On the Complexity of All $$\varepsilon $$-Best Arms Identification

We consider the question introduced by [16] of identifying all $$\varepsilon $$ -optimal arms in a finite stochastic multi-armed bandit with Gaussian rewards. give two lower bounds on sample complexity any algorithm solving problem confidence at least $$1-\delta . The first, unimprovable asymptotic regime, motivates design Track-and-Stop strategy whose average is asymptotically optimal when risk $$\delta goes to zero. Notably, we provide an efficient numerical method solve convex max-min program that appears bound. Our based complete characterization alternative instances sampling needs rule out, thus making our bound tighter than one provided [16]. second deals regime high and moderate values , characterizes behavior initial phase. It emphasizes linear dependency number arms. Finally, report simulations demonstrating algorithm’s advantage over state-of-the-art methods, even for risks.