  We consider the traffic assignment problem in nonatomic routing games where
the players' cost functions may be subject to random fluctuations (e.g.,
weather disturbances, perturbations in the underlying network, etc.). We tackle
this problem from the viewpoint of a control interface that makes routing
recommendations based solely on observed costs and without any further
knowledge of the system's governing dynamics -- such as the network's cost
functions, the distribution of any random events affecting the network, etc. In
this online setting, learning methods based on the popular exponential weights
algorithm converge to equilibrium at an $\mathcal{O}({1/\sqrt{T}})$ rate: this
rate is known to be order-optimal in stochastic networks, but it is otherwise
suboptimal in static networks. In the latter case, it is possible to achieve an
$\mathcal{O}({1/T^{2}})$ equilibrium convergence rate via the use of finely
tuned accelerated algorithms; on the other hand, these accelerated algorithms
fail to converge altogether in the presence of persistent randomness, so it is
not clear how to achieve the "best of both worlds" in terms of convergence
speed. Our paper seeks to fill this gap by proposing an adaptive routing
algortihm with the following desirable properties: $(i)$ it seamlessly
interpolates between the $\mathcal{O}({1/T^{2}})$ and
$\mathcal{O}({1/\sqrt{T}})$ rates for static and stochastic environments
respectively; $(ii)$ its convergence speed is polylogarithmic in the number of
paths in the network; ${(iii)}$ the method's per-iteration complexity and
memory requirements are both linear in the number of nodes and edges in the
network; and ${(iv)}$ it does not require any prior knowledge of the problem's
parameters.
