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- W4384652236 abstract "In the kernelized bandit problem, a learner aims to sequentially compute the optimum of a function lying in a reproducing kernel Hilbert space given only noisy evaluations at sequentially chosen points. In particular, the learner aims to minimize regret, which is a measure of the suboptimality of the choices made. Arguably the most popular algorithm is the Gaussian Process Upper Confidence Bound (GP-UCB) algorithm, which involves acting based on a simple linear estimator of the unknown function. Despite its popularity, existing analyses of GP-UCB give a suboptimal regret rate, which fails to be sublinear for many commonly used kernels such as the Mat'ern kernel. This has led to a longstanding open question: are existing regret analyses for GP-UCB tight, or can bounds be improved by using more sophisticated analytical techniques? In this work, we resolve this open question and show that GP-UCB enjoys nearly optimal regret. In particular, our results yield sublinear regret rates for the Mat'ern kernel, improving over the state-of-the-art analyses and partially resolving a COLT open problem posed by Vakili et al. Our improvements rely on a key technical contribution -- regularizing kernel ridge estimators in proportion to the smoothness of the underlying kernel $k$. Applying this key idea together with a largely overlooked concentration result in separable Hilbert spaces (for which we provide an independent, simplified derivation), we are able to provide a tighter analysis of the GP-UCB algorithm." @default.
- W4384652236 created "2023-07-19" @default.
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- W4384652236 date "2023-07-14" @default.
- W4384652236 modified "2023-09-27" @default.
- W4384652236 title "On the Sublinear Regret of GP-UCB" @default.
- W4384652236 doi "https://doi.org/10.48550/arxiv.2307.07539" @default.
- W4384652236 hasPublicationYear "2023" @default.
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