The Bias of Nonlinear Two-Time-scale Stochastic Approximation under Constant Step-Sizes

Two-timescale stochastic approximation (TTSA) is a fundamental tool for analyzing coupled iterative algorithms in reinforcement learning, optimization, and stochastic control. However, finite-time guarantees for nonlinear two-timescale schemes remain difficult to obtain, especially under constant step-sizes. In this paper, we study nonlinear TTSA with step-sizes $\alpha \gg \beta$. Under standard stability, regularity, and Markovian noise assumptions, we upper bound the mean-squared error and the bias of both iterates around their limiting equilibria. Our bounds scale as $O(\alpha +{\beta }^2\mathrm{/}{\alpha }^2)$, which we prove to be tight when $\beta \le {\alpha }^{3\mathrm{/}2}$. The analysis separates the contributions of initial conditions, fast-timescale tracking error, Markovian dependence, and timescale coupling, thereby clarifying the origin of the ${\beta }^2\mathrm{/}{\alpha }^2$ term. Our results reveal qualitative differences from the linear TTSA setting previously studied, showing that nonlinear dynamics introduce additional finite-time effects that are absent in the linear case.