An Optimal Algorithm for Adversarial Bandits with Arbitrary Delays
This provides an optimal solution for sequential decision-making under delayed feedback in adversarial environments, addressing open problems in bandit theory.
The paper tackles the problem of adversarial multi-armed bandits with arbitrary delays by proposing a new algorithm based on a hybrid regularizer in the FTRL framework, achieving an optimal regret bound of O(√(kn) + √(D log(k))) that matches the lower bound and resolves open problems from prior work.
We propose a new algorithm for adversarial multi-armed bandits with unrestricted delays. The algorithm is based on a novel hybrid regularizer applied in the Follow the Regularized Leader (FTRL) framework. It achieves $\mathcal{O}(\sqrt{kn}+\sqrt{D\log(k)})$ regret guarantee, where $k$ is the number of arms, $n$ is the number of rounds, and $D$ is the total delay. The result matches the lower bound within constants and requires no prior knowledge of $n$ or $D$. Additionally, we propose a refined tuning of the algorithm, which achieves $\mathcal{O}(\sqrt{kn}+\min_{S}|S|+\sqrt{D_{\bar S}\log(k)})$ regret guarantee, where $S$ is a set of rounds excluded from delay counting, $\bar S = [n]\setminus S$ are the counted rounds, and $D_{\bar S}$ is the total delay in the counted rounds. If the delays are highly unbalanced, the latter regret guarantee can be significantly tighter than the former. The result requires no advance knowledge of the delays and resolves an open problem of Thune et al. (2019). The new FTRL algorithm and its refined tuning are anytime and require no doubling, which resolves another open problem of Thune et al. (2019).