Best-of-Both Worlds for linear contextual bandits with paid observations

We study the problem of linear contextual bandits with paid observations, where at each round the learner selects an action in order to minimize its loss in a given context, and can then decide to pay a fixed cost to observe the loss of any arm. Building on the Follow-the-Regularized-Leader framework with efficient estimators via Matrix Geometric Resampling, we introduce a computationally efficient Best-of-Both-Worlds (BOBW) algorithm for this problem. We show that it achieves the minimax-optimal regret of $Θ(T^{2/3})$ in adversarial settings, while guaranteeing poly-logarithmic regret in (corrupted) stochastic regimes. Our approach builds on the framework from \cite{BOBWhardproblems} to design BOBW algorithms for ``hard problem'', using analysis techniques tailored for the setting that we consider.