Context-lumpable stochastic bandits

• URL: http://arxiv.org/abs/2306.13053v2
• Date: Tue, 28 Nov 2023 00:53:55 GMT
• Title: Context-lumpable stochastic bandits
• Authors: Chung-Wei Lee, Qinghua Liu, Yasin Abbasi-Yadkori, Chi Jin, Tor Lattimore, Csaba Szepesv\'ari
• Abstract summary: We consider a contextual bandit problem with $S$ contexts and $K$ actions. We give an algorithm that outputs an $epsilon$-optimal policy after using at most $widetilde O(r (S +K )/epsilon2)$ samples. In the regret setting, we give an algorithm whose cumulative regret up to time $T$ is bounded by $widetilde O(sqrtr3(S+K)T)$.
• Score: 49.024050919419366
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider a contextual bandit problem with $S$ contexts and $K$ actions. In each round $t=1,2,\dots$, the learner observes a random context and chooses an action based on its past experience. The learner then observes a random reward whose mean is a function of the context and the action for the round. Under the assumption that the contexts can be lumped into $r\le \min\{S,K\}$ groups such that the mean reward for the various actions is the same for any two contexts that are in the same group, we give an algorithm that outputs an $\epsilon$-optimal policy after using at most $\widetilde O(r (S +K )/\epsilon^2)$ samples with high probability and provide a matching $\Omega(r(S+K)/\epsilon^2)$ lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time $T$ is bounded by $\widetilde O(\sqrt{r^3(S+K)T})$. To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and $\widetilde O(\sqrt{{poly}(r)(S+K)T})$ minimax regret in the online setting for this problem. We also show our algorithms can be applied to more general low-rank bandits and get improved regret bounds in some scenarios.

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