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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.23498 |
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| _version_ | 1866914507708694528 |
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| author | An, Sunyoung Huo, Xiaoming |
| author_facet | An, Sunyoung Huo, Xiaoming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23498 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | When Does Dynamic Preconditioning Preserve the Polyak-Ruppert CLT? A Stabilization Threshold An, Sunyoung Huo, Xiaoming Statistics Theory Optimization and Control Machine Learning 62L20, 62F12, 60F05 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. |
| title | When Does Dynamic Preconditioning Preserve the Polyak-Ruppert CLT? A Stabilization Threshold |
| topic | Statistics Theory Optimization and Control Machine Learning 62L20, 62F12, 60F05 |
| url | https://arxiv.org/abs/2604.23498 |