## Electronic Journal of Statistics

### Linear Thompson sampling revisited

#### Abstract

We derive an alternative proof for the regret of Thompson sampling (TS) in the stochastic linear bandit setting. While we obtain a regret bound of order $\widetilde{O}(d^{3/2}\sqrt{T})$ as in previous results, the proof sheds new light on the functioning of the TS. We leverage the structure of the problem to show how the regret is related to the sensitivity (i.e., the gradient) of the objective function and how selecting optimal arms associated to optimistic parameters does control it. Thus we show that TS can be seen as a generic randomized algorithm where the sampling distribution is designed to have a fixed probability of being optimistic, at the cost of an additional $\sqrt{d}$ regret factor compared to a UCB-like approach. Furthermore, we show that our proof can be readily applied to regularized linear optimization and generalized linear model problems.

#### Article information

Source
Electron. J. Statist., Volume 11, Number 2 (2017), 5165-5197.

Dates
First available in Project Euclid: 15 December 2017

https://projecteuclid.org/euclid.ejs/1513306870

Digital Object Identifier
doi:10.1214/17-EJS1341SI

Mathematical Reviews number (MathSciNet)
MR3738208

Zentralblatt MATH identifier
06825043

Keywords
Linear bandit Thompson sampling

#### Citation

Abeille, Marc; Lazaric, Alessandro. Linear Thompson sampling revisited. Electron. J. Statist. 11 (2017), no. 2, 5165--5197. doi:10.1214/17-EJS1341SI. https://projecteuclid.org/euclid.ejs/1513306870

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