Improved Best-of-Both-Worlds Regret for Bandits with Delayed Feedback
This solves a theoretical gap in bandit algorithms for delayed feedback, benefiting researchers in online learning and optimization, though it is incremental as it builds on prior BoBW work.
The paper tackles the multi-armed bandit problem with adversarial delays in the Best-of-Both-Worlds framework, achieving an algorithm that matches known lower bounds up to logarithmic factors in both stochastic and adversarial settings, with regret bounds such as Õ(√KT + √D) in adversarial cases and improved terms in stochastic cases.
We study the multi-armed bandit problem with adversarially chosen delays in the Best-of-Both-Worlds (BoBW) framework, which aims to achieve near-optimal performance in both stochastic and adversarial environments. While prior work has made progress toward this goal, existing algorithms suffer from significant gaps to the known lower bounds, especially in the stochastic settings. Our main contribution is a new algorithm that, up to logarithmic factors, matches the known lower bounds in each setting individually. In the adversarial case, our algorithm achieves regret of $\widetilde{O}(\sqrt{KT} + \sqrt{D})$, which is optimal up to logarithmic terms, where $T$ is the number of rounds, $K$ is the number of arms, and $D$ is the cumulative delay. In the stochastic case, we provide a regret bound which scale as $\sum_{i:Δ_i>0}\left(\log T/Δ_i\right) + \frac{1}{K}\sum Δ_i σ_{max}$, where $Δ_i$ is the sub-optimality gap of arm $i$ and $σ_{\max}$ is the maximum number of missing observations. To the best of our knowledge, this is the first BoBW algorithm to simultaneously match the lower bounds in both stochastic and adversarial regimes in delayed environment. Moreover, even beyond the BoBW setting, our stochastic regret bound is the first to match the known lower bound under adversarial delays, improving the second term over the best known result by a factor of $K$.