Frequency-Uniform Log-Regret Reinforcement Learning for Diffusions

High-frequency control problems are often studied through time discretization, but refining the timestep $δ$ increases the discrete horizon $H=T/δ$ and can make standard finite-horizon regret guarantees deteriorate even when the physical horizon $T$ is fixed. We study episodic value-based reinforcement learning for Euler discretizations of controlled diffusions. The performance is measured as the cumulative regret over $K$ episodes relative to the optimal policy of the corresponding Euler model. Under stability and regularity conditions on value function gradients, Bellman residual derivatives, and fitted critics, we prove an $\mathcal{O}(\log K)$ regret bound with constants uniform in $δ$. For dynamics with drift linear in unknown parameters and unknown constant diffusion covariance, we verify localized versions of these conditions for a constrained, model-based controlled-diffusion fitted-$Q$ algorithm (CD-FQ) and obtain high-probability $\mathcal{O}(\log K\log\log K)$ regret for fixed and finite $T$, again with constants uniform in $δ$.