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We propose a new best-of-both-worlds algorithm for bandits with variablydelayed feedback. In contrast to prior work, which required prior knowledge ofthe maximal delay $d_{\mathrm{max}}$ and had a linear dependence of the regreton it, our algorithm can tolerate arbitrary excessive delays up to order $T$(where $T$ is the time horizon). The algorithm is based on three technicalinnovations, which may all be of independent interest: (1) We introduce thefirst implicit exploration scheme that works in best-of-both-worlds setting.(2) We introduce the first control of distribution drift that does not rely onboundedness of delays. The control is based on the implicit exploration schemeand adaptive skipping of observations with excessive delays. (3) We introduce aprocedure relating standard regret with drifted regret that does not rely onboundedness of delays. At the conceptual level, we demonstrate that complexityof best-of-both-worlds bandits with delayed feedback is characterized by theamount of information missing at the time of decision making (measured by thenumber of outstanding observations) rather than the time that the informationis missing (measured by the delays).

A Best-of-both-worlds Algorithm for Bandits with Delayed Feedback with Robustness to Excessive Delays