# Prediction with Limited Advice and Multiarmed Bandits with Paid Observations

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Yevgeny Seldin,
Peter Bartlett,
Koby Crammer,
Yasin Abbasi-Yadkori
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Proceedings of the 31st International Conference on Machine Learning, PMLR 32(1):280-287, 2014.

#### Abstract

We study two problems of online learning under restricted information access. In the first problem, \emphprediction with limited advice, we consider a game of prediction with expert advice, where on each round of the game we query the advice of a subset of M out of N experts. We present an algorithm that achieves O(\sqrt(N/M)T\ln N) regret on T rounds of this game. The second problem, the \emphmultiarmed bandit with paid observations, is a variant of the adversarial N-armed bandit game, where on round t of the game we can observe the reward of any number of arms, but each observation has a cost c. We present an algorithm that achieves O((cN\ln N)^1/3 T^2/3 + \sqrtT \ln N) regret on T rounds of this game in the worst case. Furthermore, we present a number of refinements that treat arm- and time-dependent observation costs and achieve lower regret under benign conditions. We present lower bounds that show that, apart from the logarithmic factors, the worst-case regret bounds cannot be improved.

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