Decoupled Functional Central Limit Theorems for Two-Time-Scale Stochastic Approximation

In two-time-scale stochastic approximation (SA), two iterates are updated at different rates, governed by distinct step sizes, with each update influencing the other. Previous studies have demonstrated 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-time-scale SA, offering a more precise characterization of its asymptotic behavior. Our results show that, on each time scale, the limiting dynamics 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 time scales, we introduce an innovative auxiliary sequence to eliminate the primary influence of the fast-time-scale update on the slow-time-scale update.

💡 Research Summary

This paper establishes decoupled functional central limit theorems (FCLTs) for two‑time‑scale stochastic approximation (SA), a class of iterative algorithms in which two coupled variables are updated with distinct step‑size sequences, typically denoted by αₙ (fast scale) and βₙ (slow scale) with βₙ ≪ αₙ. While prior work has shown that the pointwise errors αₙ⁻¹ᐟ²(xₙ − H(yₙ)) and βₙ⁻¹ᐟ²(yₙ − y*) converge weakly to normal distributions, no result has described the entire trajectory of these rescaled errors as continuous‑time stochastic processes.

The authors focus on the “strict” two‑time‑scale regime where βₙ/αₙ → 0. They construct two continuous‑time interpolations:

• U_fⁿ(t) = β_{⌊t/βₙ⌋}⁻¹ᐟ² (y_{⌊t/βₙ⌋} − y*), representing the slow‑scale error, and
• U_sⁿ(t) = α_{⌊t/αₙ⌋}⁻¹ᐟ² (x_{⌊t/αₙ⌋} − H(y_{⌊t/αₙ⌋})), representing the fast‑scale error.

The main technical contributions are:

1. Tightness – Lemma 5.3 proves that the families {U_fⁿ} and {U_sⁿ} are tight in C(