Near-Minimax-Optimal Rates for Asynchronous TD Learning with Markovian Data

Yang Peng

We study the last iterate of standard importance-weighted temporal-difference (TD) learning from a single trajectory of a finite Markov decision process. The trajectory follows a behavior policy $b$ with an irreducible, possibly periodic state chain; a target policy $π$ covered by $b$ is evaluated with action importance ratios bounded by $\varrho$. Write $H=(1-γ)^{-1}$, with $μ_{\min}$ the minimum stationary probability and $\mathfrak h_\star$ the largest expected hitting time of the reverse behavior state chain. We prove that the last iterate achieves sup-norm error at most $\varepsilon$ with high probability after $\widetilde O(\varrho H^3/(μ_{\min}\varepsilon^2)+\mathfrak h_\star)$ transitions for $0<\varepsilon\le1$. The statistical term retains the cubic effective-horizon dependence of synchronous TD, while the initialization cost is additive in $\mathfrak h_\star$. Setting $b=π$ and $\varrho=1$ recovers the on-policy rate. For aperiodic behavior chains, $\mathfrak h_\star=\widetilde O(t_{\rm mix}/μ_{\min})$ recovers a mixing-time bound. The proof combines an anchored local Poisson equation in reverse time to control fluctuations without a mixing-time factor, a compensation…