With a known optimality gap, streaming best-arm identification needs about log(n)/log log(n) passes with slightly sublinear memory, and O(log n) passes with a single arm of memory.
The best arm evades: Near-optimal multi-pass streaming lower bounds for pure exploration in multi-armed bandits
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Nearly Tight Bounds for Exploration in Streaming Multi-armed Bandits with Known Optimality Gap
With a known optimality gap, streaming best-arm identification needs about log(n)/log log(n) passes with slightly sublinear memory, and O(log n) passes with a single arm of memory.