  This paper considers the recently popular beyond-worst-case algorithm
analysis model which integrates machine-learned predictions with online
algorithm design. We consider the online Steiner tree problem in this model for
both directed and undirected graphs. Steiner tree is known to have strong lower
bounds in the online setting and any algorithm's worst-case guarantee is far
from desirable. This paper considers algorithms that predict which terminal
arrives online. The predictions may be incorrect and the algorithms'
performance is parameterized by the number of incorrectly predicted terminals.
These guarantees ensure that algorithms break through the online lower bounds
with good predictions and the competitive ratio gracefully degrades as the
prediction error grows. We then observe that the theory is predictive of what
will occur empirically. We show on graphs where terminals are drawn from a
distribution, the new online algorithms have strong performance even with
modestly correct predictions.
