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

2026-04-25

AI Summary

This work investigates how the stability rate of the preconditioning matrix in dynamically preconditioned stochastic approximation affects the validity of the Polyak–Ruppert central limit theorem (CLT). By introducing a precise decomposition that isolates the preconditioner’s influence, the authors decompose the averaging error into a martingale term, a Taylor remainder, and a residual term depending solely on the dynamic preconditioner. They establish, for the first time, that the CLT holds provided the stability rate satisfies β > (α+1)/2, and demonstrate that this threshold is sharp under polynomial-rate assumptions. Leveraging L² analysis in operator norm and refined estimates of the dynamic residual, the study verifies the stability of algorithms such as SA-AdaGrad, SA-RMSProp, and SA-ONS within this framework. When the stability rate exceeds the threshold, dynamically preconditioned averaged SGD satisfies the CLT and achieves a Wasserstein convergence rate of n⁻¹/⁶ under bounded inputs, offering theoretical guarantees for online statistical inference.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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 $\|M_t - M_{t-1}\|_{\mathrm{op}} \lesssim t^{-β}$ and step-size exponent $α\in (1/2, 1)$. We identify a stabilization-rate threshold $β> (α+1)/2$ 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 $β> (α+1)/2$, and we exhibit sequences satisfying those hypotheses for which it does not vanish when $β\le (α+1)/2$. A single stabilization argument certifies three SA variants - SA-AdaGrad, SA-RMSProp, and SA-ONS - with gain $ρ_t = c/t$, 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 $β= 1$ further preserves the $n^{-1/6}$ Wasserstein rate at $α^* = 2/3$. Under standard regularity conditions, Wald-type online inference remains valid for dynamically preconditioned averaged SGD whose stabilization rate exceeds the threshold.

Problem

• dynamic preconditioning
• Polyak-Ruppert averaging
• central limit theorem
• stabilization rate
• stochastic approximation

Innovation

• dynamic preconditioning
• Polyak-Ruppert averaging
• central limit theorem
• stabilization threshold
• stochastic approximation