  Contextual linear optimization (CLO) uses predictive observations to reduce
uncertainty in random cost coefficients and thereby improve average-cost
performance. An example is a stochastic shortest path with random edge costs
(e.g., traffic) and predictive features (e.g., lagged traffic, weather).
Existing work on CLO assumes the data has fully observed cost coefficient
vectors, but in many applications, we can only see the realized cost of a
historical decision, that is, just one projection of the random cost
coefficient vector, to which we refer as bandit feedback. We study a class of
algorithms for CLO with bandit feedback, which we term induced empirical risk
minimization (IERM), where we fit a predictive model to directly optimize the
downstream performance of the policy it induces. We show a fast-rate regret
bound for IERM that allows for misspecified model classes and flexible choices
of the optimization estimate, and we develop computationally tractable
surrogate losses. A byproduct of our theory of independent interest is
fast-rate regret bound for IERM with full feedback and misspecified policy
class. We compare the performance of different modeling choices numerically
using a stochastic shortest path example and provide practical insights from
the empirical results.
