{"id":1176607,"date":"2026-06-22T16:05:05","date_gmt":"2026-06-22T23:05:05","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/continuous-k-max-bandits\/"},"modified":"2026-07-10T16:48:27","modified_gmt":"2026-07-10T23:48:27","slug":"continuous-k-max-bandits","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/continuous-k-max-bandits\/","title":{"rendered":"Continuous K-Max Bandits"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\">We study the <math><mi>K<\/mi><\/math>-Max combinatorial multi-armed bandits problem with continuous outcome distributions and weak value-index feedback: each base arm has an unknown continuous outcome distribution, and in each round the learning agent selects <math><mi>K<\/mi><\/math> arms, obtains the maximum value sampled from these <math><mi>K<\/mi><\/math> arms as reward and observes this reward together with the corresponding arm index as feedback. This setting captures critical applications in recommendation systems, distributed computing, server scheduling, etc. The continuous <math><mi>K<\/mi><\/math>-Max bandits introduce unique challenges, including discretization error from continuous-to-discrete conversion, non-deterministic tie-breaking under limited feedback, and biased estimation due to partial observability. Our key contribution is the computationally efficient algorithm DCK-UCB, which combines adaptive discretization with bias-corrected confidence bounds to tackle these challenges. For general continuous distributions, we prove that DCK-UCB achieves a <math><semantics><mrow><mtext>widetilde{mathcal{O}}(T^{3\/4})<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">widetilde{mathcal{O}}(T^{3\/4})<\/annotation><\/semantics><\/math> regret upper bound, establishing the first sublinear regret guarantee for this setting. Furthermore, we identify an important special case with exponential distributions under full-bandit feedback. In this case, our proposed algorithm MLE-Exp enables <math><semantics><mrow><mtext>widetilde{mathcal{O}}(sqrt{T})<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">widetilde{mathcal{O}}(sqrt{T})<\/annotation><\/semantics><\/math> regret upper bound through maximal log-likelihood estimation, achieving near-minimax optimality.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We study the K-Max combinatorial multi-armed bandits problem with continuous outcome distributions and weak value-index feedback: each base arm has an unknown continuous outcome distribution, and in each round the learning agent selects K arms, obtains the maximum value sampled from these K arms as reward and observes this reward together with the corresponding arm 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