On Frequentist Regret of Linear Thompson Sampling
- URL: http://arxiv.org/abs/2006.06790v3
- Date: Thu, 20 Apr 2023 21:35:31 GMT
- Title: On Frequentist Regret of Linear Thompson Sampling
- Authors: Nima Hamidi, Mohsen Bayati
- Abstract summary: This paper studies the linear bandit problem, where a decision-maker chooses actions from possibly time-dependent sets of vectors in $mathbbRd$ and receives noisy rewards.
The objective is to minimize regret, the difference between the cumulative expected reward of the decision-maker and that of an oracle with access to the expected reward of each action, over a sequence of $T$ decisions.
- Score: 8.071506311915398
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: This paper studies the stochastic linear bandit problem, where a
decision-maker chooses actions from possibly time-dependent sets of vectors in
$\mathbb{R}^d$ and receives noisy rewards. The objective is to minimize regret,
the difference between the cumulative expected reward of the decision-maker and
that of an oracle with access to the expected reward of each action, over a
sequence of $T$ decisions. Linear Thompson Sampling (LinTS) is a popular
Bayesian heuristic, supported by theoretical analysis that shows its Bayesian
regret is bounded by $\widetilde{\mathcal{O}}(d\sqrt{T})$, matching minimax
lower bounds. However, previous studies demonstrate that the frequentist regret
bound for LinTS is $\widetilde{\mathcal{O}}(d\sqrt{dT})$, which requires
posterior variance inflation and is by a factor of $\sqrt{d}$ worse than the
best optimism-based algorithms. We prove that this inflation is fundamental and
that the frequentist bound of $\widetilde{\mathcal{O}}(d\sqrt{dT})$ is the best
possible, by demonstrating a randomization bias phenomenon in LinTS that can
cause linear regret without inflation.We propose a data-driven version of LinTS
that adjusts posterior inflation using observed data, which can achieve minimax
optimal frequentist regret, under additional conditions. Our analysis provides
new insights into LinTS and settles an open problem in the field.
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