Achieving Optimal Static and Dynamic Regret Simultaneously in Bandits with Deterministic Losses

In adversarial multi-armed bandits, two performance measures are commonly used: static regret, which compares the learner to the best fixed arm, and dynamic regret, which compares it to the best sequence of arms. While optimal algorithms are known for each measure individually, there is no known algorithm achieving optimal bounds for both simultaneously. \cite{marinov2021pareto} first showed that such simultaneous optimality is impossible against an \emph{adaptive} adversary. Our work takes a first step to demonstrate its possibility against an \emph{oblivious} adversary when losses are \emph{deterministic}. First, we extend the impossibility result of \cite{marinov2021pareto} to the case of deterministic losses. Then, we present an algorithm achieving optimal static and dynamic regret simultaneously against an oblivious adversary. Together, they reveal a fundamental separation between adaptive and oblivious adversaries when multiple regret benchmarks are considered simultaneously. It also provides new insight into the long open problem of simultaneously achieving optimal regret against switching benchmarks of different numbers of switches. Our algorithm uses negative static regret to compensate for the exploration overhead incurred when controlling dynamic regret, and leverages Blackwell approachability to jointly control both regrets. This yields a new model selection procedure for bandits that may be of independent interest.