  We study contextual search, a generalization of binary search in higher
dimensions, which captures settings such as feature-based dynamic pricing.
Standard formulations of this problem assume that agents act in accordance with
a specific homogeneous response model. In practice, however, some responses may
be adversarially corrupted. Existing algorithms heavily depend on the assumed
response model being (approximately) accurate for all agents and have poor
performance in the presence of even a few such arbitrary misspecifications.
  We initiate the study of contextual search when some of the agents can behave
in ways inconsistent with the underlying response model. In particular, we
provide two algorithms, one based on multidimensional binary search methods and
one based on gradient descent. We show that these algorithms attain
near-optimal regret in the absence of adversarial corruptions and their
performance degrades gracefully with the number of such agents, providing the
first results for contextual search in any adversarial noise model. Our
techniques draw inspiration from learning theory, game theory, high-dimensional
geometry, and convex analysis.
