SPAR：流式先验锚定岭回归的有限节奏刷新均衡——闭式遗忘窗口与谱校正
Finite-Cadence Refresh Equilibria for Streaming Prior-Anchored Ridge: A Closed-Form Forgetting Window and Spectral Correction
AI 导读
arXiv 论文研究漂移数据流上先验锚定岭回归的遗忘权衡，推导出锚点每 k 步刷新时的有限节奏均衡、闭式遗忘窗口 Δₖ(s) 与谱校正。最优节奏为 ω*=(4ν²Σ/c')^(1/3)，随机矩阵确定性等价与仿真误差在固定工作点 5% 以内、自选点 13% 以内。所得 SPAR 方法在线校准窗口与强度，在 27 组配置网格上遇漂移可开启窗口而不发散，核心代数与遗憾界经 Lean 4 形式化验证。
正文 · 原文
Abstract:Recursive estimators on drifting streams must forget stale data fast enough to track the target, yet slowly enough to average out noise. In prior-anchored ridge regression, this trade-off admits a closed-form spectral decomposition in which each eigen-direction interpolates between the data solution and the prior with pull weight $s_i = \rho / (\mu_i + \rho)$. When the anchor is refreshed every $k$ steps, we show that the moving-frame offset converges to a finite-cadence equilibrium that lags the pure-leaky path by an explicit $\Delta_k(s)$ while reducing estimation variance by a factor $\Psi(s, r_k)$. Minimizing the cycle-averaged risk gives $\omega^* = (4\nu^2 \Sigma / c')^{1/3}$, $c' = \sigma^2 d_{\mathrm{eff}}(\rho)$, and the plug-in strength $\rho^* = \sigma^2 / (n\delta^2)$. A random-matrix deterministic equivalent extends both laws to finite dimension, renormalizing the anchor to $\rho_{\mathrm{eff}} = \rho (e^{\zeta q} - 1) / (\zeta q)$; it matches simulation within 5% at fixed operating points and within 13% at the method's own, and rank-deficient populations reduce exactly to their active subspace. The resulting method, SPAR, calibrates window and strength online from residuals. On a 27-configuration grid the calibrated window opens on drift without divergence, and the grid defaults transfer to an hourly electricity stream and to BCI IV 2b. The core algebra, refresh chain, and regret bounds are formalized in Lean 4.
| Comments: | 67 pages, 7 figures. Lean 4 formalization and code: [this https URL](this https URL) |
| Subjects: | Machine Learning (stat.ML); Artificial Intelligence (cs.AI); Computer Science and Game Theory (cs.GT); Machine Learning (cs.LG) |
| Cite as: | arXiv:2611.14586 [stat.ML] |
| (or arXiv:2611.14586v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2611.14586 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Mrigank Shekhar Singh [view email]
[v1]
Tue, 10 Nov 2026 06:42:46 UTC (1,249 KB)
来源：arXiv cs.AI · arxiv.org