Stationary Bias and Extrapolation in Nonlinear Two-Timescale Stochastic Approximation

Constant-step stochastic approximation generally has a nonzero stationary mean error that persists under time averaging. This paper studies that error for nonlinear two-timescale recursions driven by an exogenous finite-state Markov chain. Under stated smoothness assumptions and conditions on the stationary distribution, we derive a first-order bias expansion whose error bound remains uniform as the slow step size becomes much smaller than the fast step size. Fast-manifold coordinates keep the associated covariance equation regular in this limit. For fast step η and slow step varepsilon, the expansion reveals a mixed contribution varepsilon^2/η alongside terms linear in each step size. This dependence matters for bias reduction: along power-law step-size paths, the bias exponents need not be integers, so Richardson--Romberg extrapolation requires weights matched to the path. An exactly solvable nonlinear Markov example verifies the coefficients. We verify localization for temporal-difference learning and compare finite-run extrapolation at equal update budgets. For finite runs, we bound the initialization error of tail averages on both timescales under an additional coupling assumption. In the special case of additive independent noise, signed third-moment cancellation yields a sharper remainder.