Improved Regret Bounds of (Multinomial) Logistic Bandits via Regret-to-Confidence-Set Conversion

Lee, Junghyun; Yun, Se-Young; Jun, Kwang-Sung

Selecting the best items in a dataset is a common task in data exploration. However, the concept of “best” lies in the eyes of the beholder: different users may consider different attributes more important, and hence arrive at different rankings. Nevertheless, one can remove “dominated” items and create a “representative” subset of the data, comprising the “best items” in it. A Pareto-optimal representative is guaranteed to contain the best item of each possible ranking, but it can be a large portion of data. A much smaller representative can be found if we relax the requirement to include the best item for each user, and instead just limit the users’ “regret”. Existing work defines regret as the loss in score by limiting consideration to the representative instead of the full data set, for any chosen ranking function. However, the score is often not a meaningful number and users may not understand its absolute value. Sometimes small ranges in score can include large fractions of the data set. In contrast, users do understand the notion of rank ordering. Therefore, we consider the position of the items in the ranked list for defining the regret and propose the rank-regret representative as the minimal subset of the data containing at least one of the top-k of any possible ranking function. This problem is NP-complete. We use a geometric interpretation of items to bound their ranks on ranges of functions and to utilize combinatorial geometry notions for developing effective and efficient approximation algorithms for the problem. Experiments on real datasets demonstrate that we can efficiently find small subsets with small rank-regrets.

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