When Does Dynamic Preconditioning Preserve the Polyak-Ruppert CLT? A Stabilization Threshold

Polyak-Ruppert averaging yields an asymptotically normal estimator with sandwich covariance $H^{-1}SH^{-1}$, the foundation of online inference. When the gradient step is preconditioned by a data-driven matrix $P_t$, we ask how fast $P_t$ must stabilize for the central limit theorem (CLT) to remain valid. We resolve this via an exact preconditioner-isolating decomposition of the averaged error that confines $P_t$ to a dynamic remainder $R_n$, leaving the martingale and Taylor terms preconditioner-free. Let $M_t = (P_t H)^{-1}$ denote the effective inverse drift matrix, with $\|M_t - M_{t-1}\|_{\mathrm{op}} \lesssim t^{-\beta}$ and step-size exponent $\alpha \in (1/2, 1)$. We identify a stabilization-rate threshold $\beta > (\alpha+1)/2$ and prove that, within the class of polynomial rate hypotheses used in our upper bound, it cannot be weakened: the dynamic remainder $\sqrt{n}\,R_n$ vanishes in $L^2$ whenever $\beta > (\alpha+1)/2$, and we exhibit sequences satisfying those hypotheses for which it does not vanish when $\beta \le (\alpha+1)/2$. A single stabilization argument certifies three SA variants - SA-AdaGrad, SA-RMSProp, and SA-ONS - with gain $\rho_t = c/t$, each delivering one-step $L^2(\mathrm{op})$ stabilization of order $t^{-1}$, yielding the CLT $\sqrt{n}(\bar{x}_n - x^*) \to N(0, H^{-1}SH^{-1})$; under bounded inputs the pathwise rate $\beta = 1$ further preserves the $n^{-1/6}$ Wasserstein rate at $\alpha^* = 2/3$. Under standard regularity conditions, Wald-type online inference remains valid for dynamically preconditioned averaged SGD whose stabilization rate exceeds the threshold.