Lookahead identification in adversarial bandits: accuracy and memory bounds

We study an identification problem in multi-armed bandits. In each round a learner selects one of $K$ arms and observes its reward, with the goal of eventually identifying an arm that will perform best at a {\it future} time. In adversarial environments, however, past performance may offer little information about the future, raising the question of whether meaningful identification is possible at all. In this work, we introduce \emph{lookahead identification}, a task in which the goal of the learner is to select a future prediction window and commit in advance to an arm whose average reward over that window is within $\varepsilon$ of optimal. Our analysis characterizes both the achievable accuracy of lookahead identification and the memory resources required to obtain it. From an accuracy standpoint, for any horizon $T$ we give an algorithm achieving $\varepsilon = O\bigl(1/\sqrt{\log T}\bigr)$ over $Ω(\sqrt{T})$ prediction windows. This demonstrates that, perhaps surprisingly, identification is possible in adversarial settings, despite significant lack of information. We also prove a near-matching lower bound showing that $\varepsilon = Ω\bigl(1/\log T\bigr)$ is unavoidable. We then turn to investigate the role of memory in our problem, first proving that any algorithm achieving nontrivial accuracy requires $Ω(K)$ bits of memory. Under a natural \emph{local sparsity} condition, we show that the same accuracy guarantees can be achieved using only poly-logarithmic memory.