Understanding Gap-Dependent Regret for Optimism-Based Reinforcement Learning with Linear Function Approximation

We study gap-dependent regret for reinforcement learning with linear function approximation. While prior works have established gap-dependent guarantees in this setting, existing analyses do not apply to algorithms that achieve the nearly minimax-optimal worst-case regret bound $\tilde{O}(d\sqrt{H^3K})$, where $d$ is the feature dimension, $H$ is the horizon length, and $K$ is the number of episodes. We bridge this gap by establishing the first gap-dependent regret bound for the nearly minimax-optimal algorithm LSVI-UCB++ (He et al., 2023), with an expected regret bound $\tilde{O}(d^2H^3/Δ_{\min}+d^6H^5)$, improving the dependence on both $d$ and $H$ in the leading gap-dependent term compared with previous results. To understand the exploration cost induced by optimism, we establish a structural lower bound $Ω(d^2H^3/Δ_{\min})$ for a broad class of algorithms based on persistent ellipsoidal optimism. When specialized to LSVI-UCB++, this result shows that the leading dependence of our upper bound on $d$, $H$, and $Δ_{\min}$ is tight up to logarithmic factors. Beyond this algorithmic class, we establish a general gap-dependent lower bound $Ω(dH^3\log K/Δ_{\min})$ for arbitrary learning algorithms, showing that the logarithmic dependence on $K$ and the cubic dependence on $H$ are intrinsic to gap-dependent expected regret in linear MDPs. Together, our results substantially narrow the gap between upper and lower bounds and provide a sharper characterization of gap-dependent learning and optimism-based exploration with linear function approximation.