. In a setting generalizing the multi-armed
bandit problem, after choosing a path, the decision maker learns only the
weights of those edges that belong to the chosen path. For this problem, an
algorithm is given whose average cumulative loss in n rounds exceeds that of
the best path, matched off-line to the entire sequence of the edge weights, by
a quantity that is proportional to 1/\sqrt{n} and depends only polynomially on
the number of edges of the graph. The algorithm can be implemented with linear
complexity in the number of rounds n and in the number of edges. An extension
to the so-called label efficient setting is also given, in which the decision
maker is informed about the weights of the edges corresponding to the chosen
path at a total of m << n time instances. Another extension is shown where the
decision maker competes against a time-varying path, a generalization of the
problem of tracking the best expert. A version of the multi-armed bandit
setting for shortest path is also discussed where the decision maker learns
only the total weight of the chosen path but not the weights of the individual
edges on the path. Applications to routing in packet switched networks along
with simulation results are also presented.
