发表机构
Huazhong University of Science and Technology（华中科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在预测风险下研究长记忆序列模型的资源需求，证明代数衰减记忆下最优状态复杂度为对数平方，并揭示分数长记忆改变预测几何，且预测误差消失迫使递归趋向临界性。
AI 中文摘要
长程时间依赖性对序列模型提出了一个资源问题：对于指定的预测记忆律，为了准确预测，需要多少状态、上下文或动力学临界性？我们直接在预测风险中研究这个问题。对于代数衰减的预测记忆，我们证明了指数模式和有限状态模式的匹配上下逼近界。最佳的 $r$ 模式预测误差以 $e^{-\Theta(\sqrt r)}$ 的速率衰减，因此达到预测误差 $\tau$ 需要 $r=\Theta(\log^2(1/\tau))$ 个状态或模式。早期的记忆诅咒结果建立了稳定递归模型在不同逼近概念下的广泛局限性；这里双方在一个典型预测目标上匹配，从而确定了该目标的最优资源指数。然后我们证明，真正的分数长记忆改变了几何本身。特别地，在分数积分之后测量预测误差，从长度为 $L$ 的有限上下文进行预测具有精确的 $1/L$ 前导阶，而固定的分数强度 $d$ 保持平方对数状态复杂度律。在短记忆边界附近，我们确定了相关的 $d^2$ 和 $d^4$ 尺度，并在中间区域给出了统一的构造性律。对于具有一致压缩状态动力学的非线性上下文递归，我们推导了一个指数第一混沌包络和一个显式的必要条件，该条件将预测精度与压缩余量联系起来。在代数目标上，预测误差消失迫使递归定量地趋向临界性，这是一个必要但不充分的条件。有限样本 Kullback-Leibler 计算进一步将预测几何与统计信息联系起来。与定理匹配的实验使用压缩状态空间、门控递归和注意力模型，重现了状态和稳定性的预测。
英文摘要
Long-range temporal dependence poses a resource question for sequence models: for a specified predictive-memory law, how much state, context, or dynamical criticality is required in order to forecast accurately? We study this question directly in forecasting risk. For algebraically decaying predictive memory, we prove matching upper and lower approximation bounds for exponential and finite-state modes. The best $r$-mode forecast error decays as $e^{-Θ(\sqrt r)}$, so reaching forecast error $τ$ needs $r=Θ(\log^2(1/τ))$ states or modes. Earlier curse-of-memory results establish broad limitations of stable recurrent models under different approximation notions; here both sides match for one canonical predictive target in forecast risk, which fixes the optimal resource exponent for that target. We then show that genuine fractional long memory changes the geometry itself. In particular, forecast error is measured after fractional integration, prediction from a finite context of length $L$ has an exact $1/L$ leading order, and a fixed fractional strength $d$ keeps the square-log state-complexity law. Near the short-memory boundary, we identify the relevant $d^2$ and $d^4$ scales and give a uniform constructive law in the intermediate regime. For nonlinear contextual recurrences with uniformly contractive state dynamics, we derive an exponential first-chaos envelope and an explicit necessary condition that relates forecast accuracy to the contraction margin. Vanishing forecasting error on an algebraic target forces the recurrence quantitatively toward criticality, a condition that is necessary and not by itself sufficient. Finite-sample Kullback--Leibler calculations further connect the predictive geometry to statistical information. Theorem-matched experiments with contractive state-space, gated recurrent, and attention models reproduce the state and stability predictions.
Comments49 pages, 2 figures
