Instance-dependent Stochastic Lipschitz bandit

• URL: http://arxiv.org/abs/2605.29748v1
• Date: Thu, 28 May 2026 10:46:30 GMT
• Title: Instance-dependent Stochastic Lipschitz bandit
• Abstract summary: We study the Lipschitz bandit problem, where a learner sequentially maximizes an unknown Lipschitz function $f$ over a domain.<n>Existing regret bounds are either worst-case, scaling as $tilde left ( Td+1/d+2right )$, or adaptive via the zooming dimension $d_z$.<n>We provide an analysis and an algorithm that characterizes the regret through integrals of the suboptimality gap of $f$ over its level sets.
• Score: 28.946496440127603
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the Lipschitz bandit problem, where a learner sequentially maximizes an unknown Lipschitz function $f$ over a domain $\mathcal{X} \subset [0,1]^d$ using noisy pointwise evaluations. Existing regret bounds are either worst-case, scaling as $\tildeΘ \left ( T^{d+1/d+2}\right )$, or adaptive via the zooming dimension $d_z$, yielding $\tildeΘ \left ( T^{d_z+1/d_z+2}\right )$. However, such zooming-based guarantees are only partially instance-dependent, as they depend solely on the asymptotic growth of near-optimal level sets and fail to capture finer structural properties of $f$. We provide an analysis and an algorithm that characterizes the regret through integrals of the suboptimality gap of $f$ over its level sets. This yields regret bounds that adapt to the local growth of level sets, rather than only their asymptotic behavior. As a corollary, when the set of maximizers has dimension $d^\star>0$, we obtain improved adaptive rates of order $\tilde{\mathcal{O}} \left ( T^{d_z+1 / \max(d_z,d^\star)+2}\right )$ strictly improving over classical zooming bounds in this regime. Finally, we extend our analysis to the full-information setting (Lipschitz experts) and show how some of the regularity assumptions can be relaxed.

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