  The problem of online learning with graph feedback has been extensively
studied in the literature due to its generality and potential to model various
learning tasks. Existing works mainly study the adversarial and stochastic
feedback separately. If the prior knowledge of the feedback mechanism is
unavailable or wrong, such specially designed algorithms could suffer great
loss. To avoid this problem, \citet{erez2021towards} try to optimize for both
environments. However, they assume the feedback graphs are undirected and each
vertex has a self-loop, which compromises the generality of the framework and
may not be satisfied in applications. With a general feedback graph, the
observation of an arm may not be available when this arm is pulled, which makes
the exploration more expensive and the algorithms more challenging to perform
optimally in both environments. In this work, we overcome this difficulty by a
new trade-off mechanism with a carefully-designed proportion for exploration
and exploitation. We prove the proposed algorithm simultaneously achieves
$\mathrm{poly} \log T$ regret in the stochastic setting and minimax-optimal
regret of $\tilde{O}(T^{2/3})$ in the adversarial setting where $T$ is the
horizon and $\tilde{O}$ hides parameters independent of $T$ as well as
logarithmic terms. To our knowledge, this is the first best-of-both-worlds
result for general feedback graphs.

Machine Learning