Contextual linear optimization (CLO) uses predictive contextual features to reduce uncertainty in random cost coefficients and thereby improve average-cost performance. An example is the stochastic shortest path problem with random edge costs (e.g., traffic) and contextual 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 offline learning 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.

This paper proposes a system for finding a userpsilas interests on the Internet. It is based on his browsing behaviors and the contents of his visited pages. The system has two features. One is building userpsilas browsing interests implicitly, multiple keyword vectors, one per interest. The other is that it can generate interests by selecting different time periods. Dynamical generation can adapt to the change of user interests. Experiments show that most of generated interests are matched to userpsilas real interests. The system finds their interests automatically and dynamically.