When are Bandits Robust to Misspecification?

Parametric feature-based reward models are widely employed by algorithms for decision making settings such as bandits and contextual bandits. The typical assumption under which they are analysed is realizability, i.e., that the true rewards of actions are perfectly explained by some parametric model in the class. We are, however, interested in the situation where the true rewards are (potentially significantly) misspecified with respect to the model class. For parameterized bandits and contextual bandits, we identify sufficient conditions, depending on the problem instance and model class, under which classic algorithms such as $\epsilon$-greedy and LinUCB enjoy sublinear (in the time horizon) regret guarantees under even grossly misspecified rewards. This is in contrast to existing worst-case results for misspecified bandits which show regret bounds that scale linearly with time, and shows that there can be a nontrivially large set of bandit instances that are robust to misspecification.