  We introduce efficient algorithms which achieve nearly optimal regrets for
the problem of stochastic online shortest path routing with end-to-end
feedback. The setting is a natural application of the combinatorial stochastic
bandits problem, a special case of the linear stochastic bandits problem. We
show how the difficulties posed by the large scale action set can be overcome
by the networked structure of the action set. Our approach presents a novel
connection between bandit learning and shortest path algorithms. Our main
contribution is an adaptive exploration algorithm with nearly optimal
instance-dependent regret for any directed acyclic network. We then modify it
so that nearly optimal worst case regret is achieved simultaneously. Driven by
the carefully designed Top-Two Comparison (TTC) technique, the algorithms are
efficiently implementable. We further conduct extensive numerical experiments
to show that our proposed algorithms not only achieve superior regret
performances, but also reduce the runtime drastically.
