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Is there a term or name (or better yet, strategies) for the following problem?

Take a 'standard' $k$-armed multi-armed bandit problem (stochastic real rewards, IID pulls for a given arm), but instead of maximizing the mean sum of rewards across $H$ pulls, maximize the mean maximum reward across $H$ pulls.

(I know that classically this is formulated as minimizing regret instead.)

Intuitively, strategies for solving this would have substantially more variance-seeking than strategies for a standard multi-armed bandit problem - if you're in an exploit phase with sufficient pulls left and arm A is known to have a somewhat higher mean, but somewhat lower variance, than arm B, a strategy for a normal multi-armed bandit problem would likely pick A, whereas a strategy for this would likely pick B.

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  • $\begingroup$ you may be looking for the distinction between "simplex regret" and "cumulative regret" $\endgroup$ May 8, 2023 at 19:46
  • $\begingroup$ I went to search the term 'simplex regret' and this question was on the first page of the results :/ $\endgroup$
    – TLW
    May 9, 2023 at 0:41
  • $\begingroup$ I am aware of the mathematical concept of a simplex; I do not see how it applies in this situation. I have changed the reward function, not the feasibility region. Please elaborate? $\endgroup$
    – TLW
    May 9, 2023 at 1:23
  • $\begingroup$ oh sorry, I meant to say "simple regret" $\endgroup$ May 9, 2023 at 1:26
  • $\begingroup$ Mm. Both simple and cumulative regret are measured w.r.t. the arm with the highest mean however, which may not be the optimal choice in this setting. Or am I missing subtleties here? $\endgroup$
    – TLW
    May 9, 2023 at 1:36

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After much hunting, this appears to be known as, imaginatively (/s), the Max Bandit Problem. (This is annoyingly difficult to search for, as 'max' appears in almost all descriptions of the 'standard' k-armed bandit problem.)

I have implemented the algorithm described in Kikkawa & Ohno, 2022, and it appears to suffice for my usecase.

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