arXiv:2605.07104v3 Announce Type: replace-cross Abstract: Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as $Q$-learning and linear temporal difference learning. Specifically, for a power-law learning rate $O(n^{-\eta})$ with $\eta \in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2\eta})$. For the harmonic learning rate $\alpha_n=\alpha/(n+1)$, the almost sure squared-error rate is $o(n^{-\zeta})$ for every $\zeta<\min\{1,2(1-\kappa)\alpha\}$, where $\kappa$ is the contraction factor. Thus exponents arbitrarily close to one are available when $2(1-\kappa)\alpha\geq1$. This approaches the $n^{-1}$ polynomial exponent of classical law-of-the-iterated-logarithm benchmarks. Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.