Decoupled Functional Central Limit Theorems for Two-Timescale Stochastic Approximation

In two-timescale stochastic approximation (SA), two iterates are updated at different rates, governed by distinct step sizes, with each update influencing the other. Previous studies demonstrate that the convergence rates of the error terms for these updates depend solely on their respective step sizes, a property known as decoupled convergence. However, a functional version of this decoupled convergence has not been explored. Our work fills this gap by establishing decoupled functional central limit theorems for two-timescale SA, offering a more precise characterization of its asymptotic behavior. Our results show that, on each timescale, the limiting dynamic has the same form as in standard SA, and the coupling between the two iterates enters the limit only through the associated coefficients. To achieve these results, we leverage the martingale problem approach and establish tightness as a crucial intermediate step. Furthermore, to address the interdependence between different timescales, we introduce an innovative auxiliary sequence to eliminate the primary influence of the fast timescale update on the slow timescale update. Funding: This research was supported by the National Key Research and Development Program of China [Grant 2022YFA1004002], and the National Natural Science Foundation of China [Grants 12501417 and 12350001]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2025.0876 .