Regret Bounds for Adversarial Contextual Bandits with General Function Approximation and Delayed Feedback

We present regret minimization algorithms for the contextual multi-armed bandit (CMAB) problem over $K$ actions in the presence of delayed feedback, a scenario where loss observations arrive with delays chosen by an adversary. As a preliminary result, assuming direct access to a finite policy class $\Pi$ we establish an optimal expected regret bound of $ O (\sqrt{KT \log |\Pi|} + \sqrt{D \log |\Pi|)} $ where $D$ is the sum of delays. For our main contribution, we study the general function approximation setting over a (possibly infinite) contextual loss function class $ \mathcal{F} $ with access to an online least-square regression oracle $\mathcal{O}$ over $\mathcal{F}$. In this setting, we achieve an expected regret bound of $O(\sqrt{KTR_T(\mathcal{O})} + \sqrt{ d_{\max} D \beta})$ assuming FIFO order, where $d_{\max}$ is the maximal delay, $R_T(\mathcal{O})$ is an upper bound on the oracle's regret and $\beta$ is a stability parameter associated with the oracle.