Recent finite-time analyses of nonlinear two-time-scale stochastic approximation show that under contractive assumptions the slow iterate $Y_k$ with stepsizes $β_k=Θ(k^{-1})$ and $α_k=Θ(k^{-a})$, $a\in(1/2,1)$, generally satisfies a mean-square rate of order $k^{-a}$; decoupled $k^{-1}$ rates require strong local linearity. We identify a sharp regularity-dependent boundary. In a rate-determining normal form where the slow drift contains a locally linear leakage and a nonlinear remainder of order $1+ρ$ ($ρ\in[0,1]$), the uncorrected recursion satisfies \[ \mathbb{E}\|Y_k\|^2 \le C\bigl(k^{-1}+k^{-a(1+ρ)}\bigr), \] and a matching scalar Gaussian lower bound shows that the slower term is unavoidable without modifying the update. Thus the decoupled $k^{-1}$ rate is guaranteed for the uncorrected recursion exactly when $a(1+ρ)\ge 1$. This lower bound concerns only the naive update; it is not an information-theoretic obstruction. We demonstrate this by equipping the normal-form recursion with an auxiliary online bias estimator \[ M_{k+1}=M_k+γ_k(R(X_k)-M_k),\qquad β_k\llγ_k\llα_k, \] and subtracting $M_k$ from the slow update. Under the same stability, moment, and remainder assumptions, the corrected recursion achieves $\mathbb{E}\|\widetilde Y_k\|^2=O(k^{-1})$ for every $ρ\in[0,1]$, including regimes where the uncorrected update provably suffers the slower rate. Finally, we prove localized transfer theorems that extend the phase-transition mechanism to general nonlinear TTSA in fast-manifold coordinates. The proofs are non-asymptotic and rely on two Abel-transform cancellations: one for the locally linear fast-error leakage, and one for the tracked nonlinear bias.

翻译：近期对非线性两时间尺度随机逼近的有限时间分析表明，在压缩性假设下，慢速迭代$Y_k$（步长取$β_k=Θ(k^{-1})$和$α_k=Θ(k^{-a})$，$a\in(1/2,1)$）通常满足$k^{-a}$阶均方收敛速率；解耦的$k^{-1}$阶速率需要强局部线性性。我们识别出一个尖锐的规律依赖性边界。在速率决定的标准形式中，慢速漂移包含局部线性泄漏项和$1+ρ$阶非线性余项（$ρ\in[0,1]$），未修正递归满足\[ \mathbb{E}\|Y_k\|^2 \le C\bigl(k^{-1}+k^{-a(1+ρ)}\bigr), \]且匹配的标量高斯下界表明，在不修改更新方式的情况下，较慢项不可避免。因此，当且仅当$a(1+ρ)\ge 1$时，未修正递归保证解耦$k^{-1}$速率。该下界仅针对朴素更新，并非信息论障碍。我们通过在标准形式递归中引入辅助在线偏差估计器\[ M_{k+1}=M_k+γ_k(R(X_k)-M_k),\qquad β_k\llγ_k\llα_k, \]并从慢速更新中减去$M_k$来证明这一点。在相同稳定性、矩和余项假设下，修正递归对所有$ρ\in[0,1]$达到$\mathbb{E}\|\widetilde Y_k\|^2=O(k^{-1})$，包括未修正更新被证明确实遭受较慢速率的区间。最后，我们证明局部化传递定理，将相变机制推广到快速流形坐标下的广义非线性TTSA。证明是非渐近的，依赖于两个阿贝尔变换抵消：一个针对局部线性快速误差泄漏，另一个针对跟踪的非线性偏差。