An Improved Algorithm for Adversarial Linear Contextual Bandits via Reduction
About
We present an oracle-efficient, near-optimal algorithm for linear contextual bandits with adversarial losses and stochastic action sets, only requiring a linear optimization oracle for the action sets in each round. Our approach reduces this setting to misspecification-robust adversarial linear bandits with fixed action sets. Without knowledge of the context distribution or access to a context simulator, the algorithm achieves $\widetilde{\mathcal{O}}(\min\{d^2\sqrt{T}, \sqrt{d^3T\log K}\})$ regret and runs in $\mathrm{poly}(d,T)$ time plus $\mathrm{poly}(d,T)$ calls to the linear optimization oracles, where $d$ is the feature dimension, $K$ is an upper bound on the number of actions in each round, and $T$ is number of rounds. This resolves the open question by Liu et al. (2023) on whether one can obtain $\mathrm{poly}(d)\sqrt{T}$ regret in polynomial time independent of the number of actions. For the important class of combinatorial bandits with adversarial losses and stochastic action sets, our algorithm is the first to achieve $\mathrm{poly}(d)\sqrt{T}$ regret in polynomial time, while no prior algorithm achieves even $o(T)$ regret in polynomial time to our knowledge. When a simulator is available, the regret bound can be improved to $\widetilde{\mathcal{O}}(d\sqrt{L^\star})$, where $L^\star$ is the cumulative loss of the best policy.