Near Instance Optimal Model Selection for Pure Exploration Linear Bandits

The model selection problem in the pure exploration linear bandit setting is introduced and studied in both the fixed confidence and fixed budget settings. The model selection problem considers a nested sequence of hypothesis classes of increasing complexities. Our goal is to automatically adapt to the instance-dependent complexity measure of the smallest hypothesis class containing the true model, rather than suffering from the complexity measure related to the largest hypothesis class. We provide evidence showing that a standard doubling trick over dimension fails to achieve the optimal instance-dependent sample complexity. Our algorithms define a new optimization problem based on experimental design that leverages the geometry of the action set to efficiently identify a near-optimal hypothesis class. Our fixed budget algorithm uses a novel application of a selection-validation trick in bandits. This provides a new method for the understudied fixed budget setting in linear bandits (even without the added challenge of model selection). We further generalize the model selection problem to the misspecified regime, adapting our algorithms in both fixed confidence and fixed budget settings.