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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.19981 |
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| _version_ | 1866910195007881216 |
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| author | Yao, Senhan Zhang, Longxu |
| author_facet | Yao, Senhan Zhang, Longxu |
| contents | 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 \(1/3 < α< 1\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19981 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Fundamental Limit of the Stochastic Gradient Identification Algorithm Under Non-Persistent Excitation Yao, Senhan Zhang, Longxu Optimization and Control 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 \(1/3 < α< 1\). |
| title | On the Fundamental Limit of the Stochastic Gradient Identification Algorithm Under Non-Persistent Excitation |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2511.19981 |