Asymptotically Optimal Thompson Sampling Based Policy for the Uniform Bandits and the Gaussian Bandits

Thompson sampling (TS) for the parametric stochastic multi-armed bandits has been well studied under the one-dimensional parametric models. It is often reported that TS is fairly insensitive to the choice of the prior when it comes to regret bounds. However, this property is not necessarily true when multiparameter models are considered, e.g., a Gaussian model with unknown mean and variance parameters. In this paper, we first extend the regret analysis of TS to the model of uniform distributions with unknown supports. Specifically, we show that a switch of noninformative priors drastically affects the regret in expectation. Through our analysis, the uniform prior is proven to be the optimal choice in terms of the expected regret, while the reference prior and the Jeffreys prior are found to be suboptimal, which is consistent with previous findings in the model of Gaussian distributions. However, the uniform prior is specific to the parameterization of the distributions, meaning that if an agent considers different parameterizations of the same model, the agent with the uniform prior might not always achieve the optimal performance. In light of this limitation, we propose a slightly modified TS-based policy, called TS with Truncation (TS-T), which can achieve the asymptotic optimality for the Gaussian distributions and the uniform distributions by using the reference prior and the Jeffreys prior that are invariant under one-to-one reparameterizations. The pre-processig of the posterior distribution is the key to TS-T, where we add an adaptive truncation procedure on the parameter space of the posterior distributions. Simulation results support our analysis, where TS-T shows the best performance in a finite-time horizon compared to other known optimal policies, while TS with the invariant priors performs poorly.