{"ID":2836776,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.19981","arxiv_id":"2511.19981","title":"On the Fundamental Limit of the Stochastic Gradient Identification Algorithm Under Non-Persistent Excitation","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 \\|\\varphi_i\\|^2\\). Existing theory establishes strong consistency when \\(α\\le 1/3\\), whereas the same condition with \\(α\u003e 1\\) is insufficient to guarantee strong consistency. We prove that strong consistency holds throughout the range \\(0 \\le α\u003c 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 \u003c α\u003c 1\\).","short_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...","url_abs":"https://arxiv.org/abs/2511.19981","url_pdf":"https://arxiv.org/pdf/2511.19981v3","authors":"[\"Senhan Yao\",\"Longxu Zhang\"]","published":"2025-11-25T06:49:32Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}