  This work advances randomized exploration in reinforcement learning (RL) with
function approximation modeled by linear mixture MDPs. We establish the first
prior-dependent Bayesian regret bound for RL with function approximation; and
refine the Bayesian regret analysis for posterior sampling reinforcement
learning (PSRL), presenting an upper bound of ${\mathcal{O}}(d\sqrt{H^3 T \log
T})$, where $d$ represents the dimensionality of the transition kernel, $H$ the
planning horizon, and $T$ the total number of interactions. This signifies a
methodological enhancement by optimizing the $\mathcal{O}(\sqrt{\log T})$
factor over the previous benchmark (Osband and Van Roy, 2014) specified to
linear mixture MDPs. Our approach, leveraging a value-targeted model learning
perspective, introduces a decoupling argument and a variance reduction
technique, moving beyond traditional analyses reliant on confidence sets and
concentration inequalities to formalize Bayesian regret bounds more
effectively.
