On the Fundamental Limit of the Stochastic Gradient Identification Algorithm Under Non-Persistent Excitation

Senhan Yao; Longxu Zhang

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arXiv · 2511.19981

On the Fundamental Limit of the Stochastic Gradient Identification Algorithm Under Non-Persistent Excitation

Senhan Yao·Longxu Zhang

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Abstract

Stochastic gradient (SG) methods are fundamental to system identification and machine learning, enabling online parameter estimation in large-scale and streaming-data settings. As a classical identification method, the SG algorithm has been extensively studied for decades. Under non-persistent excitation, the strongest currently available convergence result assumes that the condition number of the Fisher information matrix is $O((\log r_n)^α)$, where $r_n = 1 + \sum_{i=1}^n \|φ_i\|^2$. Existing theory establishes strong consistency when $α\le 1/3$, whereas the same condition with $α> 1$ is insufficient to guarantee strong consistency. We prove that strong consistency holds throughout the range $0 \le α< 1$. The proof is based on a new algebraic framework that yields substantially sharper matrix norm bounds. This result nearly resolves the four-decade-old Chen--Guo conjecture by establishing strong consistency throughout the previously open range $1/3 < α< 1$.

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Senhan Yao, Longxu Zhang. 2026-05-07. On the Fundamental Limit of the Stochastic Gradient Identification Algorithm Under Non-Persistent Excitation. https://arxiv.org/abs/2511.19981

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