  We study the problem of online binary classification in settings where
strategic agents can modify their observable features to receive a positive
classification. We model the set of feasible manipulations by a directed graph
over the feature space, and assume the learner only observes the manipulated
features instead of the original ones. We introduce the Strategic Littlestone
Dimension, a new combinatorial measure that captures the joint complexity of
the hypothesis class and the manipulation graph. We demonstrate that it
characterizes the instance-optimal mistake bounds for deterministic learning
algorithms in the realizable setting. We also achieve improved regret in the
agnostic setting by a refined agnostic-to-realizable reduction that accounts
for the additional challenge of not observing agents' original features.
Finally, we relax the assumption that the learner knows the manipulation graph,
instead assuming their knowledge is captured by a family of graphs. We derive
regret bounds in both the realizable setting where all agents manipulate
according to the same graph within the graph family, and the agnostic setting
where the manipulation graphs are chosen adversarially and not consistently
modeled by a single graph in the family.
