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Abstract: In this paper, we study continuous-time reinforcement learning, where the state follows an unknown stochastic differential equation but only a fixed off-policy batch of discrete-time transitions is available. Standard discrete-time Q-learning becomes problematic as the sampling interval shrinks, because differences between action values vanish. Continuous-time alternatives, such as Deep Advantage Updating and martingale q-learning, preserve the action signal but jointly learn a value function and an action-dependent critic. Building on the PhiBE-Q formulation (arXiv:2606.21925), we develop a model-free PhiBE policy iteration (PhiBE-PI) framework that uses a single state-action critic while preserving the continuous-time action-ranking scale. At each iteration, the framework learns the critic directly from sampled increments and uses it to improve the policy. The same batch is reused throughout, without explicitly estimating the drift or diffusion. We further develop an implementation under linear function approximation and establish a local end-to-end suboptimality guarantee for the learned policy. Our analysis quantifies how errors from time discretization, function approximation, and finite-sample policy evaluation propagate through policy iteration. It accounts for the dependence of successive policies on the reused data, so the guarantee does not require a fresh batch at each iteration. Moreover, we identify a bias-variance trade-off: smaller sampling intervals improve the approximation to the continuous-time problem but can increase statistical error at a fixed sample size. Thus, finer time discretization does not necessarily lead to better policy performance.

Abstract: We investigate the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in any fixed Euclidean dimension. The potential is $μ$-strongly convex and $L$-smooth, with an unknown mode in the ball of radius $μ^{-1/2}$ about the origin. We have access to unbiased stochastic oracles with the variance at most $σ^2$. For every $σ^2\ge0$ and total variation (TV) accuracy $0<\varepsilon\le1/10$, we prove that the tight complexity of sampling a distribution within $ε$-TV distance from the target distribution is \[ N^\star_{\text{TV}}=Θ\!\left(\log(1+κ)+ \frac{σ^2}{με}\right), \] where $κ:=\frac Lμ$ is the condition number. Note that this complexity bound is simultaneously tight for the condition number $κ$ and accuracy $ε$. Besides, our tight complexity bound is adaptive to noiseless setting $σ=0$, which is $ N^\star_{\text{TV}}=Θ\!\left(\log(1+κ)\right)$.